Properties

Label 9405.2.a.w.1.3
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
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: \(\Z[a_1, \ldots, a_{7}]\)
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 \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.759131 q^{2} -1.42372 q^{4} -1.00000 q^{5} +4.95189 q^{7} +2.59905 q^{8} +O(q^{10})\) \(q-0.759131 q^{2} -1.42372 q^{4} -1.00000 q^{5} +4.95189 q^{7} +2.59905 q^{8} +0.759131 q^{10} +1.00000 q^{11} +0.499163 q^{13} -3.75913 q^{14} +0.874419 q^{16} +4.34828 q^{17} -1.00000 q^{19} +1.42372 q^{20} -0.759131 q^{22} +2.78646 q^{23} +1.00000 q^{25} -0.378930 q^{26} -7.05010 q^{28} -8.84908 q^{29} +1.67449 q^{31} -5.86190 q^{32} -3.30091 q^{34} -4.95189 q^{35} -1.81215 q^{37} +0.759131 q^{38} -2.59905 q^{40} -10.5406 q^{41} +2.65924 q^{43} -1.42372 q^{44} -2.11529 q^{46} -11.6115 q^{47} +17.5212 q^{49} -0.759131 q^{50} -0.710668 q^{52} -10.6541 q^{53} -1.00000 q^{55} +12.8702 q^{56} +6.71761 q^{58} -10.0174 q^{59} -12.5990 q^{61} -1.27116 q^{62} +2.70111 q^{64} -0.499163 q^{65} +2.84997 q^{67} -6.19073 q^{68} +3.75913 q^{70} +13.3351 q^{71} -13.2131 q^{73} +1.37566 q^{74} +1.42372 q^{76} +4.95189 q^{77} -7.08995 q^{79} -0.874419 q^{80} +8.00173 q^{82} -15.1160 q^{83} -4.34828 q^{85} -2.01871 q^{86} +2.59905 q^{88} +7.25826 q^{89} +2.47180 q^{91} -3.96714 q^{92} +8.81467 q^{94} +1.00000 q^{95} +0.0194069 q^{97} -13.3009 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 6 q^{5} + 5 q^{7} + 6 q^{11} - 9 q^{13} - 18 q^{14} + 4 q^{16} + 5 q^{17} - 6 q^{19} - 8 q^{20} - 8 q^{23} + 6 q^{25} + 22 q^{26} + 10 q^{28} + 5 q^{29} - q^{31} - 15 q^{32} - 22 q^{34} - 5 q^{35} + 9 q^{37} - 25 q^{41} + 15 q^{43} + 8 q^{44} - 16 q^{46} - 24 q^{47} + 13 q^{49} - 27 q^{52} - 5 q^{53} - 6 q^{55} + 12 q^{56} + 13 q^{58} - 39 q^{59} - 11 q^{61} + 42 q^{62} - 14 q^{64} + 9 q^{65} + 24 q^{67} - 45 q^{68} + 18 q^{70} + 24 q^{71} - 26 q^{73} - q^{74} - 8 q^{76} + 5 q^{77} + 11 q^{79} - 4 q^{80} + 8 q^{82} - 39 q^{83} - 5 q^{85} - 18 q^{86} - 22 q^{89} - 26 q^{91} + 11 q^{92} - 30 q^{94} + 6 q^{95} + 22 q^{97} - 33 q^{98}+O(q^{100}) \) 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 \(0.586493\pi\)
\(3\) 0 0
\(4\) −1.42372 −0.711860
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.95189 1.87164 0.935819 0.352482i \(-0.114662\pi\)
0.935819 + 0.352482i \(0.114662\pi\)
\(8\) 2.59905 0.918904
\(9\) 0 0
\(10\) 0.759131 0.240058
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 0.499163 0.138443 0.0692214 0.997601i \(-0.477949\pi\)
0.0692214 + 0.997601i \(0.477949\pi\)
\(14\) −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 \(0.323202\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 1.42372 0.318354
\(21\) 0 0
\(22\) −0.759131 −0.161847
\(23\) 2.78646 0.581017 0.290509 0.956872i \(-0.406176\pi\)
0.290509 + 0.956872i \(0.406176\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −0.378930 −0.0743142
\(27\) 0 0
\(28\) −7.05010 −1.33234
\(29\) −8.84908 −1.64323 −0.821616 0.570041i \(-0.806927\pi\)
−0.821616 + 0.570041i \(0.806927\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −3.30091 −0.566102
\(35\) −4.95189 −0.837022
\(36\) 0 0
\(37\) −1.81215 −0.297916 −0.148958 0.988844i \(-0.547592\pi\)
−0.148958 + 0.988844i \(0.547592\pi\)
\(38\) 0.759131 0.123147
\(39\) 0 0
\(40\) −2.59905 −0.410946
\(41\) −10.5406 −1.64617 −0.823086 0.567916i \(-0.807750\pi\)
−0.823086 + 0.567916i \(0.807750\pi\)
\(42\) 0 0
\(43\) 2.65924 0.405531 0.202765 0.979227i \(-0.435007\pi\)
0.202765 + 0.979227i \(0.435007\pi\)
\(44\) −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 \(0.821509\pi\)
\(48\) 0 0
\(49\) 17.5212 2.50303
\(50\) −0.759131 −0.107357
\(51\) 0 0
\(52\) −0.710668 −0.0985519
\(53\) −10.6541 −1.46345 −0.731726 0.681598i \(-0.761285\pi\)
−0.731726 + 0.681598i \(0.761285\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 12.8702 1.71985
\(57\) 0 0
\(58\) 6.71761 0.882065
\(59\) −10.0174 −1.30416 −0.652079 0.758151i \(-0.726103\pi\)
