Properties

Label 6044.2.a.b.1.8
Level $6044$
Weight $2$
Character 6044.1
Self dual yes
Analytic conductor $48.262$
Analytic rank $0$
Dimension $63$
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] = [6044,2,Mod(1,6044)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6044, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6044.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6044 = 2^{2} \cdot 1511 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6044.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: \(48.2615829817\)
Analytic rank: \(0\)
Dimension: \(63\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Character \(\chi\) \(=\) 6044.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56682 q^{3} +0.554911 q^{5} -3.08439 q^{7} +3.58855 q^{9} +O(q^{10})\) \(q-2.56682 q^{3} +0.554911 q^{5} -3.08439 q^{7} +3.58855 q^{9} -2.32793 q^{11} -3.87679 q^{13} -1.42435 q^{15} -5.46032 q^{17} +0.936559 q^{19} +7.91705 q^{21} +3.57344 q^{23} -4.69207 q^{25} -1.51069 q^{27} -9.53204 q^{29} -10.6186 q^{31} +5.97537 q^{33} -1.71156 q^{35} -0.514996 q^{37} +9.95102 q^{39} -3.66609 q^{41} +8.72398 q^{43} +1.99132 q^{45} -4.43686 q^{47} +2.51344 q^{49} +14.0156 q^{51} -8.97080 q^{53} -1.29179 q^{55} -2.40398 q^{57} +10.1881 q^{59} -7.00928 q^{61} -11.0685 q^{63} -2.15128 q^{65} -12.7819 q^{67} -9.17235 q^{69} -3.98089 q^{71} -7.61585 q^{73} +12.0437 q^{75} +7.18023 q^{77} +16.9462 q^{79} -6.88797 q^{81} -17.1277 q^{83} -3.02999 q^{85} +24.4670 q^{87} +17.8878 q^{89} +11.9575 q^{91} +27.2559 q^{93} +0.519707 q^{95} -3.77798 q^{97} -8.35388 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 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 0
\(3\) −2.56682 −1.48195 −0.740976 0.671531i \(-0.765637\pi\)
−0.740976 + 0.671531i \(0.765637\pi\)
\(4\) 0 0
\(5\) 0.554911 0.248164 0.124082 0.992272i \(-0.460401\pi\)
0.124082 + 0.992272i \(0.460401\pi\)
\(6\) 0 0
\(7\) −3.08439 −1.16579 −0.582894 0.812548i \(-0.698080\pi\)
−0.582894 + 0.812548i \(0.698080\pi\)
\(8\) 0 0
\(9\) 3.58855 1.19618
\(10\) 0 0
\(11\) −2.32793 −0.701897 −0.350948 0.936395i \(-0.614141\pi\)
−0.350948 + 0.936395i \(0.614141\pi\)
\(12\) 0 0
\(13\) −3.87679 −1.07523 −0.537615 0.843191i \(-0.680674\pi\)
−0.537615 + 0.843191i \(0.680674\pi\)
\(14\) 0 0
\(15\) −1.42435 −0.367767
\(16\) 0 0
\(17\) −5.46032 −1.32432 −0.662161 0.749362i \(-0.730360\pi\)
−0.662161 + 0.749362i \(0.730360\pi\)
\(18\) 0 0
\(19\) 0.936559 0.214861 0.107431 0.994213i \(-0.465738\pi\)
0.107431 + 0.994213i \(0.465738\pi\)
\(20\) 0 0
\(21\) 7.91705 1.72764
\(22\) 0 0
\(23\) 3.57344 0.745113 0.372556 0.928010i \(-0.378481\pi\)
0.372556 + 0.928010i \(0.378481\pi\)
\(24\) 0 0
\(25\) −4.69207 −0.938415
\(26\) 0 0
\(27\) −1.51069 −0.290733
\(28\) 0 0
\(29\) −9.53204 −1.77006 −0.885028 0.465538i \(-0.845861\pi\)
−0.885028 + 0.465538i \(0.845861\pi\)
\(30\) 0 0
\(31\) −10.6186 −1.90715 −0.953575 0.301156i \(-0.902627\pi\)
−0.953575 + 0.301156i \(0.902627\pi\)
\(32\) 0 0
\(33\) 5.97537 1.04018
\(34\) 0 0
\(35\) −1.71156 −0.289307
\(36\) 0 0
\(37\) −0.514996 −0.0846647 −0.0423324 0.999104i \(-0.513479\pi\)
−0.0423324 + 0.999104i \(0.513479\pi\)
\(38\) 0 0
\(39\) 9.95102 1.59344
\(40\) 0 0
\(41\) −3.66609 −0.572547 −0.286274 0.958148i \(-0.592417\pi\)
−0.286274 + 0.958148i \(0.592417\pi\)
\(42\) 0 0
\(43\) 8.72398 1.33039 0.665197 0.746668i \(-0.268347\pi\)
0.665197 + 0.746668i \(0.268347\pi\)
\(44\) 0 0
\(45\) 1.99132 0.296849
\(46\) 0 0
\(47\) −4.43686 −0.647182 −0.323591 0.946197i \(-0.604890\pi\)
−0.323591 + 0.946197i \(0.604890\pi\)
\(48\) 0 0
\(49\) 2.51344 0.359063
\(50\) 0 0
\(51\) 14.0156 1.96258
\(52\) 0 0
\(53\) −8.97080 −1.23223 −0.616117 0.787655i \(-0.711295\pi\)
−0.616117 + 0.787655i \(0.711295\pi\)
\(54\) 0 0
\(55\) −1.29179 −0.174185
\(56\) 0 0
\(57\) −2.40398 −0.318414
\(58\) 0 0
\(59\) 10.1881 1.32637 0.663187 0.748453i \(-0.269203\pi\)
0.663187 + 0.748453i \(0.269203\pi\)
\(60\) 0 0
\(61\) −7.00928 −0.897447 −0.448723 0.893671i \(-0.648121\pi\)
