Properties

Label 5225.2.a.k.1.3
Level $5225$
Weight $2$
Character 5225.1
Self dual yes
Analytic conductor $41.722$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5225,2,Mod(1,5225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5225.a (trivial)

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: yes
Analytic conductor: \(41.7218350561\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.131947641.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 3x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1045)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.759131\) of defining polynomial
Character \(\chi\) \(=\) 5225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.759131 q^{2} +2.25829 q^{3} -1.42372 q^{4} -1.71434 q^{6} -4.95189 q^{7} +2.59905 q^{8} +2.09989 q^{9} +O(q^{10})\) \(q-0.759131 q^{2} +2.25829 q^{3} -1.42372 q^{4} -1.71434 q^{6} -4.95189 q^{7} +2.59905 q^{8} +2.09989 q^{9} -1.00000 q^{11} -3.21518 q^{12} -0.499163 q^{13} +3.75913 q^{14} +0.874419 q^{16} +4.34828 q^{17} -1.59409 q^{18} -1.00000 q^{19} -11.1828 q^{21} +0.759131 q^{22} +2.78646 q^{23} +5.86942 q^{24} +0.378930 q^{26} -2.03271 q^{27} +7.05010 q^{28} +8.84908 q^{29} +1.67449 q^{31} -5.86190 q^{32} -2.25829 q^{33} -3.30091 q^{34} -2.98966 q^{36} +1.81215 q^{37} +0.759131 q^{38} -1.12726 q^{39} +10.5406 q^{41} +8.48922 q^{42} -2.65924 q^{43} +1.42372 q^{44} -2.11529 q^{46} -11.6115 q^{47} +1.97470 q^{48} +17.5212 q^{49} +9.81969 q^{51} +0.710668 q^{52} -10.6541 q^{53} +1.54310 q^{54} -12.8702 q^{56} -2.25829 q^{57} -6.71761 q^{58} +10.0174 q^{59} -12.5990 q^{61} -1.27116 q^{62} -10.3984 q^{63} +2.70111 q^{64} +1.71434 q^{66} -2.84997 q^{67} -6.19073 q^{68} +6.29265 q^{69} -13.3351 q^{71} +5.45772 q^{72} +13.2131 q^{73} -1.37566 q^{74} +1.42372 q^{76} +4.95189 q^{77} +0.855735 q^{78} -7.08995 q^{79} -10.8901 q^{81} -8.00173 q^{82} -15.1160 q^{83} +15.9212 q^{84} +2.01871 q^{86} +19.9838 q^{87} -2.59905 q^{88} -7.25826 q^{89} +2.47180 q^{91} -3.96714 q^{92} +3.78150 q^{93} +8.81467 q^{94} -13.2379 q^{96} -0.0194069 q^{97} -13.3009 q^{98} -2.09989 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 8 q^{4} + 2 q^{6} - 5 q^{7} + 9 q^{9} - 6 q^{11} - 19 q^{12} + 9 q^{13} + 18 q^{14} + 4 q^{16} + 5 q^{17} - 2 q^{18} - 6 q^{19} + 3 q^{21} - 8 q^{23} - 7 q^{24} - 22 q^{26} - 30 q^{27} - 10 q^{28} - 5 q^{29} - q^{31} - 15 q^{32} + 3 q^{33} - 22 q^{34} + 12 q^{36} - 9 q^{37} - 32 q^{39} + 25 q^{41} - 11 q^{42} - 15 q^{43} - 8 q^{44} - 16 q^{46} - 24 q^{47} + 4 q^{48} + 13 q^{49} + 27 q^{52} - 5 q^{53} - 11 q^{54} - 12 q^{56} + 3 q^{57} - 13 q^{58} + 39 q^{59} - 11 q^{61} + 42 q^{62} - 38 q^{63} - 14 q^{64} - 2 q^{66} - 24 q^{67} - 45 q^{68} + 14 q^{69} - 24 q^{71} + 61 q^{72} + 26 q^{73} + q^{74} - 8 q^{76} + 5 q^{77} + 29 q^{78} + 11 q^{79} + 30 q^{81} - 8 q^{82} - 39 q^{83} + 25 q^{84} + 18 q^{86} + 16 q^{87} + 22 q^{89} - 26 q^{91} + 11 q^{92} + 6 q^{93} - 30 q^{94} - 15 q^{96} - 22 q^{97} - 33 q^{98} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.759131 −0.536787 −0.268393 0.963309i \(-0.586493\pi\)
−0.268393 + 0.963309i \(0.586493\pi\)
\(3\) 2.25829 1.30383 0.651913 0.758294i \(-0.273967\pi\)
0.651913 + 0.758294i \(0.273967\pi\)
\(4\) −1.42372 −0.711860
\(5\) 0 0
\(6\) −1.71434 −0.699877
\(7\) −4.95189 −1.87164 −0.935819 0.352482i \(-0.885338\pi\)
−0.935819 + 0.352482i \(0.885338\pi\)
\(8\) 2.59905 0.918904
\(9\) 2.09989 0.699963
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −3.21518 −0.928142
\(13\) −0.499163 −0.138443 −0.0692214 0.997601i \(-0.522051\pi\)
−0.0692214 + 0.997601i \(0.522051\pi\)
\(14\) 3.75913 1.00467
\(15\) 0 0
\(16\) 0.874419 0.218605
\(17\) 4.34828 1.05461 0.527306 0.849675i \(-0.323202\pi\)
0.527306 + 0.849675i \(0.323202\pi\)
\(18\) −1.59409 −0.375731
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −11.1828 −2.44029
\(22\) 0.759131 0.161847
\(23\) 2.78646 0.581017 0.290509 0.956872i \(-0.406176\pi\)
0.290509 + 0.956872i \(0.406176\pi\)
\(24\) 5.86942 1.19809
\(25\) 0 0
\(26\) 0.378930 0.0743142
\(27\) −2.03271 −0.391196
\(28\) 7.05010 1.33234
\(29\) 8.84908 1.64323 0.821616 0.570041i \(-0.193073\pi\)
0.821616 + 0.570041i \(0.193073\pi\)
\(30\) 0 0
\(31\) 1.67449 0.300748 0.150374 0.988629i \(-0.451952\pi\)
0.150374 + 0.988629i \(0.451952\pi\)
\(32\) −5.86190 −1.03625
\(33\) −2.25829 −0.393118
\(34\) −3.30091 −0.566102
\(35\) 0 0
\(36\) −2.98966 −0.498276
\(37\) 1.81215 0.297916 0.148958 0.988844i \(-0.452408\pi\)
0.148958 + 0.988844i \(0.452408\pi\)
\(38\) 0.759131 0.123147
\(39\) −1.12726 −0.180505
\(40\) 0 0
\(41\) 10.5406 1.64617 0.823086 0.567916i \(-0.192250\pi\)
0.823086 + 0.567916i \(0.192250\pi\)
\(42\) 8.48922 1.30992
\(43\) −2.65924 −0.405531 −0.202765 0.979227i \(-0.564993\pi\)
−0.202765 + 0.979227i \(0.564993\pi\)
\(44\) 1.42372 0.214634
\(45\) 0 0
\(46\) −2.11529 −0.311882