−0.652079 + 0.758151i \(0.726103\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 2.70111 0.337639
\(65\) −0.499163 −0.0619135
\(66\) 0 0
\(67\) 2.84997 0.348180 0.174090 0.984730i \(-0.444302\pi\)
0.174090 + 0.984730i \(0.444302\pi\)
\(68\) −6.19073 −0.750736
\(69\) 0 0
\(70\) 3.75913 0.449302
\(71\) 13.3351 1.58258 0.791291 0.611440i \(-0.209410\pi\)
0.791291 + 0.611440i \(0.209410\pi\)
\(72\) 0 0
\(73\) −13.2131 −1.54647 −0.773236 0.634118i \(-0.781363\pi\)
−0.773236 + 0.634118i \(0.781363\pi\)
\(74\) 1.37566 0.159917
\(75\) 0 0
\(76\) 1.42372 0.163312
\(77\) 4.95189 0.564320
\(78\) 0 0
\(79\) −7.08995 −0.797681 −0.398841 0.917020i \(-0.630587\pi\)
−0.398841 + 0.917020i \(0.630587\pi\)
\(80\) −0.874419 −0.0977631
\(81\) 0 0
\(82\) 8.00173 0.883643
\(83\) −15.1160 −1.65920 −0.829600 0.558359i \(-0.811431\pi\)
−0.829600 + 0.558359i \(0.811431\pi\)
\(84\) 0 0
\(85\) −4.34828 −0.471637
\(86\) −2.01871 −0.217683
\(87\) 0 0
\(88\) 2.59905 0.277060
\(89\) 7.25826 0.769374 0.384687 0.923047i \(-0.374309\pi\)
0.384687 + 0.923047i \(0.374309\pi\)
\(90\) 0 0
\(91\) 2.47180 0.259115
\(92\) −3.96714 −0.413603
\(93\) 0 0
\(94\) 8.81467 0.909164
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 0.0194069 0.00197047 0.000985234 1.00000i \(-0.499686\pi\)
0.000985234 1.00000i \(0.499686\pi\)
\(98\) −13.3009 −1.34359
\(99\) 0 0
\(100\) −1.42372 −0.142372
\(101\) −10.2786 −1.02276 −0.511379 0.859355i \(-0.670865\pi\)
−0.511379 + 0.859355i \(0.670865\pi\)
\(102\) 0 0
\(103\) −5.23959 −0.516272 −0.258136 0.966109i \(-0.583108\pi\)
−0.258136 + 0.966109i \(0.583108\pi\)
\(104\) 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 \(0.794477\pi\)
\(108\) 0 0
\(109\) −0.934463 −0.0895053 −0.0447527 0.998998i \(-0.514250\pi\)
−0.0447527 + 0.998998i \(0.514250\pi\)
\(110\) 0.759131 0.0723803
\(111\) 0 0
\(112\) 4.33003 0.409149
\(113\) −17.0120 −1.60036 −0.800179 0.599761i \(-0.795262\pi\)
−0.800179 + 0.599761i \(0.795262\pi\)
\(114\) 0 0
\(115\) −2.78646 −0.259839
\(116\) 12.5986 1.16975
\(117\) 0 0
\(118\) 7.60454 0.700054
\(119\) 21.5322 1.97385
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 9.56430 0.865911
\(123\) 0 0
\(124\) −2.38401 −0.214091
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 15.2360 1.35197 0.675987 0.736914i \(-0.263718\pi\)
0.675987 + 0.736914i \(0.263718\pi\)
\(128\) 9.67331 0.855008
\(129\) 0 0
\(130\) 0.378930 0.0332343
\(131\) 5.16429 0.451206 0.225603 0.974219i \(-0.427565\pi\)
0.225603 + 0.974219i \(0.427565\pi\)
\(132\) 0 0
\(133\) −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 \(0.292773\pi\)
\(138\) 0 0
\(139\) −1.20156 −0.101915 −0.0509573 0.998701i \(-0.516227\pi\)
−0.0509573 + 0.998701i \(0.516227\pi\)
\(140\) 7.05010 0.595842
\(141\) 0 0
\(142\) −10.1231 −0.849509
\(143\) 0.499163 0.0417421
\(144\) 0 0
\(145\) 8.84908 0.734876
\(146\) 10.0304 0.830126
\(147\) 0 0
\(148\) 2.58000 0.212074
\(149\) −3.25736 −0.266853 −0.133427 0.991059i \(-0.542598\pi\)
−0.133427 + 0.991059i \(0.542598\pi\)
\(150\) 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
\(153\) 0 0
\(154\) −3.75913 −0.302919
\(155\) −1.67449 −0.134499
\(156\) 0 0
\(157\) 2.08129 0.166105 0.0830524 0.996545i \(-0.473533\pi\)
0.0830524 + 0.996545i \(0.473533\pi\)
\(158\) 5.38220 0.428185
\(159\) 0 0
\(160\) 5.86190 0.463424
\(161\) 13.7982 1.08745
\(162\) 0 0
\(163\) 13.2889 1.04087 0.520434 0.853902i \(-0.325770\pi\)
0.520434 + 0.853902i \(0.325770\pi\)
\(164\) 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 \(0.859693\pi\)
\(168\) 0 0
\(169\) −12.7508 −0.980834
\(170\) 3.30091 0.253168
\(171\) 0 0
\(172\) −3.78602 −0.288681
\(173\) −11.6401 −0.884983 −0.442491 0.896773i \(-0.645905\pi\)
−0.442491 + 0.896773i \(0.645905\pi\)
\(174\) 0 0
\(175\) 4.95189 0.374327
\(176\) 0.874419 0.0659118
\(177\) 0 0
\(178\) −5.50997 −0.412990
\(179\) 15.0945 1.12822 0.564110 0.825700i \(-0.309219\pi\)