−0.448723 + 0.893671i \(0.648121\pi\)
\(62\) 0 0
\(63\) −11.0685 −1.39450
\(64\) 0 0
\(65\) −2.15128 −0.266833
\(66\) 0 0
\(67\) −12.7819 −1.56156 −0.780782 0.624804i \(-0.785179\pi\)
−0.780782 + 0.624804i \(0.785179\pi\)
\(68\) 0 0
\(69\) −9.17235 −1.10422
\(70\) 0 0
\(71\) −3.98089 −0.472444 −0.236222 0.971699i \(-0.575909\pi\)
−0.236222 + 0.971699i \(0.575909\pi\)
\(72\) 0 0
\(73\) −7.61585 −0.891368 −0.445684 0.895190i \(-0.647039\pi\)
−0.445684 + 0.895190i \(0.647039\pi\)
\(74\) 0 0
\(75\) 12.0437 1.39069
\(76\) 0 0
\(77\) 7.18023 0.818263
\(78\) 0 0
\(79\) 16.9462 1.90660 0.953299 0.302030i \(-0.0976642\pi\)
0.953299 + 0.302030i \(0.0976642\pi\)
\(80\) 0 0
\(81\) −6.88797 −0.765330
\(82\) 0 0
\(83\) −17.1277 −1.88001 −0.940005 0.341160i \(-0.889180\pi\)
−0.940005 + 0.341160i \(0.889180\pi\)
\(84\) 0 0
\(85\) −3.02999 −0.328649
\(86\) 0 0
\(87\) 24.4670 2.62314
\(88\) 0 0
\(89\) 17.8878 1.89610 0.948052 0.318117i \(-0.103050\pi\)
0.948052 + 0.318117i \(0.103050\pi\)
\(90\) 0 0
\(91\) 11.9575 1.25349
\(92\) 0 0
\(93\) 27.2559 2.82630
\(94\) 0 0
\(95\) 0.519707 0.0533208
\(96\) 0 0
\(97\) −3.77798 −0.383596 −0.191798 0.981434i \(-0.561432\pi\)
−0.191798 + 0.981434i \(0.561432\pi\)
\(98\) 0 0
\(99\) −8.35388 −0.839597
\(100\) 0 0
\(101\) −7.82851 −0.778966 −0.389483 0.921034i \(-0.627346\pi\)
−0.389483 + 0.921034i \(0.627346\pi\)
\(102\) 0 0
\(103\) 3.39080 0.334105 0.167052 0.985948i \(-0.446575\pi\)
0.167052 + 0.985948i \(0.446575\pi\)
\(104\) 0 0
\(105\) 4.39326 0.428738
\(106\) 0 0
\(107\) −4.71638 −0.455950 −0.227975 0.973667i \(-0.573210\pi\)
−0.227975 + 0.973667i \(0.573210\pi\)
\(108\) 0 0
\(109\) −7.54113 −0.722309 −0.361155 0.932506i \(-0.617617\pi\)
−0.361155 + 0.932506i \(0.617617\pi\)
\(110\) 0 0
\(111\) 1.32190 0.125469
\(112\) 0 0
\(113\) −13.7860 −1.29688 −0.648440 0.761266i \(-0.724578\pi\)
−0.648440 + 0.761266i \(0.724578\pi\)
\(114\) 0 0
\(115\) 1.98294 0.184910
\(116\) 0 0
\(117\) −13.9121 −1.28617
\(118\) 0 0
\(119\) 16.8417 1.54388
\(120\) 0 0
\(121\) −5.58075 −0.507341
\(122\) 0 0
\(123\) 9.41018 0.848488
\(124\) 0 0
\(125\) −5.37824 −0.481044
\(126\) 0 0
\(127\) −6.16426 −0.546990 −0.273495 0.961873i \(-0.588180\pi\)
−0.273495 + 0.961873i \(0.588180\pi\)
\(128\) 0 0
\(129\) −22.3929 −1.97158
\(130\) 0 0
\(131\) 2.03971 0.178210 0.0891052 0.996022i \(-0.471599\pi\)
0.0891052 + 0.996022i \(0.471599\pi\)
\(132\) 0 0
\(133\) −2.88871 −0.250483
\(134\) 0 0
\(135\) −0.838300 −0.0721494
\(136\) 0 0
\(137\) 2.60734 0.222760 0.111380 0.993778i \(-0.464473\pi\)
0.111380 + 0.993778i \(0.464473\pi\)
\(138\) 0 0
\(139\) −22.3646 −1.89694 −0.948472 0.316862i \(-0.897371\pi\)
−0.948472 + 0.316862i \(0.897371\pi\)
\(140\) 0 0
\(141\) 11.3886 0.959093
\(142\) 0 0
\(143\) 9.02490 0.754700
\(144\) 0 0
\(145\) −5.28944 −0.439264
\(146\) 0 0
\(147\) −6.45154 −0.532114
\(148\) 0 0
\(149\) 7.85018 0.643112 0.321556 0.946891i \(-0.395794\pi\)
0.321556 + 0.946891i \(0.395794\pi\)
\(150\) 0 0
\(151\) 16.2576 1.32302 0.661511 0.749935i \(-0.269915\pi\)
0.661511 + 0.749935i \(0.269915\pi\)
\(152\) 0 0
\(153\) −19.5946 −1.58413
\(154\) 0 0
\(155\) −5.89236 −0.473285
\(156\) 0 0
\(157\) −1.14315 −0.0912330 −0.0456165 0.998959i \(-0.514525\pi\)
−0.0456165 + 0.998959i \(0.514525\pi\)
\(158\) 0 0
\(159\) 23.0264 1.82611
\(160\) 0 0
\(161\) −11.0219 −0.868644
\(162\) 0 0
\(163\) 1.25700 0.0984561 0.0492281 0.998788i \(-0.484324\pi\)
0.0492281 + 0.998788i \(0.484324\pi\)
\(164\) 0 0
\(165\) 3.31580 0.258134
\(166\) 0 0
\(167\) 2.71447 0.210052 0.105026 0.994469i \(-0.466507\pi\)
0.105026 + 0.994469i \(0.466507\pi\)
\(168\) 0 0
\(169\) 2.02953 0.156118
\(170\) 0 0
\(171\) 3.36089 0.257014
\(172\) 0 0
\(173\) −7.22921 −0.549627 −0.274813 0.961498i \(-0.588616\pi\)
−0.274813 + 0.961498i \(0.588616\pi\)
\(174\) 0 0
\(175\) 14.4722 1.09399
\(176\) 0 0
\(177\) −26.1509 −1.96562
\(178\) 0 0
\(179\) 12.1822 0.910537 0.455268 0.890354i \(-0.349543\pi\)
0.455268 + 0.890354i \(0.349543\pi\)
\(180\) 0 0
\(181\) 18.1105 1.34614 0.673072 0.739577i \(-0.264975\pi\)