\(47\) −11.6115 −1.69372 −0.846858 0.531819i \(-0.821509\pi\)
−0.846858 + 0.531819i \(0.821509\pi\)
\(48\) 1.97470 0.285023
\(49\) 17.5212 2.50303
\(50\) 0 0
\(51\) 9.81969 1.37503
\(52\) 0.710668 0.0985519
\(53\) −10.6541 −1.46345 −0.731726 0.681598i \(-0.761285\pi\)
−0.731726 + 0.681598i \(0.761285\pi\)
\(54\) 1.54310 0.209989
\(55\) 0 0
\(56\) −12.8702 −1.71985
\(57\) −2.25829 −0.299118
\(58\) −6.71761 −0.882065
\(59\) 10.0174 1.30416 0.652079 0.758151i \(-0.273897\pi\)
0.652079 + 0.758151i \(0.273897\pi\)
\(60\) 0 0
\(61\) −12.5990 −1.61314 −0.806569 0.591140i \(-0.798678\pi\)
−0.806569 + 0.591140i \(0.798678\pi\)
\(62\) −1.27116 −0.161438
\(63\) −10.3984 −1.31008
\(64\) 2.70111 0.337639
\(65\) 0 0
\(66\) 1.71434 0.211021
\(67\) −2.84997 −0.348180 −0.174090 0.984730i \(-0.555698\pi\)
−0.174090 + 0.984730i \(0.555698\pi\)
\(68\) −6.19073 −0.750736
\(69\) 6.29265 0.757546
\(70\) 0 0
\(71\) −13.3351 −1.58258 −0.791291 0.611440i \(-0.790590\pi\)
−0.791291 + 0.611440i \(0.790590\pi\)
\(72\) 5.45772 0.643199
\(73\) 13.2131 1.54647 0.773236 0.634118i \(-0.218637\pi\)
0.773236 + 0.634118i \(0.218637\pi\)
\(74\) −1.37566 −0.159917
\(75\) 0 0
\(76\) 1.42372 0.163312
\(77\) 4.95189 0.564320
\(78\) 0.855735 0.0968929
\(79\) −7.08995 −0.797681 −0.398841 0.917020i \(-0.630587\pi\)
−0.398841 + 0.917020i \(0.630587\pi\)
\(80\) 0 0
\(81\) −10.8901 −1.21001
\(82\) −8.00173 −0.883643
\(83\) −15.1160 −1.65920 −0.829600 0.558359i \(-0.811431\pi\)
−0.829600 + 0.558359i \(0.811431\pi\)
\(84\) 15.9212 1.73715
\(85\) 0 0
\(86\) 2.01871 0.217683
\(87\) 19.9838 2.14249
\(88\) −2.59905 −0.277060
\(89\) −7.25826 −0.769374 −0.384687 0.923047i \(-0.625691\pi\)
−0.384687 + 0.923047i \(0.625691\pi\)
\(90\) 0 0
\(91\) 2.47180 0.259115
\(92\) −3.96714 −0.413603
\(93\) 3.78150 0.392123
\(94\) 8.81467 0.909164
\(95\) 0 0
\(96\) −13.2379 −1.35109
\(97\) −0.0194069 −0.00197047 −0.000985234 1.00000i \(-0.500314\pi\)
−0.000985234 1.00000i \(0.500314\pi\)
\(98\) −13.3009 −1.34359
\(99\) −2.09989 −0.211047
\(100\) 0 0
\(101\) 10.2786 1.02276 0.511379 0.859355i \(-0.329135\pi\)
0.511379 + 0.859355i \(0.329135\pi\)
\(102\) −7.45443 −0.738098
\(103\) 5.23959 0.516272 0.258136 0.966109i \(-0.416892\pi\)
0.258136 + 0.966109i \(0.416892\pi\)
\(104\) −1.29735 −0.127216
\(105\) 0 0
\(106\) 8.08786 0.785562
\(107\) −16.5236 −1.59739 −0.798697 0.601733i \(-0.794477\pi\)
−0.798697 + 0.601733i \(0.794477\pi\)
\(108\) 2.89402 0.278477
\(109\) −0.934463 −0.0895053 −0.0447527 0.998998i \(-0.514250\pi\)
−0.0447527 + 0.998998i \(0.514250\pi\)
\(110\) 0 0
\(111\) 4.09237 0.388431
\(112\) −4.33003 −0.409149
\(113\) −17.0120 −1.60036 −0.800179 0.599761i \(-0.795262\pi\)
−0.800179 + 0.599761i \(0.795262\pi\)
\(114\) 1.71434 0.160563
\(115\) 0 0
\(116\) −12.5986 −1.16975
\(117\) −1.04819 −0.0969048
\(118\) −7.60454 −0.700054
\(119\) −21.5322 −1.97385
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 9.56430 0.865911
\(123\) 23.8039 2.14632
\(124\) −2.38401 −0.214091
\(125\) 0 0
\(126\) 7.89376 0.703232
\(127\) −15.2360 −1.35197 −0.675987 0.736914i \(-0.736282\pi\)
−0.675987 + 0.736914i \(0.736282\pi\)
\(128\) 9.67331 0.855008
\(129\) −6.00535 −0.528741
\(130\) 0 0
\(131\) −5.16429 −0.451206 −0.225603 0.974219i \(-0.572435\pi\)
−0.225603 + 0.974219i \(0.572435\pi\)
\(132\) 3.21518 0.279845
\(133\) 4.95189 0.429383
\(134\) 2.16350 0.186898
\(135\) 0 0
\(136\) 11.3014 0.969087
\(137\) 14.1861 1.21200 0.606001 0.795464i \(-0.292773\pi\)
0.606001 + 0.795464i \(0.292773\pi\)
\(138\) −4.77694 −0.406640
\(139\) −1.20156 −0.101915 −0.0509573 0.998701i \(-0.516227\pi\)
−0.0509573 + 0.998701i \(0.516227\pi\)
\(140\) 0 0
\(141\) −26.2222 −2.20831
\(142\) 10.1231 0.849509
\(143\) 0.499163 0.0417421
\(144\) 1.83618 0.153015
\(145\) 0 0
\(146\) −10.0304 −0.830126
\(147\) 39.5680 3.26351
\(148\) −2.58000 −0.212074
\(149\) 3.25736 0.266853 0.133427 0.991059i \(-0.457402\pi\)
0.133427 + 0.991059i \(0.457402\pi\)
\(150\) 0 0
\(151\) 14.9900 1.21987 0.609933 0.792453i \(-0.291196\pi\)
0.609933 + 0.792453i \(0.291196\pi\)
\(152\) −2.59905 −0.210811
\(153\) 9.13090 0.738190
\(154\) −3.75913 −0.302919
\(155\) 0 0
\(156\) 1.60490 0.128495
\(157\) −2.08129 −0.166105 −0.0830524 0.996545i \(-0.526467\pi\)
−0.0830524 + 0.996545i \(0.526467\pi\)
\(158\) 5.38220 0.428185
\(159\) −24.0601 −1.90809
\(160\) 0 0
\(161\) −13.7982 −1.08745
\(162\) 8.26704 0.649520
\(163\) −13.2889 −1.04087 −0.520434 0.853902i \(-0.674230\pi\)
−0.520434 + 0.853902i \(0.674230\pi\)
\(164\) −15.0069 −1.17184
\(165\) 0 0
\(166\) 11.4750 0.890636
\(167\) −23.3753 −1.80883 −0.904416 0.426652i \(-0.859693\pi\)
−0.904416 + 0.426652i \(0.859693\pi\)
\(168\) −29.0647 −2.24239
\(169\) −12.7508 −0.980834
\(170\) 0 0
\(171\) −2.09989 −0.160583
\(172\) 3.78602 0.288681
\(173\) −11.6401 −0.884983 −0.442491 0.896773i \(-0.645905\pi\)
−0.442491 + 0.896773i \(0.645905\pi\)
\(174\) −15.1703 −1.15006