0.564110 + 0.825700i \(0.309219\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 7.24216 0.533899
\(185\) 1.81215 0.133232
\(186\) 0 0
\(187\) 4.34828 0.317978
\(188\) 16.5316 1.20569
\(189\) 0 0
\(190\) −0.759131 −0.0550731
\(191\) −3.60614 −0.260931 −0.130466 0.991453i \(-0.541647\pi\)
−0.130466 + 0.991453i \(0.541647\pi\)
\(192\) 0 0
\(193\) −24.4479 −1.75980 −0.879900 0.475158i \(-0.842391\pi\)
−0.879900 + 0.475158i \(0.842391\pi\)
\(194\) −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 \(0.314597\pi\)
\(198\) 0 0
\(199\) −19.3689 −1.37302 −0.686512 0.727119i \(-0.740859\pi\)
−0.686512 + 0.727119i \(0.740859\pi\)
\(200\) 2.59905 0.183781
\(201\) 0 0
\(202\) 7.80280 0.549003
\(203\) −43.8196 −3.07554
\(204\) 0 0
\(205\) 10.5406 0.736191
\(206\) 3.97753 0.277128
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 12.5436 0.857460
\(215\) −2.65924 −0.181359
\(216\) 0 0
\(217\) 8.29191 0.562891
\(218\) 0.709380 0.0480453
\(219\) 0 0
\(220\) 1.42372 0.0959872
\(221\) 2.17050 0.146003
\(222\) 0 0
\(223\) −9.46847 −0.634055 −0.317028 0.948416i \(-0.602685\pi\)
−0.317028 + 0.948416i \(0.602685\pi\)
\(224\) −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 \(0.512776\pi\)
\(228\) 0 0
\(229\) −14.5551 −0.961831 −0.480916 0.876767i \(-0.659696\pi\)
−0.480916 + 0.876767i \(0.659696\pi\)
\(230\) 2.11529 0.139478
\(231\) 0 0
\(232\) −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 0
\(235\) 11.6115 0.757453
\(236\) 14.2620 0.928378
\(237\) 0 0
\(238\) −16.3457 −1.05954
\(239\) 3.88964 0.251600 0.125800 0.992056i \(-0.459850\pi\)
0.125800 + 0.992056i \(0.459850\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 17.9375 1.14833
\(245\) −17.5212 −1.11939
\(246\) 0 0
\(247\) −0.499163 −0.0317610
\(248\) 4.35210 0.276358
\(249\) 0 0
\(250\) 0.759131 0.0480117
\(251\) 2.97829 0.187988 0.0939940 0.995573i \(-0.470037\pi\)
0.0939940 + 0.995573i \(0.470037\pi\)
\(252\) 0 0
\(253\) 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 \(0.561566\pi\)
\(258\) 0 0
\(259\) −8.97357 −0.557591
\(260\) 0.710668 0.0440737
\(261\) 0 0
\(262\) −3.92037 −0.242202
\(263\) 18.8622 1.16310 0.581548 0.813512i \(-0.302447\pi\)
0.581548 + 0.813512i \(0.302447\pi\)
\(264\) 0 0
\(265\) 10.6541 0.654476
\(266\) 3.75913 0.230487
\(267\) 0 0
\(268\) −4.05756 −0.247855
\(269\) 0.262397 0.0159986 0.00799932 0.999968i \(-0.497454\pi\)
0.00799932 + 0.999968i \(0.497454\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −10.7691 −0.650587
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) 22.1763 1.33244 0.666222 0.745754i \(-0.267910\pi\)
0.666222 + 0.745754i \(0.267910\pi\)
\(278\) 0.912138 0.0547064
\(279\) 0 0
\(280\) −12.8702 −0.769142
\(281\) 15.9540 0.951737 0.475868 0.879517i \(-0.342134\pi\)
0.475868 + 0.879517i \(0.342134\pi\)
\(282\) 0 0
\(283\) −2.99709 −0.178159 −0.0890793 0.996025i \(-0.528392\pi\)
−0.0890793 + 0.996025i \(0.528392\pi\)
\(284\) −18.9854 −1.12658
\(285\) 0 0
\(286\) −0.378930 −0.0224066
\(287\) −52.1961 −3.08104
\(288\) 0 0
\(289\) 1.90752 0.112207
\(290\) −6.71761 −0.394472
\(291\) 0 0
\(292\) 18.8117 1.10087
\(293\) −2.71694 −0.158725 −0.0793625 0.996846i \(-0.525288\pi\)
−0.0793625 + 0.996846i \(0.525288\pi\)
\(294\) 0 0
\(295\) 10.0174 0.583237
\(296\) −4.70988 −0.273756
\(297\) 0 0
\(298\) 2.47276 0.143243
\(299\) 1.39090 0.0804376
\(300\) 0 0
\(301\) 13.1683 0.759006
\(302\) −11.3793 −0.654808
\(303\) 0 0
\(304\) −0.874419 −0.0501514
\(305\) 12.5990 0.721418
\(306\) 0 0
\(307\) −4.88778 −0.278961 −0.139480 0.990225i \(-0.544543\pi\)
−0.139480 + 0.990225i \(0.544543\pi\)
\(308\) −7.05010 −0.401717
\(309\) 0 0
\(310\) 1.27116 0.0721971
\(311\) 16.6843 0.946082 0.473041 0.881040i \(-0.343156\pi\)
0.473041 + 0.881040i \(0.343156\pi\)
\(312\) 0 0
\(313\) 10.5924 0.598716 0.299358 0.954141i \(-0.403227\pi\)
0.299358 + 0.954141i \(0.403227\pi\)
\(314\) −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 \(0.512767\pi\)
\(318\) 0 0
\(319\) −8.84908 −0.495453
\(320\) −2.70111 −0.150997