0.673072 + 0.739577i \(0.264975\pi\)
\(182\) 0 0
\(183\) 17.9915 1.32997
\(184\) 0 0
\(185\) −0.285777 −0.0210107
\(186\) 0 0
\(187\) 12.7112 0.929537
\(188\) 0 0
\(189\) 4.65956 0.338933
\(190\) 0 0
\(191\) −14.7887 −1.07007 −0.535035 0.844830i \(-0.679702\pi\)
−0.535035 + 0.844830i \(0.679702\pi\)
\(192\) 0 0
\(193\) −0.276911 −0.0199325 −0.00996626 0.999950i \(-0.503172\pi\)
−0.00996626 + 0.999950i \(0.503172\pi\)
\(194\) 0 0
\(195\) 5.52193 0.395434
\(196\) 0 0
\(197\) 13.9680 0.995180 0.497590 0.867412i \(-0.334218\pi\)
0.497590 + 0.867412i \(0.334218\pi\)
\(198\) 0 0
\(199\) 9.74910 0.691095 0.345548 0.938401i \(-0.387693\pi\)
0.345548 + 0.938401i \(0.387693\pi\)
\(200\) 0 0
\(201\) 32.8089 2.31416
\(202\) 0 0
\(203\) 29.4005 2.06351
\(204\) 0 0
\(205\) −2.03435 −0.142085
\(206\) 0 0
\(207\) 12.8234 0.891291
\(208\) 0 0
\(209\) −2.18024 −0.150811
\(210\) 0 0
\(211\) −23.9954 −1.65191 −0.825957 0.563733i \(-0.809365\pi\)
−0.825957 + 0.563733i \(0.809365\pi\)
\(212\) 0 0
\(213\) 10.2182 0.700140
\(214\) 0 0
\(215\) 4.84103 0.330156
\(216\) 0 0
\(217\) 32.7517 2.22333
\(218\) 0 0
\(219\) 19.5485 1.32096
\(220\) 0 0
\(221\) 21.1685 1.42395
\(222\) 0 0
\(223\) −27.2107 −1.82217 −0.911083 0.412224i \(-0.864752\pi\)
−0.911083 + 0.412224i \(0.864752\pi\)
\(224\) 0 0
\(225\) −16.8377 −1.12252
\(226\) 0 0
\(227\) 1.67544 0.111203 0.0556014 0.998453i \(-0.482292\pi\)
0.0556014 + 0.998453i \(0.482292\pi\)
\(228\) 0 0
\(229\) −3.81367 −0.252015 −0.126007 0.992029i \(-0.540216\pi\)
−0.126007 + 0.992029i \(0.540216\pi\)
\(230\) 0 0
\(231\) −18.4303 −1.21263
\(232\) 0 0
\(233\) −2.08496 −0.136590 −0.0682951 0.997665i \(-0.521756\pi\)
−0.0682951 + 0.997665i \(0.521756\pi\)
\(234\) 0 0
\(235\) −2.46206 −0.160607
\(236\) 0 0
\(237\) −43.4978 −2.82549
\(238\) 0 0
\(239\) 7.41693 0.479761 0.239881 0.970802i \(-0.422892\pi\)
0.239881 + 0.970802i \(0.422892\pi\)
\(240\) 0 0
\(241\) −15.8223 −1.01920 −0.509601 0.860411i \(-0.670207\pi\)
−0.509601 + 0.860411i \(0.670207\pi\)
\(242\) 0 0
\(243\) 22.2122 1.42492
\(244\) 0 0
\(245\) 1.39474 0.0891065
\(246\) 0 0
\(247\) −3.63085 −0.231025
\(248\) 0 0
\(249\) 43.9637 2.78609
\(250\) 0 0
\(251\) −0.414111 −0.0261385 −0.0130692 0.999915i \(-0.504160\pi\)
−0.0130692 + 0.999915i \(0.504160\pi\)
\(252\) 0 0
\(253\) −8.31870 −0.522992
\(254\) 0 0
\(255\) 7.77743 0.487042
\(256\) 0 0
\(257\) −12.0037 −0.748771 −0.374385 0.927273i \(-0.622146\pi\)
−0.374385 + 0.927273i \(0.622146\pi\)
\(258\) 0 0
\(259\) 1.58845 0.0987012
\(260\) 0 0
\(261\) −34.2062 −2.11731
\(262\) 0 0
\(263\) −10.9766 −0.676846 −0.338423 0.940994i \(-0.609894\pi\)
−0.338423 + 0.940994i \(0.609894\pi\)
\(264\) 0 0
\(265\) −4.97800 −0.305796
\(266\) 0 0
\(267\) −45.9147 −2.80993
\(268\) 0 0
\(269\) −23.3354 −1.42278 −0.711391 0.702797i \(-0.751934\pi\)
−0.711391 + 0.702797i \(0.751934\pi\)
\(270\) 0 0
\(271\) 21.0795 1.28049 0.640243 0.768172i \(-0.278834\pi\)
0.640243 + 0.768172i \(0.278834\pi\)
\(272\) 0 0
\(273\) −30.6928 −1.85761
\(274\) 0 0
\(275\) 10.9228 0.658670
\(276\) 0 0
\(277\) 6.84605 0.411339 0.205670 0.978621i \(-0.434063\pi\)
0.205670 + 0.978621i \(0.434063\pi\)
\(278\) 0 0
\(279\) −38.1052 −2.28130
\(280\) 0 0
\(281\) 14.6724 0.875280 0.437640 0.899150i \(-0.355814\pi\)
0.437640 + 0.899150i \(0.355814\pi\)
\(282\) 0 0
\(283\) −0.112425 −0.00668295 −0.00334147 0.999994i \(-0.501064\pi\)
−0.00334147 + 0.999994i \(0.501064\pi\)
\(284\) 0 0
\(285\) −1.33399 −0.0790189
\(286\) 0 0
\(287\) 11.3076 0.667469
\(288\) 0 0
\(289\) 12.8151 0.753828
\(290\) 0 0
\(291\) 9.69739 0.568471
\(292\) 0 0
\(293\) 19.7849 1.15585 0.577924 0.816091i \(-0.303863\pi\)
0.577924 + 0.816091i \(0.303863\pi\)
\(294\) 0 0
\(295\) 5.65348 0.329158
\(296\) 0 0
\(297\) 3.51678 0.204065
\(298\) 0 0
\(299\) −13.8535 −0.801167
\(300\) 0 0
\(301\) −26.9081 −1.55096
\(302\) 0 0
\(303\) 20.0943 1.15439
\(304\) 0 0
\(305\) −3.88953 −0.222714
\(306\) 0 0
\(307\) 14.1127 0.805455 0.402728 0.915320i \(-0.368062\pi\)