\(175\) 0 0
\(176\) −0.874419 −0.0659118
\(177\) 22.6223 1.70040
\(178\) 5.50997 0.412990
\(179\) −15.0945 −1.12822 −0.564110 0.825700i \(-0.690781\pi\)
−0.564110 + 0.825700i \(0.690781\pi\)
\(180\) 0 0
\(181\) −15.0754 −1.12055 −0.560273 0.828308i \(-0.689304\pi\)
−0.560273 + 0.828308i \(0.689304\pi\)
\(182\) −1.87642 −0.139089
\(183\) −28.4523 −2.10325
\(184\) 7.24216 0.533899
\(185\) 0 0
\(186\) −2.87065 −0.210487
\(187\) −4.34828 −0.317978
\(188\) 16.5316 1.20569
\(189\) 10.0658 0.732177
\(190\) 0 0
\(191\) 3.60614 0.260931 0.130466 0.991453i \(-0.458353\pi\)
0.130466 + 0.991453i \(0.458353\pi\)
\(192\) 6.09991 0.440223
\(193\) 24.4479 1.75980 0.879900 0.475158i \(-0.157609\pi\)
0.879900 + 0.475158i \(0.157609\pi\)
\(194\) 0.0147323 0.00105772
\(195\) 0 0
\(196\) −24.9453 −1.78180
\(197\) 15.4415 1.10016 0.550082 0.835111i \(-0.314597\pi\)
0.550082 + 0.835111i \(0.314597\pi\)
\(198\) 1.59409 0.113287
\(199\) −19.3689 −1.37302 −0.686512 0.727119i \(-0.740859\pi\)
−0.686512 + 0.727119i \(0.740859\pi\)
\(200\) 0 0
\(201\) −6.43607 −0.453966
\(202\) −7.80280 −0.549003
\(203\) −43.8196 −3.07554
\(204\) −13.9805 −0.978830
\(205\) 0 0
\(206\) −3.97753 −0.277128
\(207\) 5.85126 0.406691
\(208\) −0.436477 −0.0302643
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 12.4253 0.855393 0.427697 0.903922i \(-0.359325\pi\)
0.427697 + 0.903922i \(0.359325\pi\)
\(212\) 15.1685 1.04177
\(213\) −30.1145 −2.06341
\(214\) 12.5436 0.857460
\(215\) 0 0
\(216\) −5.28313 −0.359471
\(217\) −8.29191 −0.562891
\(218\) 0.709380 0.0480453
\(219\) 29.8390 2.01633
\(220\) 0 0
\(221\) −2.17050 −0.146003
\(222\) −3.10664 −0.208504
\(223\) 9.46847 0.634055 0.317028 0.948416i \(-0.397315\pi\)
0.317028 + 0.948416i \(0.397315\pi\)
\(224\) 29.0275 1.93948
\(225\) 0 0
\(226\) 12.9144 0.859051
\(227\) −1.20914 −0.0802532 −0.0401266 0.999195i \(-0.512776\pi\)
−0.0401266 + 0.999195i \(0.512776\pi\)
\(228\) 3.21518 0.212930
\(229\) −14.5551 −0.961831 −0.480916 0.876767i \(-0.659696\pi\)
−0.480916 + 0.876767i \(0.659696\pi\)
\(230\) 0 0
\(231\) 11.1828 0.735775
\(232\) 22.9992 1.50997
\(233\) 14.0283 0.919022 0.459511 0.888172i \(-0.348025\pi\)
0.459511 + 0.888172i \(0.348025\pi\)
\(234\) 0.795711 0.0520172
\(235\) 0 0
\(236\) −14.2620 −0.928378
\(237\) −16.0112 −1.04004
\(238\) 16.3457 1.05954
\(239\) −3.88964 −0.251600 −0.125800 0.992056i \(-0.540150\pi\)
−0.125800 + 0.992056i \(0.540150\pi\)
\(240\) 0 0
\(241\) −10.6210 −0.684160 −0.342080 0.939671i \(-0.611131\pi\)
−0.342080 + 0.939671i \(0.611131\pi\)
\(242\) −0.759131 −0.0487988
\(243\) −18.4950 −1.18645
\(244\) 17.9375 1.14833
\(245\) 0 0
\(246\) −18.0703 −1.15212
\(247\) 0.499163 0.0317610
\(248\) 4.35210 0.276358
\(249\) −34.1364 −2.16331
\(250\) 0 0
\(251\) −2.97829 −0.187988 −0.0939940 0.995573i \(-0.529963\pi\)
−0.0939940 + 0.995573i \(0.529963\pi\)
\(252\) 14.8044 0.932592
\(253\) −2.78646 −0.175183
\(254\) 11.5661 0.725721
\(255\) 0 0
\(256\) −12.7455 −0.796596
\(257\) −6.16276 −0.384423 −0.192211 0.981354i \(-0.561566\pi\)
−0.192211 + 0.981354i \(0.561566\pi\)
\(258\) 4.55885 0.283821
\(259\) −8.97357 −0.557591
\(260\) 0 0
\(261\) 18.5821 1.15020
\(262\) 3.92037 0.242202
\(263\) 18.8622 1.16310 0.581548 0.813512i \(-0.302447\pi\)
0.581548 + 0.813512i \(0.302447\pi\)
\(264\) −5.86942 −0.361238
\(265\) 0 0
\(266\) −3.75913 −0.230487
\(267\) −16.3913 −1.00313
\(268\) 4.05756 0.247855
\(269\) −0.262397 −0.0159986 −0.00799932 0.999968i \(-0.502546\pi\)
−0.00799932 + 0.999968i \(0.502546\pi\)
\(270\) 0 0
\(271\) 27.2164 1.65328 0.826638 0.562734i \(-0.190250\pi\)
0.826638 + 0.562734i \(0.190250\pi\)
\(272\) 3.80222 0.230543
\(273\) 5.58204 0.337841
\(274\) −10.7691 −0.650587
\(275\) 0 0
\(276\) −8.95897 −0.539266
\(277\) −22.1763 −1.33244 −0.666222 0.745754i \(-0.732090\pi\)
−0.666222 + 0.745754i \(0.732090\pi\)
\(278\) 0.912138 0.0547064
\(279\) 3.51625 0.210513
\(280\) 0 0
\(281\) −15.9540 −0.951737 −0.475868 0.879517i \(-0.657866\pi\)
−0.475868 + 0.879517i \(0.657866\pi\)
\(282\) 19.9061 1.18539
\(283\) 2.99709 0.178159 0.0890793 0.996025i \(-0.471608\pi\)
0.0890793 + 0.996025i \(0.471608\pi\)
\(284\) 18.9854 1.12658
\(285\) 0 0
\(286\) −0.378930 −0.0224066
\(287\) −52.1961 −3.08104
\(288\) −12.3093 −0.725335
\(289\) 1.90752 0.112207
\(290\) 0 0
\(291\) −0.0438264 −0.00256915
\(292\) −18.8117 −1.10087
\(293\) −2.71694 −0.158725 −0.0793625 0.996846i \(-0.525288\pi\)
−0.0793625 + 0.996846i \(0.525288\pi\)
\(294\) −30.0373 −1.75181
\(295\) 0 0
\(296\) 4.70988 0.273756
\(297\) 2.03271 0.117950
\(298\) −2.47276 −0.143243
\(299\) −1.39090 −0.0804376
\(300\) 0 0
\(301\) 13.1683 0.759006
\(302\) −11.3793 −0.654808
\(303\) 23.2121 1.33350
\(304\) −0.874419 −0.0501514
\(305\) 0 0
\(306\) −6.93155 −0.396250
\(307\) 4.88778 0.278961 0.139480 0.990225i \(-0.455457\pi\)
0.139480 + 0.990225i \(0.455457\pi\)
\(308\) −7.05010 −0.401717