\(321\) 0 0
\(322\) −10.4747 −0.583731
\(323\) −4.34828 −0.241945
\(324\) 0 0
\(325\) 0.499163 0.0276886
\(326\) −10.0880 −0.558724
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) 17.7449 0.970957
\(335\) −2.84997 −0.155711
\(336\) 0 0
\(337\) −22.1330 −1.20566 −0.602830 0.797869i \(-0.705960\pi\)
−0.602830 + 0.797869i \(0.705960\pi\)
\(338\) 9.67956 0.526498
\(339\) 0 0
\(340\) 6.19073 0.335740
\(341\) 1.67449 0.0906790
\(342\) 0 0
\(343\) 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 \(0.514598\pi\)
\(348\) 0 0
\(349\) −3.58382 −0.191837 −0.0959186 0.995389i \(-0.530579\pi\)
−0.0959186 + 0.995389i \(0.530579\pi\)
\(350\) −3.75913 −0.200934
\(351\) 0 0
\(352\) −5.86190 −0.312440
\(353\) 2.41512 0.128544 0.0642719 0.997932i \(-0.479528\pi\)
0.0642719 + 0.997932i \(0.479528\pi\)
\(354\) 0 0
\(355\) −13.3351 −0.707752
\(356\) −10.3337 −0.547686
\(357\) 0 0
\(358\) −11.4587 −0.605613
\(359\) 18.6345 0.983489 0.491745 0.870739i \(-0.336359\pi\)
0.491745 + 0.870739i \(0.336359\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 11.4442 0.601494
\(363\) 0 0
\(364\) −3.51915 −0.184453
\(365\) 13.2131 0.691603
\(366\) 0 0
\(367\) −1.21394 −0.0633673 −0.0316837 0.999498i \(-0.510087\pi\)
−0.0316837 + 0.999498i \(0.510087\pi\)
\(368\) 2.43654 0.127013
\(369\) 0 0
\(370\) −1.37566 −0.0715172
\(371\) −52.7579 −2.73905
\(372\) 0 0
\(373\) −30.3171 −1.56976 −0.784881 0.619646i \(-0.787276\pi\)
−0.784881 + 0.619646i \(0.787276\pi\)
\(374\) −3.30091 −0.170686
\(375\) 0 0
\(376\) −30.1790 −1.55636
\(377\) −4.41713 −0.227494
\(378\) 0 0
\(379\) 15.3776 0.789897 0.394948 0.918703i \(-0.370762\pi\)
0.394948 + 0.918703i \(0.370762\pi\)
\(380\) −1.42372 −0.0730353
\(381\) 0 0
\(382\) 2.73753 0.140064
\(383\) −1.62199 −0.0828798 −0.0414399 0.999141i \(-0.513195\pi\)
−0.0414399 + 0.999141i \(0.513195\pi\)
\(384\) 0 0
\(385\) −4.95189 −0.252372
\(386\) 18.5592 0.944638
\(387\) 0 0
\(388\) −0.0276299 −0.00140270
\(389\) 33.6559 1.70642 0.853210 0.521567i \(-0.174652\pi\)
0.853210 + 0.521567i \(0.174652\pi\)
\(390\) 0 0
\(391\) 12.1163 0.612748
\(392\) 45.5385 2.30004
\(393\) 0 0
\(394\) −11.7221 −0.590553
\(395\) 7.08995 0.356734
\(396\) 0 0
\(397\) −6.60727 −0.331609 −0.165805 0.986159i \(-0.553022\pi\)
−0.165805 + 0.986159i \(0.553022\pi\)
\(398\) 14.7035 0.737021
\(399\) 0 0
\(400\) 0.874419 0.0437210
\(401\) −0.424693 −0.0212082 −0.0106041 0.999944i \(-0.503375\pi\)
−0.0106041 + 0.999944i \(0.503375\pi\)
\(402\) 0 0
\(403\) 0.835845 0.0416364
\(404\) 14.6339 0.728061
\(405\) 0 0
\(406\) 33.2648 1.65091
\(407\) −1.81215 −0.0898250
\(408\) 0 0
\(409\) 2.86729 0.141779 0.0708893 0.997484i \(-0.477416\pi\)
0.0708893 + 0.997484i \(0.477416\pi\)
\(410\) −8.00173 −0.395177
\(411\) 0 0
\(412\) 7.45971 0.367513
\(413\) −49.6052 −2.44091
\(414\) 0 0
\(415\) 15.1160 0.742017
\(416\) −2.92604 −0.143461
\(417\) 0 0
\(418\) 0.759131 0.0371303
\(419\) −26.1407 −1.27706 −0.638529 0.769597i \(-0.720457\pi\)
−0.638529 + 0.769597i \(0.720457\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −27.6906 −1.34477
\(425\) 4.34828 0.210922
\(426\) 0 0
\(427\) −62.3889 −3.01921
\(428\) 23.5249 1.13712
\(429\) 0 0
\(430\) 2.01871 0.0973510
\(431\) 4.12884 0.198879 0.0994397 0.995044i \(-0.468295\pi\)
0.0994397 + 0.995044i \(0.468295\pi\)
\(432\) 0 0
\(433\) 2.75298 0.132300 0.0661499 0.997810i \(-0.478928\pi\)
0.0661499 + 0.997810i \(0.478928\pi\)
\(434\) −6.29464 −0.302153
\(435\) 0 0
\(436\) 1.33041 0.0637153
\(437\) −2.78646 −0.133294
\(438\) 0 0
\(439\) 20.8996 0.997485 0.498743 0.866750i \(-0.333795\pi\)
0.498743 + 0.866750i \(0.333795\pi\)
\(440\) −2.59905 −0.123905
\(441\) 0 0
\(442\) −1.64769 −0.0783727
\(443\) 7.56488 0.359418 0.179709 0.983720i \(-0.442484\pi\)
0.179709 + 0.983720i \(0.442484\pi\)
\(444\) 0 0
\(445\) −7.25826 −0.344074
\(446\) 7.18781 0.340352
\(447\) 0 0
\(448\) 13.3756 0.631938
\(449\) 0.886578 0.0418402 0.0209201 0.999781i \(-0.493340\pi\)