0.402728 + 0.915320i \(0.368062\pi\)
\(308\) 0 0
\(309\) −8.70355 −0.495128
\(310\) 0 0
\(311\) −12.3561 −0.700652 −0.350326 0.936628i \(-0.613929\pi\)
−0.350326 + 0.936628i \(0.613929\pi\)
\(312\) 0 0
\(313\) −28.8084 −1.62835 −0.814173 0.580622i \(-0.802809\pi\)
−0.814173 + 0.580622i \(0.802809\pi\)
\(314\) 0 0
\(315\) −6.14202 −0.346063
\(316\) 0 0
\(317\) 13.6046 0.764113 0.382056 0.924139i \(-0.375216\pi\)
0.382056 + 0.924139i \(0.375216\pi\)
\(318\) 0 0
\(319\) 22.1899 1.24240
\(320\) 0 0
\(321\) 12.1061 0.675696
\(322\) 0 0
\(323\) −5.11391 −0.284546
\(324\) 0 0
\(325\) 18.1902 1.00901
\(326\) 0 0
\(327\) 19.3567 1.07043
\(328\) 0 0
\(329\) 13.6850 0.754477
\(330\) 0 0
\(331\) 23.0221 1.26541 0.632705 0.774393i \(-0.281945\pi\)
0.632705 + 0.774393i \(0.281945\pi\)
\(332\) 0 0
\(333\) −1.84809 −0.101274
\(334\) 0 0
\(335\) −7.09284 −0.387523
\(336\) 0 0
\(337\) 9.91881 0.540312 0.270156 0.962817i \(-0.412925\pi\)
0.270156 + 0.962817i \(0.412925\pi\)
\(338\) 0 0
\(339\) 35.3862 1.92191
\(340\) 0 0
\(341\) 24.7192 1.33862
\(342\) 0 0
\(343\) 13.8383 0.747197
\(344\) 0 0
\(345\) −5.08984 −0.274028
\(346\) 0 0
\(347\) 23.4888 1.26094 0.630472 0.776212i \(-0.282861\pi\)
0.630472 + 0.776212i \(0.282861\pi\)
\(348\) 0 0
\(349\) 29.5814 1.58346 0.791728 0.610873i \(-0.209182\pi\)
0.791728 + 0.610873i \(0.209182\pi\)
\(350\) 0 0
\(351\) 5.85664 0.312605
\(352\) 0 0
\(353\) −0.523893 −0.0278840 −0.0139420 0.999903i \(-0.504438\pi\)
−0.0139420 + 0.999903i \(0.504438\pi\)
\(354\) 0 0
\(355\) −2.20904 −0.117244
\(356\) 0 0
\(357\) −43.2296 −2.28795
\(358\) 0 0
\(359\) −16.6871 −0.880713 −0.440357 0.897823i \(-0.645148\pi\)
−0.440357 + 0.897823i \(0.645148\pi\)
\(360\) 0 0
\(361\) −18.1229 −0.953835
\(362\) 0 0
\(363\) 14.3248 0.751855
\(364\) 0 0
\(365\) −4.22612 −0.221205
\(366\) 0 0
\(367\) −22.3795 −1.16820 −0.584100 0.811682i \(-0.698552\pi\)
−0.584100 + 0.811682i \(0.698552\pi\)
\(368\) 0 0
\(369\) −13.1559 −0.684871
\(370\) 0 0
\(371\) 27.6694 1.43652
\(372\) 0 0
\(373\) −8.76809 −0.453994 −0.226997 0.973895i \(-0.572891\pi\)
−0.226997 + 0.973895i \(0.572891\pi\)
\(374\) 0 0
\(375\) 13.8050 0.712885
\(376\) 0 0
\(377\) 36.9538 1.90322
\(378\) 0 0
\(379\) 22.8397 1.17319 0.586597 0.809879i \(-0.300467\pi\)
0.586597 + 0.809879i \(0.300467\pi\)
\(380\) 0 0
\(381\) 15.8225 0.810613
\(382\) 0 0
\(383\) 21.1904 1.08278 0.541389 0.840772i \(-0.317899\pi\)
0.541389 + 0.840772i \(0.317899\pi\)
\(384\) 0 0
\(385\) 3.98439 0.203063
\(386\) 0 0
\(387\) 31.3064 1.59139
\(388\) 0 0
\(389\) 28.7870 1.45956 0.729778 0.683684i \(-0.239623\pi\)
0.729778 + 0.683684i \(0.239623\pi\)
\(390\) 0 0
\(391\) −19.5121 −0.986769
\(392\) 0 0
\(393\) −5.23557 −0.264099
\(394\) 0 0
\(395\) 9.40364 0.473148
\(396\) 0 0
\(397\) −3.73363 −0.187385 −0.0936927 0.995601i \(-0.529867\pi\)
−0.0936927 + 0.995601i \(0.529867\pi\)
\(398\) 0 0
\(399\) 7.41479 0.371204
\(400\) 0 0
\(401\) −22.4215 −1.11968 −0.559839 0.828602i \(-0.689137\pi\)
−0.559839 + 0.828602i \(0.689137\pi\)
\(402\) 0 0
\(403\) 41.1660 2.05062
\(404\) 0 0
\(405\) −3.82221 −0.189927
\(406\) 0 0
\(407\) 1.19887 0.0594259
\(408\) 0 0
\(409\) 3.57884 0.176962 0.0884812 0.996078i \(-0.471799\pi\)
0.0884812 + 0.996078i \(0.471799\pi\)
\(410\) 0 0
\(411\) −6.69257 −0.330120
\(412\) 0 0
\(413\) −31.4240 −1.54627
\(414\) 0 0
\(415\) −9.50436 −0.466551
\(416\) 0 0
\(417\) 57.4059 2.81118
\(418\) 0 0
\(419\) −1.09920 −0.0536996 −0.0268498 0.999639i \(-0.508548\pi\)
−0.0268498 + 0.999639i \(0.508548\pi\)
\(420\) 0 0
\(421\) 9.99246 0.487002 0.243501 0.969901i \(-0.421704\pi\)
0.243501 + 0.969901i \(0.421704\pi\)
\(422\) 0 0
\(423\) −15.9219 −0.774148
\(424\) 0 0
\(425\) 25.6202 1.24276
\(426\) 0 0
\(427\) 21.6193 1.04623
\(428\) 0 0
\(429\) −23.1653 −1.11843
\(430\) 0 0
\(431\) 15.3302 0.738430 0.369215 0.929344i \(-0.379627\pi\)
0.369215 + 0.929344i \(0.379627\pi\)
\(432\) 0 0
\(433\) −22.9759 −1.10415 −0.552076 0.833793i \(-0.686164\pi\)
−0.552076 + 0.833793i \(0.686164\pi\)
\(434\) 0 0