\(309\) 11.8325 0.673129
\(310\) 0 0
\(311\) −16.6843 −0.946082 −0.473041 0.881040i \(-0.656844\pi\)
−0.473041 + 0.881040i \(0.656844\pi\)
\(312\) −2.92980 −0.165867
\(313\) −10.5924 −0.598716 −0.299358 0.954141i \(-0.596773\pi\)
−0.299358 + 0.954141i \(0.596773\pi\)
\(314\) 1.57997 0.0891628
\(315\) 0 0
\(316\) 10.0941 0.567838
\(317\) −1.42787 −0.0801974 −0.0400987 0.999196i \(-0.512767\pi\)
−0.0400987 + 0.999196i \(0.512767\pi\)
\(318\) 18.2648 1.02424
\(319\) −8.84908 −0.495453
\(320\) 0 0
\(321\) −37.3151 −2.08272
\(322\) 10.4747 0.583731
\(323\) −4.34828 −0.241945
\(324\) 15.5045 0.861361
\(325\) 0 0
\(326\) 10.0880 0.558724
\(327\) −2.11029 −0.116699
\(328\) 27.3957 1.51267
\(329\) 57.4990 3.17002
\(330\) 0 0
\(331\) −3.79639 −0.208669 −0.104334 0.994542i \(-0.533271\pi\)
−0.104334 + 0.994542i \(0.533271\pi\)
\(332\) 21.5210 1.18112
\(333\) 3.80532 0.208530
\(334\) 17.7449 0.970957
\(335\) 0 0
\(336\) −9.77847 −0.533459
\(337\) 22.1330 1.20566 0.602830 0.797869i \(-0.294040\pi\)
0.602830 + 0.797869i \(0.294040\pi\)
\(338\) 9.67956 0.526498
\(339\) −38.4182 −2.08659
\(340\) 0 0
\(341\) −1.67449 −0.0906790
\(342\) 1.59409 0.0861986
\(343\) −52.0997 −2.81312
\(344\) −6.91151 −0.372643
\(345\) 0 0
\(346\) 8.83639 0.475047
\(347\) −1.70802 −0.0916913 −0.0458457 0.998949i \(-0.514598\pi\)
−0.0458457 + 0.998949i \(0.514598\pi\)
\(348\) −28.4514 −1.52515
\(349\) −3.58382 −0.191837 −0.0959186 0.995389i \(-0.530579\pi\)
−0.0959186 + 0.995389i \(0.530579\pi\)
\(350\) 0 0
\(351\) 1.01465 0.0541583
\(352\) 5.86190 0.312440
\(353\) 2.41512 0.128544 0.0642719 0.997932i \(-0.479528\pi\)
0.0642719 + 0.997932i \(0.479528\pi\)
\(354\) −17.1733 −0.912749
\(355\) 0 0
\(356\) 10.3337 0.547686
\(357\) −48.6260 −2.57356
\(358\) 11.4587 0.605613
\(359\) −18.6345 −0.983489 −0.491745 0.870739i \(-0.663641\pi\)
−0.491745 + 0.870739i \(0.663641\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 11.4442 0.601494
\(363\) 2.25829 0.118530
\(364\) −3.51915 −0.184453
\(365\) 0 0
\(366\) 21.5990 1.12900
\(367\) 1.21394 0.0633673 0.0316837 0.999498i \(-0.489913\pi\)
0.0316837 + 0.999498i \(0.489913\pi\)
\(368\) 2.43654 0.127013
\(369\) 22.1342 1.15226
\(370\) 0 0
\(371\) 52.7579 2.73905
\(372\) −5.38380 −0.279137
\(373\) 30.3171 1.56976 0.784881 0.619646i \(-0.212724\pi\)
0.784881 + 0.619646i \(0.212724\pi\)
\(374\) 3.30091 0.170686
\(375\) 0 0
\(376\) −30.1790 −1.55636
\(377\) −4.41713 −0.227494
\(378\) −7.64124 −0.393023
\(379\) 15.3776 0.789897 0.394948 0.918703i \(-0.370762\pi\)
0.394948 + 0.918703i \(0.370762\pi\)
\(380\) 0 0
\(381\) −34.4073 −1.76274
\(382\) −2.73753 −0.140064
\(383\) −1.62199 −0.0828798 −0.0414399 0.999141i \(-0.513195\pi\)
−0.0414399 + 0.999141i \(0.513195\pi\)
\(384\) 21.8452 1.11478
\(385\) 0 0
\(386\) −18.5592 −0.944638
\(387\) −5.58411 −0.283856
\(388\) 0.0276299 0.00140270
\(389\) −33.6559 −1.70642 −0.853210 0.521567i \(-0.825348\pi\)
−0.853210 + 0.521567i \(0.825348\pi\)
\(390\) 0 0
\(391\) 12.1163 0.612748
\(392\) 45.5385 2.30004
\(393\) −11.6625 −0.588295
\(394\) −11.7221 −0.590553
\(395\) 0 0
\(396\) 2.98966 0.150236
\(397\) 6.60727 0.331609 0.165805 0.986159i \(-0.446978\pi\)
0.165805 + 0.986159i \(0.446978\pi\)
\(398\) 14.7035 0.737021
\(399\) 11.1828 0.559841
\(400\) 0 0
\(401\) 0.424693 0.0212082 0.0106041 0.999944i \(-0.496625\pi\)
0.0106041 + 0.999944i \(0.496625\pi\)
\(402\) 4.88582 0.243683
\(403\) −0.835845 −0.0416364
\(404\) −14.6339 −0.728061
\(405\) 0 0
\(406\) 33.2648 1.65091
\(407\) −1.81215 −0.0898250
\(408\) 25.5219 1.26352
\(409\) 2.86729 0.141779 0.0708893 0.997484i \(-0.477416\pi\)
0.0708893 + 0.997484i \(0.477416\pi\)
\(410\) 0 0
\(411\) 32.0364 1.58024
\(412\) −7.45971 −0.367513
\(413\) −49.6052 −2.44091
\(414\) −4.44187 −0.218306
\(415\) 0 0
\(416\) 2.92604 0.143461
\(417\) −2.71347 −0.132879
\(418\) −0.759131 −0.0371303
\(419\) 26.1407 1.27706 0.638529 0.769597i \(-0.279543\pi\)
0.638529 + 0.769597i \(0.279543\pi\)
\(420\) 0 0
\(421\) 15.0032 0.731211 0.365606 0.930770i \(-0.380862\pi\)
0.365606 + 0.930770i \(0.380862\pi\)
\(422\) −9.43243 −0.459164
\(423\) −24.3829 −1.18554
\(424\) −27.6906 −1.34477
\(425\) 0 0
\(426\) 22.8608 1.10761
\(427\) 62.3889 3.01921
\(428\) 23.5249 1.13712
\(429\) 1.12726 0.0544244
\(430\) 0 0
\(431\) −4.12884 −0.198879 −0.0994397 0.995044i \(-0.531705\pi\)
−0.0994397 + 0.995044i \(0.531705\pi\)
\(432\) −1.77744 −0.0855173
\(433\) −2.75298 −0.132300 −0.0661499 0.997810i \(-0.521072\pi\)
−0.0661499 + 0.997810i \(0.521072\pi\)
\(434\) 6.29464 0.302153
\(435\) 0 0
\(436\) 1.33041 0.0637153
\(437\) −2.78646 −0.133294
\(438\) −22.6517 −1.08234
\(439\) 20.8996 0.997485 0.498743 0.866750i \(-0.333795\pi\)
0.498743 + 0.866750i \(0.333795\pi\)
\(440\) 0 0
\(441\) 36.7926 1.75203
\(442\) 1.64769 0.0783727
\(443\) 7.56488 0.359418 0.179709 0.983720i \(-0.442484\pi\)