0.0209201 + 0.999781i \(0.493340\pi\)
\(450\) 0 0
\(451\) −10.5406 −0.496340
\(452\) 24.2204 1.13923
\(453\) 0 0
\(454\) 0.917893 0.0430788
\(455\) −2.47180 −0.115880
\(456\) 0 0
\(457\) 11.1530 0.521716 0.260858 0.965377i \(-0.415995\pi\)
0.260858 + 0.965377i \(0.415995\pi\)
\(458\) 11.0493 0.516298
\(459\) 0 0
\(460\) 3.96714 0.184969
\(461\) −42.0232 −1.95721 −0.978607 0.205737i \(-0.934041\pi\)
−0.978607 + 0.205737i \(0.934041\pi\)
\(462\) 0 0
\(463\) 16.2714 0.756197 0.378098 0.925765i \(-0.376578\pi\)
0.378098 + 0.925765i \(0.376578\pi\)
\(464\) −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 \(0.241419\pi\)
\(468\) 0 0
\(469\) 14.1127 0.651666
\(470\) −8.81467 −0.406591
\(471\) 0 0
\(472\) −26.0358 −1.19840
\(473\) 2.65924 0.122272
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −30.6558 −1.40511
\(477\) 0 0
\(478\) −2.95274 −0.135055
\(479\) 8.96367 0.409561 0.204780 0.978808i \(-0.434352\pi\)
0.204780 + 0.978808i \(0.434352\pi\)
\(480\) 0 0
\(481\) −0.904558 −0.0412443
\(482\) 8.06275 0.367248
\(483\) 0 0
\(484\) −1.42372 −0.0647146
\(485\) −0.0194069 −0.000881220 0
\(486\) 0 0
\(487\) 7.53795 0.341577 0.170788 0.985308i \(-0.445369\pi\)
0.170788 + 0.985308i \(0.445369\pi\)
\(488\) −32.7455 −1.48232
\(489\) 0 0
\(490\) 13.3009 0.600872
\(491\) 0.0747970 0.00337554 0.00168777 0.999999i \(-0.499463\pi\)
0.00168777 + 0.999999i \(0.499463\pi\)
\(492\) 0 0
\(493\) −38.4783 −1.73297
\(494\) 0.378930 0.0170489
\(495\) 0 0
\(496\) 1.46421 0.0657450
\(497\) 66.0337 2.96202
\(498\) 0 0
\(499\) 14.9395 0.668784 0.334392 0.942434i \(-0.391469\pi\)
0.334392 + 0.942434i \(0.391469\pi\)
\(500\) 1.42372 0.0636707
\(501\) 0 0
\(502\) −2.26091 −0.100909
\(503\) −26.9195 −1.20028 −0.600140 0.799895i \(-0.704888\pi\)
−0.600140 + 0.799895i \(0.704888\pi\)
\(504\) 0 0
\(505\) 10.2786 0.457392
\(506\) −2.11529 −0.0940360
\(507\) 0 0
\(508\) −21.6918 −0.962416
\(509\) 4.53071 0.200820 0.100410 0.994946i \(-0.467985\pi\)
0.100410 + 0.994946i \(0.467985\pi\)
\(510\) 0 0
\(511\) −65.4296 −2.89444
\(512\) −9.67109 −0.427406
\(513\) 0 0
\(514\) 4.67834 0.206353
\(515\) 5.23959 0.230884
\(516\) 0 0
\(517\) −11.6115 −0.510675
\(518\) 6.81212 0.299307
\(519\) 0 0
\(520\) −1.29735 −0.0568925
\(521\) 3.01018 0.131879 0.0659393 0.997824i \(-0.478996\pi\)
0.0659393 + 0.997824i \(0.478996\pi\)
\(522\) 0 0
\(523\) −20.3779 −0.891063 −0.445531 0.895266i \(-0.646985\pi\)
−0.445531 + 0.895266i \(0.646985\pi\)
\(524\) −7.35251 −0.321196
\(525\) 0 0
\(526\) −14.3189 −0.624334
\(527\) 7.28117 0.317173
\(528\) 0 0
\(529\) −15.2356 −0.662419
\(530\) −8.08786 −0.351314
\(531\) 0 0
\(532\) 7.05010 0.305661
\(533\) −5.26150 −0.227901
\(534\) 0 0
\(535\) 16.5236 0.714376
\(536\) 7.40723 0.319943
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) −25.4892 −1.09284
\(545\) 0.934463 0.0400280
\(546\) 0 0
\(547\) 4.16030 0.177881 0.0889407 0.996037i \(-0.471652\pi\)
0.0889407 + 0.996037i \(0.471652\pi\)
\(548\) −20.1971 −0.862776
\(549\) 0 0
\(550\) −0.759131 −0.0323695
\(551\) 8.84908 0.376983
\(552\) 0 0
\(553\) −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 \(0.519120\pi\)
\(558\) 0 0
\(559\) 1.32739 0.0561428
\(560\) −4.33003 −0.182977
\(561\) 0 0
\(562\) −12.1112 −0.510880
\(563\) −20.3978 −0.859663 −0.429831 0.902909i \(-0.641427\pi\)
−0.429831 + 0.902909i \(0.641427\pi\)
\(564\) 0 0
\(565\) 17.0120 0.715702
\(566\) 2.27518 0.0956331
\(567\) 0 0
\(568\) 34.6585 1.45424
\(569\) 2.20481 0.0924303 0.0462152 0.998932i \(-0.485284\pi\)
0.0462152 + 0.998932i \(0.485284\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 39.6237 1.65386
\(575\) 2.78646 0.116203
\(576\) 0 0
\(577\) −17.1388 −0.713496 −0.356748 0.934201i \(-0.616115\pi\)
−0.356748 + 0.934201i \(0.616115\pi\)
\(578\) −1.44806 −0.0602313
\(579\) 0 0
\(580\) −12.5986 −0.523129
\(581\) −74.8529 −3.10542
\(582\) 0 0
\(583\) −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 \(0.582489\pi\)
\(588\) 0 0
\(589\) −1.67449 −0.0689963