\(435\) 13.5770 0.650968
\(436\) 0 0
\(437\) 3.34673 0.160096
\(438\) 0 0
\(439\) 16.3190 0.778862 0.389431 0.921056i \(-0.372672\pi\)
0.389431 + 0.921056i \(0.372672\pi\)
\(440\) 0 0
\(441\) 9.01960 0.429505
\(442\) 0 0
\(443\) 15.9969 0.760034 0.380017 0.924979i \(-0.375918\pi\)
0.380017 + 0.924979i \(0.375918\pi\)
\(444\) 0 0
\(445\) 9.92614 0.470544
\(446\) 0 0
\(447\) −20.1500 −0.953061
\(448\) 0 0
\(449\) 15.8861 0.749711 0.374856 0.927083i \(-0.377692\pi\)
0.374856 + 0.927083i \(0.377692\pi\)
\(450\) 0 0
\(451\) 8.53440 0.401869
\(452\) 0 0
\(453\) −41.7302 −1.96066
\(454\) 0 0
\(455\) 6.63537 0.311071
\(456\) 0 0
\(457\) 15.0891 0.705836 0.352918 0.935654i \(-0.385189\pi\)
0.352918 + 0.935654i \(0.385189\pi\)
\(458\) 0 0
\(459\) 8.24886 0.385024
\(460\) 0 0
\(461\) 10.7375 0.500095 0.250048 0.968234i \(-0.419554\pi\)
0.250048 + 0.968234i \(0.419554\pi\)
\(462\) 0 0
\(463\) 1.24530 0.0578738 0.0289369 0.999581i \(-0.490788\pi\)
0.0289369 + 0.999581i \(0.490788\pi\)
\(464\) 0 0
\(465\) 15.1246 0.701386
\(466\) 0 0
\(467\) −32.4050 −1.49952 −0.749762 0.661708i \(-0.769832\pi\)
−0.749762 + 0.661708i \(0.769832\pi\)
\(468\) 0 0
\(469\) 39.4245 1.82045
\(470\) 0 0
\(471\) 2.93425 0.135203
\(472\) 0 0
\(473\) −20.3088 −0.933800
\(474\) 0 0
\(475\) −4.39441 −0.201629
\(476\) 0 0
\(477\) −32.1921 −1.47398
\(478\) 0 0
\(479\) 19.6726 0.898864 0.449432 0.893315i \(-0.351626\pi\)
0.449432 + 0.893315i \(0.351626\pi\)
\(480\) 0 0
\(481\) 1.99653 0.0910340
\(482\) 0 0
\(483\) 28.2911 1.28729
\(484\) 0 0
\(485\) −2.09644 −0.0951946
\(486\) 0 0
\(487\) −30.2470 −1.37062 −0.685311 0.728251i \(-0.740334\pi\)
−0.685311 + 0.728251i \(0.740334\pi\)
\(488\) 0 0
\(489\) −3.22650 −0.145907
\(490\) 0 0
\(491\) −10.0640 −0.454180 −0.227090 0.973874i \(-0.572921\pi\)
−0.227090 + 0.973874i \(0.572921\pi\)
\(492\) 0 0
\(493\) 52.0480 2.34412
\(494\) 0 0
\(495\) −4.63566 −0.208357
\(496\) 0 0
\(497\) 12.2786 0.550770
\(498\) 0 0
\(499\) 13.3205 0.596309 0.298155 0.954518i \(-0.403629\pi\)
0.298155 + 0.954518i \(0.403629\pi\)
\(500\) 0 0
\(501\) −6.96754 −0.311287
\(502\) 0 0
\(503\) −24.1857 −1.07839 −0.539193 0.842182i \(-0.681271\pi\)
−0.539193 + 0.842182i \(0.681271\pi\)
\(504\) 0 0
\(505\) −4.34413 −0.193311
\(506\) 0 0
\(507\) −5.20944 −0.231360
\(508\) 0 0
\(509\) 26.3150 1.16639 0.583196 0.812332i \(-0.301802\pi\)
0.583196 + 0.812332i \(0.301802\pi\)
\(510\) 0 0
\(511\) 23.4902 1.03915
\(512\) 0 0
\(513\) −1.41485 −0.0624673
\(514\) 0 0
\(515\) 1.88159 0.0829128
\(516\) 0 0
\(517\) 10.3287 0.454255
\(518\) 0 0
\(519\) 18.5561 0.814520
\(520\) 0 0
\(521\) −10.9083 −0.477901 −0.238950 0.971032i \(-0.576803\pi\)
−0.238950 + 0.971032i \(0.576803\pi\)
\(522\) 0 0
\(523\) 15.4765 0.676740 0.338370 0.941013i \(-0.390124\pi\)
0.338370 + 0.941013i \(0.390124\pi\)
\(524\) 0 0
\(525\) −37.1474 −1.62125
\(526\) 0 0
\(527\) 57.9807 2.52568
\(528\) 0 0
\(529\) −10.2306 −0.444807
\(530\) 0 0
\(531\) 36.5604 1.58659
\(532\) 0 0
\(533\) 14.2127 0.615620
\(534\) 0 0
\(535\) −2.61717 −0.113150
\(536\) 0 0
\(537\) −31.2693 −1.34937
\(538\) 0 0
\(539\) −5.85111 −0.252025
\(540\) 0 0
\(541\) 35.1806 1.51253 0.756267 0.654264i \(-0.227021\pi\)
0.756267 + 0.654264i \(0.227021\pi\)
\(542\) 0 0
\(543\) −46.4864 −1.99492
\(544\) 0 0
\(545\) −4.18466 −0.179251
\(546\) 0 0
\(547\) 18.4408 0.788471 0.394236 0.919009i \(-0.371009\pi\)
0.394236 + 0.919009i \(0.371009\pi\)
\(548\) 0 0
\(549\) −25.1531 −1.07351
\(550\) 0 0
\(551\) −8.92733 −0.380317
\(552\) 0 0
\(553\) −52.2687 −2.22269
\(554\) 0 0
\(555\) 0.733536 0.0311369
\(556\) 0 0
\(557\) −2.35913 −0.0999596 −0.0499798 0.998750i \(-0.515916\pi\)
−0.0499798 + 0.998750i \(0.515916\pi\)
\(558\) 0 0
\(559\) −33.8211 −1.43048
\(560\) 0 0
\(561\) −32.6274 −1.37753
\(562\) 0 0
\(563\) −28.0266 −1.18118 −0.590590 0.806972i \(-0.701105\pi\)
−0.590590 + 0.806972i \(0.701105\pi\)
\(564\) 0 0
\(565\) −7.65001 −0.321839
\(566\) 0 0
\(567\) 21.2452 0.892213
\(568\) 0 0
\(569\) 20.7194 0.868601 0.434300 0.900768i \(-0.356996\pi\)