0.179709 + 0.983720i \(0.442484\pi\)
\(444\) −5.82639 −0.276508
\(445\) 0 0
\(446\) −7.18781 −0.340352
\(447\) 7.35607 0.347930
\(448\) −13.3756 −0.631938
\(449\) −0.886578 −0.0418402 −0.0209201 0.999781i \(-0.506660\pi\)
−0.0209201 + 0.999781i \(0.506660\pi\)
\(450\) 0 0
\(451\) −10.5406 −0.496340
\(452\) 24.2204 1.13923
\(453\) 33.8517 1.59049
\(454\) 0.917893 0.0430788
\(455\) 0 0
\(456\) −5.86942 −0.274861
\(457\) −11.1530 −0.521716 −0.260858 0.965377i \(-0.584005\pi\)
−0.260858 + 0.965377i \(0.584005\pi\)
\(458\) 11.0493 0.516298
\(459\) −8.83880 −0.412560
\(460\) 0 0
\(461\) 42.0232 1.95721 0.978607 0.205737i \(-0.0659591\pi\)
0.978607 + 0.205737i \(0.0659591\pi\)
\(462\) −8.48922 −0.394954
\(463\) −16.2714 −0.756197 −0.378098 0.925765i \(-0.623422\pi\)
−0.378098 + 0.925765i \(0.623422\pi\)
\(464\) 7.73781 0.359219
\(465\) 0 0
\(466\) −10.6493 −0.493319
\(467\) 31.3740 1.45182 0.725909 0.687791i \(-0.241419\pi\)
0.725909 + 0.687791i \(0.241419\pi\)
\(468\) 1.49232 0.0689827
\(469\) 14.1127 0.651666
\(470\) 0 0
\(471\) −4.70016 −0.216572
\(472\) 26.0358 1.19840
\(473\) 2.65924 0.122272
\(474\) 12.1546 0.558279
\(475\) 0 0
\(476\) 30.6558 1.40511
\(477\) −22.3724 −1.02436
\(478\) 2.95274 0.135055
\(479\) −8.96367 −0.409561 −0.204780 0.978808i \(-0.565648\pi\)
−0.204780 + 0.978808i \(0.565648\pi\)
\(480\) 0 0
\(481\) −0.904558 −0.0412443
\(482\) 8.06275 0.367248
\(483\) −31.1605 −1.41785
\(484\) −1.42372 −0.0647146
\(485\) 0 0
\(486\) 14.0401 0.636872
\(487\) −7.53795 −0.341577 −0.170788 0.985308i \(-0.554631\pi\)
−0.170788 + 0.985308i \(0.554631\pi\)
\(488\) −32.7455 −1.48232
\(489\) −30.0103 −1.35711
\(490\) 0 0
\(491\) −0.0747970 −0.00337554 −0.00168777 0.999999i \(-0.500537\pi\)
−0.00168777 + 0.999999i \(0.500537\pi\)
\(492\) −33.8901 −1.52788
\(493\) 38.4783 1.73297
\(494\) −0.378930 −0.0170489
\(495\) 0 0
\(496\) 1.46421 0.0657450
\(497\) 66.0337 2.96202
\(498\) 25.9140 1.16123
\(499\) 14.9395 0.668784 0.334392 0.942434i \(-0.391469\pi\)
0.334392 + 0.942434i \(0.391469\pi\)
\(500\) 0 0
\(501\) −52.7882 −2.35840
\(502\) 2.26091 0.100909
\(503\) −26.9195 −1.20028 −0.600140 0.799895i \(-0.704888\pi\)
−0.600140 + 0.799895i \(0.704888\pi\)
\(504\) −27.0260 −1.20383
\(505\) 0 0
\(506\) 2.11529 0.0940360
\(507\) −28.7951 −1.27884
\(508\) 21.6918 0.962416
\(509\) −4.53071 −0.200820 −0.100410 0.994946i \(-0.532015\pi\)
−0.100410 + 0.994946i \(0.532015\pi\)
\(510\) 0 0
\(511\) −65.4296 −2.89444
\(512\) −9.67109 −0.427406
\(513\) 2.03271 0.0897465
\(514\) 4.67834 0.206353
\(515\) 0 0
\(516\) 8.54993 0.376390
\(517\) 11.6115 0.510675
\(518\) 6.81212 0.299307
\(519\) −26.2868 −1.15386
\(520\) 0 0
\(521\) −3.01018 −0.131879 −0.0659393 0.997824i \(-0.521004\pi\)
−0.0659393 + 0.997824i \(0.521004\pi\)
\(522\) −14.1062 −0.617413
\(523\) 20.3779 0.891063 0.445531 0.895266i \(-0.353015\pi\)
0.445531 + 0.895266i \(0.353015\pi\)
\(524\) 7.35251 0.321196
\(525\) 0 0
\(526\) −14.3189 −0.624334
\(527\) 7.28117 0.317173
\(528\) −1.97470 −0.0859376
\(529\) −15.2356 −0.662419
\(530\) 0 0
\(531\) 21.0355 0.912862
\(532\) −7.05010 −0.305661
\(533\) −5.26150 −0.227901
\(534\) 12.4431 0.538467
\(535\) 0 0
\(536\) −7.40723 −0.319943
\(537\) −34.0879 −1.47100
\(538\) 0.199194 0.00858786
\(539\) −17.5212 −0.754691
\(540\) 0 0
\(541\) −19.9886 −0.859377 −0.429689 0.902977i \(-0.641377\pi\)
−0.429689 + 0.902977i \(0.641377\pi\)
\(542\) −20.6608 −0.887457
\(543\) −34.0447 −1.46100
\(544\) −25.4892 −1.09284
\(545\) 0 0
\(546\) −4.23750 −0.181348
\(547\) −4.16030 −0.177881 −0.0889407 0.996037i \(-0.528348\pi\)
−0.0889407 + 0.996037i \(0.528348\pi\)
\(548\) −20.1971 −0.862776
\(549\) −26.4565 −1.12914
\(550\) 0 0
\(551\) −8.84908 −0.376983
\(552\) 16.3549 0.696111
\(553\) 35.1086 1.49297
\(554\) 16.8347 0.715238
\(555\) 0 0
\(556\) 1.71068 0.0725490
\(557\) −2.83353 −0.120061 −0.0600303 0.998197i \(-0.519120\pi\)
−0.0600303 + 0.998197i \(0.519120\pi\)
\(558\) −2.66930 −0.113000
\(559\) 1.32739 0.0561428
\(560\) 0 0
\(561\) −9.81969 −0.414588
\(562\) 12.1112 0.510880
\(563\) −20.3978 −0.859663 −0.429831 0.902909i \(-0.641427\pi\)
−0.429831 + 0.902909i \(0.641427\pi\)
\(564\) 37.3331 1.57201
\(565\) 0 0
\(566\) −2.27518 −0.0956331
\(567\) 53.9267 2.26471
\(568\) −34.6585 −1.45424
\(569\) −2.20481 −0.0924303 −0.0462152 0.998932i \(-0.514716\pi\)
−0.0462152 + 0.998932i \(0.514716\pi\)
\(570\) 0 0
\(571\) −33.3008 −1.39360 −0.696798 0.717268i \(-0.745392\pi\)
−0.696798 + 0.717268i \(0.745392\pi\)
\(572\) −0.710668 −0.0297145
\(573\) 8.14372 0.340209
\(574\) 39.6237 1.65386
\(575\) 0 0
\(576\) 5.67204 0.236335
\(577\) 17.1388 0.713496 0.356748 0.934201i \(-0.383885\pi\)
0.356748 + 0.934201i \(0.383885\pi\)
\(578\) −1.44806 −0.0602313
\(579\) 55.2106 2.29447
\(580\) 0 0
\(581\) 74.8529 3.10542
\(582\) 0.0332700 0.00137908
\(583\) 10.6541 0.441248
\(584\) 34.3414 1.42106
\(585\) 0 0