\(590\) −7.60454 −0.313074
\(591\) 0 0
\(592\) −1.58458 −0.0651259
\(593\) −11.0150 −0.452332 −0.226166 0.974089i \(-0.572619\pi\)
−0.226166 + 0.974089i \(0.572619\pi\)
\(594\) 0 0
\(595\) −21.5322 −0.882733
\(596\) 4.63756 0.189962
\(597\) 0 0
\(598\) −1.05587 −0.0431778
\(599\) 0.143692 0.00587109 0.00293555 0.999996i \(-0.499066\pi\)
0.00293555 + 0.999996i \(0.499066\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −21.3415 −0.868374
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 44.0261 1.78696 0.893481 0.449101i \(-0.148255\pi\)
0.893481 + 0.449101i \(0.148255\pi\)
\(608\) 5.86190 0.237732
\(609\) 0 0
\(610\) −9.56430 −0.387247
\(611\) −5.79604 −0.234483
\(612\) 0 0
\(613\) −7.70425 −0.311172 −0.155586 0.987822i \(-0.549727\pi\)
−0.155586 + 0.987822i \(0.549727\pi\)
\(614\) 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 \(0.649443\pi\)
\(618\) 0 0
\(619\) −19.9890 −0.803425 −0.401712 0.915766i \(-0.631585\pi\)
−0.401712 + 0.915766i \(0.631585\pi\)
\(620\) 2.38401 0.0957442
\(621\) 0 0
\(622\) −12.6656 −0.507844
\(623\) 35.9421 1.43999
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −8.04100 −0.321383
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 1.08394 0.0430489
\(635\) −15.2360 −0.604621
\(636\) 0 0
\(637\) 8.74592 0.346526
\(638\) 6.71761 0.265953
\(639\) 0 0
\(640\) −9.67331 −0.382371
\(641\) −0.415256 −0.0164016 −0.00820081 0.999966i \(-0.502610\pi\)
−0.00820081 + 0.999966i \(0.502610\pi\)
\(642\) 0 0
\(643\) 22.0655 0.870180 0.435090 0.900387i \(-0.356717\pi\)
0.435090 + 0.900387i \(0.356717\pi\)
\(644\) −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 \(0.782526\pi\)
\(648\) 0 0
\(649\) −10.0174 −0.393218
\(650\) −0.378930 −0.0148628
\(651\) 0 0
\(652\) −18.9197 −0.740952
\(653\) −34.3144 −1.34283 −0.671413 0.741084i \(-0.734312\pi\)
−0.671413 + 0.741084i \(0.734312\pi\)
\(654\) 0 0
\(655\) −5.16429 −0.201786
\(656\) −9.21695 −0.359861
\(657\) 0 0
\(658\) 43.6493 1.70163
\(659\) −12.6713 −0.493604 −0.246802 0.969066i \(-0.579380\pi\)
−0.246802 + 0.969066i \(0.579380\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −39.2873 −1.52464
\(665\) 4.95189 0.192026
\(666\) 0 0
\(667\) −24.6576 −0.954746
\(668\) 33.2798 1.28764
\(669\) 0 0
\(670\) 2.16350 0.0835834
\(671\) −12.5990 −0.486380
\(672\) 0 0
\(673\) −20.7136 −0.798449 −0.399225 0.916853i \(-0.630721\pi\)
−0.399225 + 0.916853i \(0.630721\pi\)
\(674\) 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 \(0.428464\pi\)
\(678\) 0 0
\(679\) 0.0961006 0.00368800
\(680\) −11.3014 −0.433389
\(681\) 0 0
\(682\) −1.27116 −0.0486753
\(683\) 21.3445 0.816727 0.408363 0.912820i \(-0.366100\pi\)
0.408363 + 0.912820i \(0.366100\pi\)
\(684\) 0 0
\(685\) −14.1861 −0.542024
\(686\) −39.5505 −1.51005
\(687\) 0 0
\(688\) 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
\(693\) 0 0
\(694\) 1.29661 0.0492187
\(695\) 1.20156 0.0455776
\(696\) 0 0
\(697\) −45.8337 −1.73607
\(698\) 2.72059 0.102976
\(699\) 0 0
\(700\) −7.05010 −0.266469
\(701\) 16.3009 0.615677 0.307839 0.951439i \(-0.400394\pi\)
0.307839 + 0.951439i \(0.400394\pi\)
\(702\) 0 0
\(703\) 1.81215 0.0683466
\(704\) 2.70111 0.101802
\(705\) 0 0
\(706\) −1.83339 −0.0690006
\(707\) −50.8985 −1.91423
\(708\) 0 0
\(709\) 43.5634 1.63606 0.818030 0.575176i \(-0.195066\pi\)
0.818030 + 0.575176i \(0.195066\pi\)
\(710\) 10.1231 0.379912
\(711\) 0 0
\(712\) 18.8646 0.706980
\(713\) 4.66591 0.174740
\(714\) 0 0
\(715\) −0.499163 −0.0186676
\(716\) −21.4904 −0.803134
\(717\) 0 0
\(718\) −14.1460 −0.527924
\(719\) 11.2024 0.417779 0.208889 0.977939i \(-0.433015\pi\)
0.208889 + 0.977939i \(0.433015\pi\)
\(720\) 0 0
\(721\) −25.9459 −0.966274
\(722\) −0.759131 −0.0282519
\(723\) 0 0
\(724\) 21.4632 0.797672
\(725\) −8.84908 −0.328647
\(726\) 0 0
\(727\) 40.2384 1.49236 0.746180 0.665744i \(-0.231886\pi\)
0.746180 + 0.665744i \(0.231886\pi\)
\(728\) 6.42433 0.238101
\(729\) 0 0
\(730\) −10.0304 −0.371243
\(731\) 11.5631 0.427677