0.434300 + 0.900768i \(0.356996\pi\)
\(570\) 0 0
\(571\) 10.7555 0.450105 0.225053 0.974347i \(-0.427745\pi\)
0.225053 + 0.974347i \(0.427745\pi\)
\(572\) 0 0
\(573\) 37.9598 1.58579
\(574\) 0 0
\(575\) −16.7668 −0.699225
\(576\) 0 0
\(577\) −10.1338 −0.421874 −0.210937 0.977500i \(-0.567651\pi\)
−0.210937 + 0.977500i \(0.567651\pi\)
\(578\) 0 0
\(579\) 0.710781 0.0295390
\(580\) 0 0
\(581\) 52.8285 2.19169
\(582\) 0 0
\(583\) 20.8834 0.864901
\(584\) 0 0
\(585\) −7.71996 −0.319181
\(586\) 0 0
\(587\) −35.4466 −1.46304 −0.731520 0.681820i \(-0.761189\pi\)
−0.731520 + 0.681820i \(0.761189\pi\)
\(588\) 0 0
\(589\) −9.94491 −0.409773
\(590\) 0 0
\(591\) −35.8533 −1.47481
\(592\) 0 0
\(593\) 12.7828 0.524928 0.262464 0.964942i \(-0.415465\pi\)
0.262464 + 0.964942i \(0.415465\pi\)
\(594\) 0 0
\(595\) 9.34566 0.383135
\(596\) 0 0
\(597\) −25.0242 −1.02417
\(598\) 0 0
\(599\) −47.2185 −1.92930 −0.964649 0.263539i \(-0.915110\pi\)
−0.964649 + 0.263539i \(0.915110\pi\)
\(600\) 0 0
\(601\) −11.7888 −0.480874 −0.240437 0.970665i \(-0.577291\pi\)
−0.240437 + 0.970665i \(0.577291\pi\)
\(602\) 0 0
\(603\) −45.8686 −1.86791
\(604\) 0 0
\(605\) −3.09682 −0.125904
\(606\) 0 0
\(607\) 0.288794 0.0117218 0.00586090 0.999983i \(-0.498134\pi\)
0.00586090 + 0.999983i \(0.498134\pi\)
\(608\) 0 0
\(609\) −75.4657 −3.05803
\(610\) 0 0
\(611\) 17.2008 0.695869
\(612\) 0 0
\(613\) −33.5326 −1.35437 −0.677184 0.735814i \(-0.736800\pi\)
−0.677184 + 0.735814i \(0.736800\pi\)
\(614\) 0 0
\(615\) 5.22181 0.210564
\(616\) 0 0
\(617\) −8.99337 −0.362060 −0.181030 0.983478i \(-0.557943\pi\)
−0.181030 + 0.983478i \(0.557943\pi\)
\(618\) 0 0
\(619\) −8.14986 −0.327571 −0.163785 0.986496i \(-0.552370\pi\)
−0.163785 + 0.986496i \(0.552370\pi\)
\(620\) 0 0
\(621\) −5.39836 −0.216629
\(622\) 0 0
\(623\) −55.1729 −2.21046
\(624\) 0 0
\(625\) 20.4759 0.819037
\(626\) 0 0
\(627\) 5.59628 0.223494
\(628\) 0 0
\(629\) 2.81204 0.112123
\(630\) 0 0
\(631\) 19.8085 0.788562 0.394281 0.918990i \(-0.370994\pi\)
0.394281 + 0.918990i \(0.370994\pi\)
\(632\) 0 0
\(633\) 61.5919 2.44806
\(634\) 0 0
\(635\) −3.42062 −0.135743
\(636\) 0 0
\(637\) −9.74410 −0.386075
\(638\) 0 0
\(639\) −14.2856 −0.565130
\(640\) 0 0
\(641\) −1.17319 −0.0463381 −0.0231690 0.999732i \(-0.507376\pi\)
−0.0231690 + 0.999732i \(0.507376\pi\)
\(642\) 0 0
\(643\) 42.6757 1.68297 0.841483 0.540284i \(-0.181683\pi\)
0.841483 + 0.540284i \(0.181683\pi\)
\(644\) 0 0
\(645\) −12.4260 −0.489275
\(646\) 0 0
\(647\) 5.91727 0.232632 0.116316 0.993212i \(-0.462891\pi\)
0.116316 + 0.993212i \(0.462891\pi\)
\(648\) 0 0
\(649\) −23.7171 −0.930978
\(650\) 0 0
\(651\) −84.0677 −3.29487
\(652\) 0 0
\(653\) −18.3871 −0.719542 −0.359771 0.933041i \(-0.617145\pi\)
−0.359771 + 0.933041i \(0.617145\pi\)
\(654\) 0 0
\(655\) 1.13186 0.0442254
\(656\) 0 0
\(657\) −27.3298 −1.06624
\(658\) 0 0
\(659\) −6.79925 −0.264861 −0.132431 0.991192i \(-0.542278\pi\)
−0.132431 + 0.991192i \(0.542278\pi\)
\(660\) 0 0
\(661\) 20.5054 0.797568 0.398784 0.917045i \(-0.369432\pi\)
0.398784 + 0.917045i \(0.369432\pi\)
\(662\) 0 0
\(663\) −54.3357 −2.11022
\(664\) 0 0
\(665\) −1.60298 −0.0621608
\(666\) 0 0
\(667\) −34.0621 −1.31889
\(668\) 0 0
\(669\) 69.8449 2.70036
\(670\) 0 0
\(671\) 16.3171 0.629915
\(672\) 0 0
\(673\) −12.7833 −0.492760 −0.246380 0.969173i \(-0.579241\pi\)
−0.246380 + 0.969173i \(0.579241\pi\)
\(674\) 0 0
\(675\) 7.08828 0.272828
\(676\) 0 0
\(677\) 3.63706 0.139784 0.0698919 0.997555i \(-0.477735\pi\)
0.0698919 + 0.997555i \(0.477735\pi\)
\(678\) 0 0
\(679\) 11.6528 0.447192
\(680\) 0 0
\(681\) −4.30055 −0.164797
\(682\) 0 0
\(683\) 20.3533 0.778796 0.389398 0.921070i \(-0.372683\pi\)
0.389398 + 0.921070i \(0.372683\pi\)
\(684\) 0 0
\(685\) 1.44684 0.0552810
\(686\) 0 0
\(687\) 9.78900 0.373474
\(688\) 0 0
\(689\) 34.7780 1.32493
\(690\) 0 0
\(691\) −48.6620 −1.85119 −0.925595 0.378516i \(-0.876435\pi\)
−0.925595 + 0.378516i \(0.876435\pi\)
\(692\) 0 0
\(693\) 25.7666 0.978792
\(694\) 0 0