\(586\) 2.06251 0.0852015
\(587\) −12.4171 −0.512510 −0.256255 0.966609i \(-0.582489\pi\)
−0.256255 + 0.966609i \(0.582489\pi\)
\(588\) −56.3337 −2.32316
\(589\) −1.67449 −0.0689963
\(590\) 0 0
\(591\) 34.8715 1.43442
\(592\) 1.58458 0.0651259
\(593\) −11.0150 −0.452332 −0.226166 0.974089i \(-0.572619\pi\)
−0.226166 + 0.974089i \(0.572619\pi\)
\(594\) −1.54310 −0.0633140
\(595\) 0 0
\(596\) −4.63756 −0.189962
\(597\) −43.7406 −1.79018
\(598\) 1.05587 0.0431778
\(599\) −0.143692 −0.00587109 −0.00293555 0.999996i \(-0.500934\pi\)
−0.00293555 + 0.999996i \(0.500934\pi\)
\(600\) 0 0
\(601\) −22.1248 −0.902490 −0.451245 0.892400i \(-0.649020\pi\)
−0.451245 + 0.892400i \(0.649020\pi\)
\(602\) −9.99644 −0.407424
\(603\) −5.98463 −0.243713
\(604\) −21.3415 −0.868374
\(605\) 0 0
\(606\) −17.6210 −0.715805
\(607\) −44.0261 −1.78696 −0.893481 0.449101i \(-0.851745\pi\)
−0.893481 + 0.449101i \(0.851745\pi\)
\(608\) 5.86190 0.237732
\(609\) −98.9576 −4.00996
\(610\) 0 0
\(611\) 5.79604 0.234483
\(612\) −12.9999 −0.525488
\(613\) 7.70425 0.311172 0.155586 0.987822i \(-0.450273\pi\)
0.155586 + 0.987822i \(0.450273\pi\)
\(614\) −3.71047 −0.149742
\(615\) 0 0
\(616\) 12.8702 0.518556
\(617\) −22.4763 −0.904864 −0.452432 0.891799i \(-0.649443\pi\)
−0.452432 + 0.891799i \(0.649443\pi\)
\(618\) −8.98244 −0.361327
\(619\) −19.9890 −0.803425 −0.401712 0.915766i \(-0.631585\pi\)
−0.401712 + 0.915766i \(0.631585\pi\)
\(620\) 0 0
\(621\) −5.66408 −0.227292
\(622\) 12.6656 0.507844
\(623\) 35.9421 1.43999
\(624\) −0.985694 −0.0394593
\(625\) 0 0
\(626\) 8.04100 0.321383
\(627\) 2.25829 0.0901876
\(628\) 2.96317 0.118243
\(629\) 7.87974 0.314186
\(630\) 0 0
\(631\) 4.75533 0.189307 0.0946534 0.995510i \(-0.469826\pi\)
0.0946534 + 0.995510i \(0.469826\pi\)
\(632\) −18.4271 −0.732992
\(633\) 28.0600 1.11528
\(634\) 1.08394 0.0430489
\(635\) 0 0
\(636\) 34.2548 1.35829
\(637\) −8.74592 −0.346526
\(638\) 6.71761 0.265953
\(639\) −28.0022 −1.10775
\(640\) 0 0
\(641\) 0.415256 0.0164016 0.00820081 0.999966i \(-0.497390\pi\)
0.00820081 + 0.999966i \(0.497390\pi\)
\(642\) 28.3270 1.11798
\(643\) −22.0655 −0.870180 −0.435090 0.900387i \(-0.643283\pi\)
−0.435090 + 0.900387i \(0.643283\pi\)
\(644\) 19.6448 0.774115
\(645\) 0 0
\(646\) 3.30091 0.129873
\(647\) −39.4539 −1.55109 −0.775547 0.631290i \(-0.782526\pi\)
−0.775547 + 0.631290i \(0.782526\pi\)
\(648\) −28.3040 −1.11189
\(649\) −10.0174 −0.393218
\(650\) 0 0
\(651\) −18.7256 −0.733913
\(652\) 18.9197 0.740952
\(653\) −34.3144 −1.34283 −0.671413 0.741084i \(-0.734312\pi\)
−0.671413 + 0.741084i \(0.734312\pi\)
\(654\) 1.60199 0.0626427
\(655\) 0 0
\(656\) 9.21695 0.359861
\(657\) 27.7460 1.08247
\(658\) −43.6493 −1.70163
\(659\) 12.6713 0.493604 0.246802 0.969066i \(-0.420620\pi\)
0.246802 + 0.969066i \(0.420620\pi\)
\(660\) 0 0
\(661\) 27.5364 1.07104 0.535520 0.844522i \(-0.320115\pi\)
0.535520 + 0.844522i \(0.320115\pi\)
\(662\) 2.88196 0.112011
\(663\) −4.90162 −0.190363
\(664\) −39.2873 −1.52464
\(665\) 0 0
\(666\) −2.88873 −0.111936
\(667\) 24.6576 0.954746
\(668\) 33.2798 1.28764
\(669\) 21.3826 0.826698
\(670\) 0 0
\(671\) 12.5990 0.486380
\(672\) 65.5526 2.52875
\(673\) 20.7136 0.798449 0.399225 0.916853i \(-0.369279\pi\)
0.399225 + 0.916853i \(0.369279\pi\)
\(674\) −16.8018 −0.647183
\(675\) 0 0
\(676\) 18.1536 0.698216
\(677\) 11.5968 0.445701 0.222850 0.974853i \(-0.428464\pi\)
0.222850 + 0.974853i \(0.428464\pi\)
\(678\) 29.1644 1.12005
\(679\) 0.0961006 0.00368800
\(680\) 0 0
\(681\) −2.73058 −0.104636
\(682\) 1.27116 0.0486753
\(683\) 21.3445 0.816727 0.408363 0.912820i \(-0.366100\pi\)
0.408363 + 0.912820i \(0.366100\pi\)
\(684\) 2.98966 0.114312
\(685\) 0 0
\(686\) 39.5505 1.51005
\(687\) −32.8698 −1.25406
\(688\) −2.32529 −0.0886509
\(689\) 5.31813 0.202604
\(690\) 0 0
\(691\) 36.1885 1.37668 0.688338 0.725390i \(-0.258340\pi\)
0.688338 + 0.725390i \(0.258340\pi\)
\(692\) 16.5723 0.629984
\(693\) 10.3984 0.395003
\(694\) 1.29661 0.0492187
\(695\) 0 0
\(696\) 51.9390 1.96874
\(697\) 45.8337 1.73607
\(698\) 2.72059 0.102976
\(699\) 31.6799 1.19824
\(700\) 0 0
\(701\) −16.3009 −0.615677 −0.307839 0.951439i \(-0.599606\pi\)
−0.307839 + 0.951439i \(0.599606\pi\)
\(702\) −0.770256 −0.0290714
\(703\) −1.81215 −0.0683466
\(704\) −2.70111 −0.101802
\(705\) 0 0
\(706\) −1.83339 −0.0690006
\(707\) −50.8985 −1.91423
\(708\) −32.2078 −1.21044
\(709\) 43.5634 1.63606 0.818030 0.575176i \(-0.195066\pi\)
0.818030 + 0.575176i \(0.195066\pi\)
\(710\) 0 0
\(711\) −14.8881 −0.558348
\(712\) −18.8646 −0.706980
\(713\) 4.66591 0.174740
\(714\) 36.9135 1.38145
\(715\) 0 0
\(716\) 21.4904 0.803134
\(717\) −8.78394 −0.328042
\(718\) 14.1460 0.527924
\(719\) −11.2024 −0.417779 −0.208889 0.977939i \(-0.566985\pi\)
−0.208889 + 0.977939i \(0.566985\pi\)
\(720\) 0 0
\(721\) −25.9459 −0.966274
\(722\) −0.759131 −0.0282519
\(723\) −23.9854 −0.892026