\(732\) 0 0
\(733\) −12.7812 −0.472084 −0.236042 0.971743i \(-0.575850\pi\)
−0.236042 + 0.971743i \(0.575850\pi\)
\(734\) 0.921542 0.0340147
\(735\) 0 0
\(736\) −16.3340 −0.602078
\(737\) 2.84997 0.104980
\(738\) 0 0
\(739\) −2.61487 −0.0961895 −0.0480948 0.998843i \(-0.515315\pi\)
−0.0480948 + 0.998843i \(0.515315\pi\)
\(740\) −2.58000 −0.0948426
\(741\) 0 0
\(742\) 40.0502 1.47029
\(743\) 26.0025 0.953937 0.476969 0.878920i \(-0.341736\pi\)
0.476969 + 0.878920i \(0.341736\pi\)
\(744\) 0 0
\(745\) 3.25736 0.119340
\(746\) 23.0147 0.842628
\(747\) 0 0
\(748\) −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
\(753\) 0 0
\(754\) 3.35318 0.122116
\(755\) −14.9900 −0.545541
\(756\) 0 0
\(757\) −0.0174245 −0.000633303 0 −0.000316651 1.00000i \(-0.500101\pi\)
−0.000316651 1.00000i \(0.500101\pi\)
\(758\) −11.6736 −0.424006
\(759\) 0 0
\(760\) 2.59905 0.0942775
\(761\) −22.6593 −0.821398 −0.410699 0.911771i \(-0.634715\pi\)
−0.410699 + 0.911771i \(0.634715\pi\)
\(762\) 0 0
\(763\) −4.62735 −0.167521
\(764\) 5.13413 0.185746
\(765\) 0 0
\(766\) 1.23130 0.0444888
\(767\) −5.00032 −0.180551
\(768\) 0 0
\(769\) −43.4932 −1.56841 −0.784203 0.620504i \(-0.786928\pi\)
−0.784203 + 0.620504i \(0.786928\pi\)
\(770\) 3.75913 0.135470
\(771\) 0 0
\(772\) 34.8070 1.25273
\(773\) −51.7597 −1.86167 −0.930833 0.365444i \(-0.880917\pi\)
−0.930833 + 0.365444i \(0.880917\pi\)
\(774\) 0 0
\(775\) 1.67449 0.0601496
\(776\) 0.0504394 0.00181067
\(777\) 0 0
\(778\) −25.5492 −0.915984
\(779\) 10.5406 0.377658
\(780\) 0 0
\(781\) 13.3351 0.477166
\(782\) −9.19786 −0.328915
\(783\) 0 0
\(784\) 15.3209 0.547174
\(785\) −2.08129 −0.0742843
\(786\) 0 0
\(787\) 30.7685 1.09678 0.548389 0.836223i \(-0.315241\pi\)
0.548389 + 0.836223i \(0.315241\pi\)
\(788\) −21.9844 −0.783163
\(789\) 0 0
\(790\) −5.38220 −0.191490
\(791\) −84.2417 −2.99529
\(792\) 0 0
\(793\) −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 \(0.542038\pi\)
\(798\) 0 0
\(799\) −50.4902 −1.78621
\(800\) −5.86190 −0.207250
\(801\) 0 0
\(802\) 0.322398 0.0113843
\(803\) −13.2131 −0.466279
\(804\) 0 0
\(805\) −13.7982 −0.486324
\(806\) −0.634516 −0.0223499
\(807\) 0 0
\(808\) −26.7146 −0.939817
\(809\) 48.6194 1.70937 0.854683 0.519150i \(-0.173752\pi\)
0.854683 + 0.519150i \(0.173752\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 1.37566 0.0482169
\(815\) −13.2889 −0.465490
\(816\) 0 0
\(817\) −2.65924 −0.0930351
\(818\) −2.17665 −0.0761049
\(819\) 0 0
\(820\) −15.0069 −0.524065
\(821\) −34.4147 −1.20108 −0.600541 0.799594i \(-0.705048\pi\)
−0.600541 + 0.799594i \(0.705048\pi\)
\(822\) 0 0
\(823\) 10.2755 0.358183 0.179091 0.983832i \(-0.442684\pi\)
0.179091 + 0.983832i \(0.442684\pi\)
\(824\) −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 \(0.272128\pi\)
\(828\) 0 0
\(829\) −7.42020 −0.257714 −0.128857 0.991663i \(-0.541131\pi\)
−0.128857 + 0.991663i \(0.541131\pi\)
\(830\) −11.4750 −0.398305
\(831\) 0 0
\(832\) 1.34829 0.0467437
\(833\) 76.1870 2.63972
\(834\) 0 0
\(835\) 23.3753 0.808934
\(836\) 1.42372 0.0492404
\(837\) 0 0
\(838\) 19.8442 0.685508
\(839\) −11.4778 −0.396259 −0.198129 0.980176i \(-0.563487\pi\)
−0.198129 + 0.980176i \(0.563487\pi\)
\(840\) 0 0
\(841\) 49.3062 1.70021
\(842\) −11.3894 −0.392505
\(843\) 0 0
\(844\) −17.6902 −0.608920
\(845\) 12.7508 0.438642
\(846\) 0 0
\(847\) 4.95189 0.170149
\(848\) −9.31615 −0.319918
\(849\) 0 0
\(850\) −3.30091 −0.113220
\(851\) −5.04949 −0.173094
\(852\) 0 0
\(853\) −34.4829 −1.18067 −0.590336 0.807158i \(-0.701005\pi\)
−0.590336 + 0.807158i \(0.701005\pi\)
\(854\) 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 0
\(859\) −27.3349 −0.932653 −0.466327 0.884613i \(-0.654423\pi\)
−0.466327 + 0.884613i \(0.654423\pi\)
\(860\) 3.78602 0.129102
\(861\) 0 0
\(862\) −3.13433 −0.106756
\(863\) 7.77083 0.264522 0.132261 0.991215i \(-0.457776\pi\)
0.132261 + 0.991215i \(0.457776\pi\)
\(864\) 0 0
\(865\) 11.6401 0.395776
\(866\) −2.08987 −0.0710167
\(867\) 0 0