\(695\) −12.4104 −0.470753
\(696\) 0 0
\(697\) 20.0180 0.758237
\(698\) 0 0
\(699\) 5.35171 0.202420
\(700\) 0 0
\(701\) 20.4517 0.772449 0.386225 0.922405i \(-0.373779\pi\)
0.386225 + 0.922405i \(0.373779\pi\)
\(702\) 0 0
\(703\) −0.482324 −0.0181912
\(704\) 0 0
\(705\) 6.31966 0.238012
\(706\) 0 0
\(707\) 24.1461 0.908109
\(708\) 0 0
\(709\) 6.27477 0.235654 0.117827 0.993034i \(-0.462407\pi\)
0.117827 + 0.993034i \(0.462407\pi\)
\(710\) 0 0
\(711\) 60.8123 2.28064
\(712\) 0 0
\(713\) −37.9447 −1.42104
\(714\) 0 0
\(715\) 5.00802 0.187289
\(716\) 0 0
\(717\) −19.0379 −0.710983
\(718\) 0 0
\(719\) −3.68371 −0.137379 −0.0686896 0.997638i \(-0.521882\pi\)
−0.0686896 + 0.997638i \(0.521882\pi\)
\(720\) 0 0
\(721\) −10.4585 −0.389496
\(722\) 0 0
\(723\) 40.6128 1.51041
\(724\) 0 0
\(725\) 44.7251 1.66105
\(726\) 0 0
\(727\) 43.2670 1.60468 0.802342 0.596865i \(-0.203587\pi\)
0.802342 + 0.596865i \(0.203587\pi\)
\(728\) 0 0
\(729\) −36.3508 −1.34633
\(730\) 0 0
\(731\) −47.6357 −1.76187
\(732\) 0 0
\(733\) −48.0408 −1.77443 −0.887214 0.461358i \(-0.847362\pi\)
−0.887214 + 0.461358i \(0.847362\pi\)
\(734\) 0 0
\(735\) −3.58003 −0.132052
\(736\) 0 0
\(737\) 29.7555 1.09606
\(738\) 0 0
\(739\) −50.7683 −1.86754 −0.933770 0.357873i \(-0.883502\pi\)
−0.933770 + 0.357873i \(0.883502\pi\)
\(740\) 0 0
\(741\) 9.31972 0.342369
\(742\) 0 0
\(743\) −26.1239 −0.958391 −0.479196 0.877708i \(-0.659072\pi\)
−0.479196 + 0.877708i \(0.659072\pi\)
\(744\) 0 0
\(745\) 4.35615 0.159597
\(746\) 0 0
\(747\) −61.4636 −2.24884
\(748\) 0 0
\(749\) 14.5471 0.531541
\(750\) 0 0
\(751\) −42.7938 −1.56157 −0.780784 0.624800i \(-0.785180\pi\)
−0.780784 + 0.624800i \(0.785180\pi\)
\(752\) 0 0
\(753\) 1.06295 0.0387360
\(754\) 0 0
\(755\) 9.02151 0.328326
\(756\) 0 0
\(757\) 3.95763 0.143843 0.0719213 0.997410i \(-0.477087\pi\)
0.0719213 + 0.997410i \(0.477087\pi\)
\(758\) 0 0
\(759\) 21.3526 0.775050
\(760\) 0 0
\(761\) −7.82342 −0.283599 −0.141799 0.989895i \(-0.545289\pi\)
−0.141799 + 0.989895i \(0.545289\pi\)
\(762\) 0 0
\(763\) 23.2598 0.842060
\(764\) 0 0
\(765\) −10.8733 −0.393124
\(766\) 0 0
\(767\) −39.4971 −1.42616
\(768\) 0 0
\(769\) −11.3209 −0.408241 −0.204121 0.978946i \(-0.565433\pi\)
−0.204121 + 0.978946i \(0.565433\pi\)
\(770\) 0 0
\(771\) 30.8113 1.10964
\(772\) 0 0
\(773\) −0.977789 −0.0351686 −0.0175843 0.999845i \(-0.505598\pi\)
−0.0175843 + 0.999845i \(0.505598\pi\)
\(774\) 0 0
\(775\) 49.8231 1.78970
\(776\) 0 0
\(777\) −4.07725 −0.146270
\(778\) 0 0
\(779\) −3.43351 −0.123018
\(780\) 0 0
\(781\) 9.26722 0.331607
\(782\) 0 0
\(783\) 14.4000 0.514614
\(784\) 0 0
\(785\) −0.634345 −0.0226407
\(786\) 0 0
\(787\) −49.3163 −1.75794 −0.878968 0.476881i \(-0.841767\pi\)
−0.878968 + 0.476881i \(0.841767\pi\)
\(788\) 0 0
\(789\) 28.1749 1.00305
\(790\) 0 0
\(791\) 42.5214 1.51189
\(792\) 0 0
\(793\) 27.1736 0.964961
\(794\) 0 0
\(795\) 12.7776 0.453175
\(796\) 0 0
\(797\) 7.74001 0.274165 0.137083 0.990560i \(-0.456227\pi\)
0.137083 + 0.990560i \(0.456227\pi\)
\(798\) 0 0
\(799\) 24.2266 0.857077
\(800\) 0 0
\(801\) 64.1912 2.26809
\(802\) 0 0
\(803\) 17.7292 0.625648
\(804\) 0 0
\(805\) −6.11615 −0.215566
\(806\) 0 0
\(807\) 59.8976 2.10849
\(808\) 0 0
\(809\) −53.0453 −1.86497 −0.932487 0.361204i \(-0.882366\pi\)
−0.932487 + 0.361204i \(0.882366\pi\)
\(810\) 0 0
\(811\) 19.6585 0.690302 0.345151 0.938547i \(-0.387828\pi\)
0.345151 + 0.938547i \(0.387828\pi\)
\(812\) 0 0
\(813\) −54.1071 −1.89762
\(814\) 0 0
\(815\) 0.697525 0.0244332
\(816\) 0 0
\(817\) 8.17053 0.285851
\(818\) 0 0
\(819\) 42.9102 1.49940
\(820\) 0 0
\(821\) 29.4328 1.02721 0.513606 0.858026i \(-0.328309\pi\)
0.513606 + 0.858026i \(0.328309\pi\)
\(822\) 0 0
\(823\) 42.2205 1.47172 0.735858 0.677136i \(-0.236779\pi\)
0.735858 + 0.677136i \(0.236779\pi\)
\(824\) 0 0
\(825\) −28.0369 −0.976118
\(826\) 0 0
\(827\) −9.48340 −0.329770 −0.164885 0.986313i \(-0.552725\pi\)
−0.164885 + 0.986313i \(0.552725\pi\)
\(828\) 0 0
\(829\) 37.5596 1.30450 0.652250 0.758004i \(-0.273825\pi\)