\(724\) 21.4632 0.797672
\(725\) 0 0
\(726\) −1.71434 −0.0636251
\(727\) −40.2384 −1.49236 −0.746180 0.665744i \(-0.768114\pi\)
−0.746180 + 0.665744i \(0.768114\pi\)
\(728\) 6.42433 0.238101
\(729\) −9.09668 −0.336914
\(730\) 0 0
\(731\) −11.5631 −0.427677
\(732\) 40.5081 1.49722
\(733\) 12.7812 0.472084 0.236042 0.971743i \(-0.424150\pi\)
0.236042 + 0.971743i \(0.424150\pi\)
\(734\) −0.921542 −0.0340147
\(735\) 0 0
\(736\) −16.3340 −0.602078
\(737\) 2.84997 0.104980
\(738\) −16.8028 −0.618518
\(739\) −2.61487 −0.0961895 −0.0480948 0.998843i \(-0.515315\pi\)
−0.0480948 + 0.998843i \(0.515315\pi\)
\(740\) 0 0
\(741\) 1.12726 0.0414108
\(742\) −40.0502 −1.47029
\(743\) 26.0025 0.953937 0.476969 0.878920i \(-0.341736\pi\)
0.476969 + 0.878920i \(0.341736\pi\)
\(744\) 9.82831 0.360323
\(745\) 0 0
\(746\) −23.0147 −0.842628
\(747\) −31.7420 −1.16138
\(748\) 6.19073 0.226356
\(749\) 81.8229 2.98974
\(750\) 0 0
\(751\) −50.2644 −1.83417 −0.917086 0.398689i \(-0.869465\pi\)
−0.917086 + 0.398689i \(0.869465\pi\)
\(752\) −10.1533 −0.370255
\(753\) −6.72585 −0.245104
\(754\) 3.35318 0.122116
\(755\) 0 0
\(756\) −14.3308 −0.521208
\(757\) 0.0174245 0.000633303 0 0.000316651 1.00000i \(-0.499899\pi\)
0.000316651 1.00000i \(0.499899\pi\)
\(758\) −11.6736 −0.424006
\(759\) −6.29265 −0.228409
\(760\) 0 0
\(761\) 22.6593 0.821398 0.410699 0.911771i \(-0.365285\pi\)
0.410699 + 0.911771i \(0.365285\pi\)
\(762\) 26.1196 0.946215
\(763\) 4.62735 0.167521
\(764\) −5.13413 −0.185746
\(765\) 0 0
\(766\) 1.23130 0.0444888
\(767\) −5.00032 −0.180551
\(768\) −28.7832 −1.03862
\(769\) −43.4932 −1.56841 −0.784203 0.620504i \(-0.786928\pi\)
−0.784203 + 0.620504i \(0.786928\pi\)
\(770\) 0 0
\(771\) −13.9173 −0.501220
\(772\) −34.8070 −1.25273
\(773\) −51.7597 −1.86167 −0.930833 0.365444i \(-0.880917\pi\)
−0.930833 + 0.365444i \(0.880917\pi\)
\(774\) 4.23907 0.152370
\(775\) 0 0
\(776\) −0.0504394 −0.00181067
\(777\) −20.2650 −0.727001
\(778\) 25.5492 0.915984
\(779\) −10.5406 −0.377658
\(780\) 0 0
\(781\) 13.3351 0.477166
\(782\) −9.19786 −0.328915
\(783\) −17.9876 −0.642826
\(784\) 15.3209 0.547174
\(785\) 0 0
\(786\) 8.85336 0.315789
\(787\) −30.7685 −1.09678 −0.548389 0.836223i \(-0.684759\pi\)
−0.548389 + 0.836223i \(0.684759\pi\)
\(788\) −21.9844 −0.783163
\(789\) 42.5965 1.51647
\(790\) 0 0
\(791\) 84.2417 2.99529
\(792\) −5.45772 −0.193932
\(793\) 6.28896 0.223327
\(794\) −5.01578 −0.178003
\(795\) 0 0
\(796\) 27.5759 0.977401
\(797\) −7.43505 −0.263363 −0.131682 0.991292i \(-0.542038\pi\)
−0.131682 + 0.991292i \(0.542038\pi\)
\(798\) −8.48922 −0.300515
\(799\) −50.4902 −1.78621
\(800\) 0 0
\(801\) −15.2415 −0.538533
\(802\) −0.322398 −0.0113843
\(803\) −13.2131 −0.466279
\(804\) 9.16317 0.323160
\(805\) 0 0
\(806\) 0.634516 0.0223499
\(807\) −0.592570 −0.0208595
\(808\) 26.7146 0.939817
\(809\) −48.6194 −1.70937 −0.854683 0.519150i \(-0.826248\pi\)
−0.854683 + 0.519150i \(0.826248\pi\)
\(810\) 0 0
\(811\) 4.14175 0.145436 0.0727182 0.997353i \(-0.476833\pi\)
0.0727182 + 0.997353i \(0.476833\pi\)
\(812\) 62.3869 2.18935
\(813\) 61.4625 2.15559
\(814\) 1.37566 0.0482169
\(815\) 0 0
\(816\) 8.58653 0.300589
\(817\) 2.65924 0.0930351
\(818\) −2.17665 −0.0761049
\(819\) 5.19050 0.181371
\(820\) 0 0
\(821\) 34.4147 1.20108 0.600541 0.799594i \(-0.294952\pi\)
0.600541 + 0.799594i \(0.294952\pi\)
\(822\) −24.3199 −0.848252
\(823\) −10.2755 −0.358183 −0.179091 0.983832i \(-0.557316\pi\)
−0.179091 + 0.983832i \(0.557316\pi\)
\(824\) 13.6180 0.474404
\(825\) 0 0
\(826\) 37.6568 1.31025
\(827\) 37.7462 1.31256 0.656282 0.754516i \(-0.272128\pi\)
0.656282 + 0.754516i \(0.272128\pi\)
\(828\) −8.33056 −0.289507
\(829\) −7.42020 −0.257714 −0.128857 0.991663i \(-0.541131\pi\)
−0.128857 + 0.991663i \(0.541131\pi\)
\(830\) 0 0
\(831\) −50.0805 −1.73727
\(832\) −1.34829 −0.0467437
\(833\) 76.1870 2.63972
\(834\) 2.05988 0.0713277
\(835\) 0 0
\(836\) −1.42372 −0.0492404
\(837\) −3.40377 −0.117651
\(838\) −19.8442 −0.685508
\(839\) 11.4778 0.396259 0.198129 0.980176i \(-0.436513\pi\)
0.198129 + 0.980176i \(0.436513\pi\)
\(840\) 0 0
\(841\) 49.3062 1.70021
\(842\) −11.3894 −0.392505
\(843\) −36.0288 −1.24090
\(844\) −17.6902 −0.608920
\(845\) 0 0
\(846\) 18.5098 0.636381
\(847\) −4.95189 −0.170149
\(848\) −9.31615 −0.319918
\(849\) 6.76831 0.232288
\(850\) 0 0
\(851\) 5.04949 0.173094
\(852\) 42.8746 1.46886
\(853\) 34.4829 1.18067 0.590336 0.807158i \(-0.298995\pi\)
0.590336 + 0.807158i \(0.298995\pi\)
\(854\) −47.3613 −1.62067
\(855\) 0 0
\(856\) −42.9456 −1.46785
\(857\) −14.4282 −0.492859 −0.246429 0.969161i \(-0.579257\pi\)
−0.246429 + 0.969161i \(0.579257\pi\)
\(858\) −0.855735 −0.0292143
\(859\) −27.3349 −0.932653 −0.466327 0.884613i \(-0.654423\pi\)
−0.466327 + 0.884613i \(0.654423\pi\)
\(860\) 0 0
\(861\) −117.874 −4.01714
\(862\) 3.13433 0.106756
\(863\) 7.77083 0.264522 0.132261 0.991215i \(-0.457776\pi\)