\(868\) −11.8054 −0.400700
\(869\) −7.08995 −0.240510
\(870\) 0 0
\(871\) 1.42260 0.0482029
\(872\) −2.42872 −0.0822468
\(873\) 0 0
\(874\) 2.11529 0.0715507
\(875\) −4.95189 −0.167404
\(876\) 0 0
\(877\) 33.4134 1.12829 0.564145 0.825676i \(-0.309206\pi\)
0.564145 + 0.825676i \(0.309206\pi\)
\(878\) −15.8656 −0.535437
\(879\) 0 0
\(880\) −0.874419 −0.0294767
\(881\) 38.4814 1.29647 0.648236 0.761439i \(-0.275507\pi\)
0.648236 + 0.761439i \(0.275507\pi\)
\(882\) 0 0
\(883\) 6.45348 0.217177 0.108589 0.994087i \(-0.465367\pi\)
0.108589 + 0.994087i \(0.465367\pi\)
\(884\) −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 \(0.165577\pi\)
\(888\) 0 0
\(889\) 75.4468 2.53040
\(890\) 5.50997 0.184695
\(891\) 0 0
\(892\) 13.4804 0.451359
\(893\) 11.6115 0.388565
\(894\) 0 0
\(895\) −15.0945 −0.504555
\(896\) 47.9011 1.60026
\(897\) 0 0
\(898\) −0.673029 −0.0224593
\(899\) −14.8177 −0.494199
\(900\) 0 0
\(901\) −46.3270 −1.54338
\(902\) 8.00173 0.266429
\(903\) 0 0
\(904\) −44.2152 −1.47057
\(905\) 15.0754 0.501123
\(906\) 0 0
\(907\) −51.3590 −1.70535 −0.852673 0.522445i \(-0.825020\pi\)
−0.852673 + 0.522445i \(0.825020\pi\)
\(908\) 1.72147 0.0571290
\(909\) 0 0
\(910\) 1.87642 0.0622026
\(911\) 30.5365 1.01172 0.505859 0.862616i \(-0.331175\pi\)
0.505859 + 0.862616i \(0.331175\pi\)
\(912\) 0 0
\(913\) −15.1160 −0.500267
\(914\) −8.46660 −0.280050
\(915\) 0 0
\(916\) 20.7225 0.684689
\(917\) 25.5730 0.844495
\(918\) 0 0
\(919\) −44.4425 −1.46602 −0.733011 0.680217i \(-0.761886\pi\)
−0.733011 + 0.680217i \(0.761886\pi\)
\(920\) −7.24216 −0.238767
\(921\) 0 0
\(922\) 31.9011 1.05061
\(923\) 6.65637 0.219097
\(924\) 0 0
\(925\) −1.81215 −0.0595832
\(926\) −12.3521 −0.405916
\(927\) 0 0
\(928\) 51.8724 1.70280
\(929\) 4.42046 0.145030 0.0725152 0.997367i \(-0.476897\pi\)
0.0725152 + 0.997367i \(0.476897\pi\)
\(930\) 0 0
\(931\) −17.5212 −0.574234
\(932\) −19.9723 −0.654215
\(933\) 0 0
\(934\) −23.8170 −0.779316
\(935\) −4.34828 −0.142204
\(936\) 0 0
\(937\) 57.9019 1.89157 0.945786 0.324789i \(-0.105293\pi\)
0.945786 + 0.324789i \(0.105293\pi\)
\(938\) −10.7134 −0.349806
\(939\) 0 0
\(940\) −16.5316 −0.539200
\(941\) 46.1004 1.50283 0.751415 0.659830i \(-0.229372\pi\)
0.751415 + 0.659830i \(0.229372\pi\)
\(942\) 0 0
\(943\) −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 \(0.661077\pi\)
\(948\) 0 0
\(949\) −6.59547 −0.214098
\(950\) 0.759131 0.0246295
\(951\) 0 0
\(952\) 55.9633 1.81378
\(953\) 9.41874 0.305103 0.152551 0.988296i \(-0.451251\pi\)
0.152551 + 0.988296i \(0.451251\pi\)
\(954\) 0 0
\(955\) 3.60614 0.116692
\(956\) −5.53775 −0.179104
\(957\) 0 0
\(958\) −6.80460 −0.219847
\(959\) 70.2481 2.26843
\(960\) 0 0
\(961\) −28.1961 −0.909551
\(962\) 0.686678 0.0221394
\(963\) 0 0
\(964\) 15.1214 0.487026
\(965\) 24.4479 0.787007
\(966\) 0 0
\(967\) 51.5972 1.65925 0.829627 0.558318i \(-0.188553\pi\)
0.829627 + 0.558318i \(0.188553\pi\)
\(968\) 2.59905 0.0835367
\(969\) 0 0
\(970\) 0.0147323 0.000473027 0
\(971\) 4.76616 0.152953 0.0764767 0.997071i \(-0.475633\pi\)
0.0764767 + 0.997071i \(0.475633\pi\)
\(972\) 0 0
\(973\) −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 \(0.492516\pi\)
\(978\) 0 0
\(979\) 7.25826 0.231975
\(980\) 24.9453 0.796847
\(981\) 0 0
\(982\) −0.0567807 −0.00181195
\(983\) −8.77114 −0.279756 −0.139878 0.990169i \(-0.544671\pi\)
−0.139878 + 0.990169i \(0.544671\pi\)
\(984\) 0 0
\(985\) −15.4415 −0.492008
\(986\) 29.2100 0.930237
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) −50.1283 −1.58997
\(995\) 19.3689 0.614035
\(996\) 0 0
\(997\) 13.8558 0.438818 0.219409 0.975633i \(-0.429587\pi\)
0.219409 + 0.975633i \(0.429587\pi\)
\(998\) −11.3410 −0.358994
\(999\) 0 0
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 9405.2.a.w.1.3 6
3.2 odd 2 1045.2.a.g.1.4 6
15.14 odd 2 5225.2.a.k.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1045.2.a.g.1.4 6 3.2 odd 2
5225.2.a.k.1.3 6 15.14 odd 2
9405.2.a.w.1.3 6 1.1 even 1 trivial