0.132261 + 0.991215i \(0.457776\pi\)
\(864\) 11.9156 0.405376
\(865\) 0 0
\(866\) 2.08987 0.0710167
\(867\) 4.30774 0.146299
\(868\) 11.8054 0.400700
\(869\) 7.08995 0.240510
\(870\) 0 0
\(871\) 1.42260 0.0482029
\(872\) −2.42872 −0.0822468
\(873\) −0.0407523 −0.00137925
\(874\) 2.11529 0.0715507
\(875\) 0 0
\(876\) −42.4824 −1.43535
\(877\) −33.4134 −1.12829 −0.564145 0.825676i \(-0.690794\pi\)
−0.564145 + 0.825676i \(0.690794\pi\)
\(878\) −15.8656 −0.535437
\(879\) −6.13564 −0.206950
\(880\) 0 0
\(881\) −38.4814 −1.29647 −0.648236 0.761439i \(-0.724493\pi\)
−0.648236 + 0.761439i \(0.724493\pi\)
\(882\) −27.9304 −0.940464
\(883\) −6.45348 −0.217177 −0.108589 0.994087i \(-0.534633\pi\)
−0.108589 + 0.994087i \(0.534633\pi\)
\(884\) 3.09018 0.103934
\(885\) 0 0
\(886\) −5.74273 −0.192931
\(887\) 51.6865 1.73546 0.867732 0.497032i \(-0.165577\pi\)
0.867732 + 0.497032i \(0.165577\pi\)
\(888\) 10.6363 0.356930
\(889\) 75.4468 2.53040
\(890\) 0 0
\(891\) 10.8901 0.364833
\(892\) −13.4804 −0.451359
\(893\) 11.6115 0.388565
\(894\) −5.58422 −0.186764
\(895\) 0 0
\(896\) −47.9011 −1.60026
\(897\) −3.14105 −0.104877
\(898\) 0.673029 0.0224593
\(899\) 14.8177 0.494199
\(900\) 0 0
\(901\) −46.3270 −1.54338
\(902\) 8.00173 0.266429
\(903\) 29.7378 0.989612
\(904\) −44.2152 −1.47057
\(905\) 0 0
\(906\) −25.6979 −0.853756
\(907\) 51.3590 1.70535 0.852673 0.522445i \(-0.174980\pi\)
0.852673 + 0.522445i \(0.174980\pi\)
\(908\) 1.72147 0.0571290
\(909\) 21.5839 0.715894
\(910\) 0 0
\(911\) −30.5365 −1.01172 −0.505859 0.862616i \(-0.668825\pi\)
−0.505859 + 0.862616i \(0.668825\pi\)
\(912\) −1.97470 −0.0653887
\(913\) 15.1160 0.500267
\(914\) 8.46660 0.280050
\(915\) 0 0
\(916\) 20.7225 0.684689
\(917\) 25.5730 0.844495
\(918\) 6.70981 0.221457
\(919\) −44.4425 −1.46602 −0.733011 0.680217i \(-0.761886\pi\)
−0.733011 + 0.680217i \(0.761886\pi\)
\(920\) 0 0
\(921\) 11.0380 0.363716
\(922\) −31.9011 −1.05061
\(923\) 6.65637 0.219097
\(924\) −15.9212 −0.523769
\(925\) 0 0
\(926\) 12.3521 0.405916
\(927\) 11.0026 0.361371
\(928\) −51.8724 −1.70280
\(929\) −4.42046 −0.145030 −0.0725152 0.997367i \(-0.523103\pi\)
−0.0725152 + 0.997367i \(0.523103\pi\)
\(930\) 0 0
\(931\) −17.5212 −0.574234
\(932\) −19.9723 −0.654215
\(933\) −37.6781 −1.23353
\(934\) −23.8170 −0.779316
\(935\) 0 0
\(936\) −2.72429 −0.0890462
\(937\) −57.9019 −1.89157 −0.945786 0.324789i \(-0.894707\pi\)
−0.945786 + 0.324789i \(0.894707\pi\)
\(938\) −10.7134 −0.349806
\(939\) −23.9207 −0.780622
\(940\) 0 0
\(941\) −46.1004 −1.50283 −0.751415 0.659830i \(-0.770628\pi\)
−0.751415 + 0.659830i \(0.770628\pi\)
\(942\) 3.56803 0.116253
\(943\) 29.3711 0.956455
\(944\) 8.75943 0.285095
\(945\) 0 0
\(946\) −2.01871 −0.0656340
\(947\) −29.8328 −0.969434 −0.484717 0.874671i \(-0.661077\pi\)
−0.484717 + 0.874671i \(0.661077\pi\)
\(948\) 22.7954 0.740362
\(949\) −6.59547 −0.214098
\(950\) 0 0
\(951\) −3.22456 −0.104563
\(952\) −55.9633 −1.81378
\(953\) 9.41874 0.305103 0.152551 0.988296i \(-0.451251\pi\)
0.152551 + 0.988296i \(0.451251\pi\)
\(954\) 16.9836 0.549864
\(955\) 0 0
\(956\) 5.53775 0.179104
\(957\) −19.9838 −0.645985
\(958\) 6.80460 0.219847
\(959\) −70.2481 −2.26843
\(960\) 0 0
\(961\) −28.1961 −0.909551
\(962\) 0.686678 0.0221394
\(963\) −34.6977 −1.11812
\(964\) 15.1214 0.487026
\(965\) 0 0
\(966\) 23.6549 0.761083
\(967\) −51.5972 −1.65925 −0.829627 0.558318i \(-0.811447\pi\)
−0.829627 + 0.558318i \(0.811447\pi\)
\(968\) 2.59905 0.0835367
\(969\) −9.81969 −0.315454
\(970\) 0 0
\(971\) −4.76616 −0.152953 −0.0764767 0.997071i \(-0.524367\pi\)
−0.0764767 + 0.997071i \(0.524367\pi\)
\(972\) 26.3317 0.844589
\(973\) 5.94997 0.190747
\(974\) 5.72229 0.183354
\(975\) 0 0
\(976\) −11.0168 −0.352640
\(977\) 1.46971 0.0470203 0.0235101 0.999724i \(-0.492516\pi\)
0.0235101 + 0.999724i \(0.492516\pi\)
\(978\) 22.7817 0.728479
\(979\) 7.25826 0.231975
\(980\) 0 0
\(981\) −1.96227 −0.0626504
\(982\) 0.0567807 0.00181195
\(983\) −8.77114 −0.279756 −0.139878 0.990169i \(-0.544671\pi\)
−0.139878 + 0.990169i \(0.544671\pi\)
\(984\) 61.8675 1.97226
\(985\) 0 0
\(986\) −29.2100 −0.930237
\(987\) 129.850 4.13316
\(988\) −0.710668 −0.0226094
\(989\) −7.40987 −0.235620
\(990\) 0 0
\(991\) 29.7744 0.945816 0.472908 0.881112i \(-0.343204\pi\)
0.472908 + 0.881112i \(0.343204\pi\)
\(992\) −9.81572 −0.311650
\(993\) −8.57337 −0.272068
\(994\) −50.1283 −1.58997
\(995\) 0 0
\(996\) 48.6007 1.53997
\(997\) −13.8558 −0.438818 −0.219409 0.975633i \(-0.570413\pi\)
−0.219409 + 0.975633i \(0.570413\pi\)
\(998\) −11.3410 −0.358994
\(999\) −3.68359 −0.116543
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 5225.2.a.k.1.3 6
5.4 even 2 1045.2.a.g.1.4 6
15.14 odd 2 9405.2.a.w.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.4 6 5.4 even 2
5225.2.a.k.1.3 6 1.1 even 1 trivial
9405.2.a.w.1.3 6 15.14 odd 2