0.652250 + 0.758004i \(0.273825\pi\)
\(830\) 0 0
\(831\) −17.5725 −0.609585
\(832\) 0 0
\(833\) −13.7242 −0.475515
\(834\) 0 0
\(835\) 1.50629 0.0521272
\(836\) 0 0
\(837\) 16.0414 0.554471
\(838\) 0 0
\(839\) −31.4541 −1.08591 −0.542957 0.839760i \(-0.682695\pi\)
−0.542957 + 0.839760i \(0.682695\pi\)
\(840\) 0 0
\(841\) 61.8599 2.13310
\(842\) 0 0
\(843\) −37.6613 −1.29712
\(844\) 0 0
\(845\) 1.12621 0.0387429
\(846\) 0 0
\(847\) 17.2132 0.591452
\(848\) 0 0
\(849\) 0.288573 0.00990381
\(850\) 0 0
\(851\) −1.84030 −0.0630848
\(852\) 0 0
\(853\) −32.4057 −1.10955 −0.554775 0.832001i \(-0.687196\pi\)
−0.554775 + 0.832001i \(0.687196\pi\)
\(854\) 0 0
\(855\) 1.86499 0.0637814
\(856\) 0 0
\(857\) −17.3441 −0.592463 −0.296232 0.955116i \(-0.595730\pi\)
−0.296232 + 0.955116i \(0.595730\pi\)
\(858\) 0 0
\(859\) −20.4271 −0.696962 −0.348481 0.937316i \(-0.613302\pi\)
−0.348481 + 0.937316i \(0.613302\pi\)
\(860\) 0 0
\(861\) −29.0246 −0.989157
\(862\) 0 0
\(863\) −39.3866 −1.34074 −0.670368 0.742029i \(-0.733864\pi\)
−0.670368 + 0.742029i \(0.733864\pi\)
\(864\) 0 0
\(865\) −4.01157 −0.136397
\(866\) 0 0
\(867\) −32.8939 −1.11714
\(868\) 0 0
\(869\) −39.4496 −1.33823
\(870\) 0 0
\(871\) 49.5530 1.67904
\(872\) 0 0
\(873\) −13.5575 −0.458851
\(874\) 0 0
\(875\) 16.5886 0.560796
\(876\) 0 0
\(877\) −45.0536 −1.52135 −0.760676 0.649131i \(-0.775132\pi\)
−0.760676 + 0.649131i \(0.775132\pi\)
\(878\) 0 0
\(879\) −50.7843 −1.71291
\(880\) 0 0
\(881\) 1.50542 0.0507190 0.0253595 0.999678i \(-0.491927\pi\)
0.0253595 + 0.999678i \(0.491927\pi\)
\(882\) 0 0
\(883\) −0.834866 −0.0280955 −0.0140477 0.999901i \(-0.504472\pi\)
−0.0140477 + 0.999901i \(0.504472\pi\)
\(884\) 0 0
\(885\) −14.5114 −0.487797
\(886\) 0 0
\(887\) −46.5394 −1.56264 −0.781319 0.624132i \(-0.785453\pi\)
−0.781319 + 0.624132i \(0.785453\pi\)
\(888\) 0 0
\(889\) 19.0130 0.637675
\(890\) 0 0
\(891\) 16.0347 0.537183
\(892\) 0 0
\(893\) −4.15538 −0.139054
\(894\) 0 0
\(895\) 6.76001 0.225962
\(896\) 0 0
\(897\) 35.5593 1.18729
\(898\) 0 0
\(899\) 101.217 3.37576
\(900\) 0 0
\(901\) 48.9834 1.63187
\(902\) 0 0
\(903\) 69.0682 2.29845
\(904\) 0 0
\(905\) 10.0497 0.334064
\(906\) 0 0
\(907\) −14.1824 −0.470920 −0.235460 0.971884i \(-0.575660\pi\)
−0.235460 + 0.971884i \(0.575660\pi\)
\(908\) 0 0
\(909\) −28.0930 −0.931785
\(910\) 0 0
\(911\) −60.3156 −1.99834 −0.999172 0.0406736i \(-0.987050\pi\)
−0.999172 + 0.0406736i \(0.987050\pi\)
\(912\) 0 0
\(913\) 39.8721 1.31957
\(914\) 0 0
\(915\) 9.98371 0.330051
\(916\) 0 0
\(917\) −6.29126 −0.207756
\(918\) 0 0
\(919\) 5.21812 0.172130 0.0860649 0.996290i \(-0.472571\pi\)
0.0860649 + 0.996290i \(0.472571\pi\)
\(920\) 0 0
\(921\) −36.2248 −1.19365
\(922\) 0 0
\(923\) 15.4331 0.507986
\(924\) 0 0
\(925\) 2.41640 0.0794506
\(926\) 0 0
\(927\) 12.1680 0.399650
\(928\) 0 0
\(929\) −22.8884 −0.750944 −0.375472 0.926834i \(-0.622519\pi\)
−0.375472 + 0.926834i \(0.622519\pi\)
\(930\) 0 0
\(931\) 2.35399 0.0771488
\(932\) 0 0
\(933\) 31.7159 1.03833
\(934\) 0 0
\(935\) 7.05360 0.230677
\(936\) 0 0
\(937\) −12.5289 −0.409301 −0.204651 0.978835i \(-0.565606\pi\)
−0.204651 + 0.978835i \(0.565606\pi\)
\(938\) 0 0
\(939\) 73.9459 2.41313
\(940\) 0 0
\(941\) −36.3085 −1.18362 −0.591811 0.806076i \(-0.701587\pi\)
−0.591811 + 0.806076i \(0.701587\pi\)
\(942\) 0 0
\(943\) −13.1005 −0.426612
\(944\) 0 0
\(945\) 2.58564 0.0841109
\(946\) 0 0
\(947\) −16.8027 −0.546014 −0.273007 0.962012i \(-0.588018\pi\)
−0.273007 + 0.962012i \(0.588018\pi\)
\(948\) 0 0
\(949\) 29.5251 0.958425
\(950\) 0 0
\(951\) −34.9206 −1.13238
\(952\) 0 0
\(953\) −10.9959 −0.356192 −0.178096 0.984013i \(-0.556994\pi\)
−0.178096 + 0.984013i \(0.556994\pi\)
\(954\) 0 0
\(955\) −8.20639 −0.265553
\(956\) 0 0
\(957\) −56.9574 −1.84117
\(958\) 0 0
\(959\) −8.04205 −0.259691
\(960\) 0 0
\(961\) 81.7538 2.63722
\(962\) 0 0
\(963\) −16.9250 −0.545399
\(964\) 0 0
\(965\) −0.153661 −0.00494653
\(966\) 0 0
\(967\) −50.2574 −1.61617 −0.808085 0.589067i \(-0.799496\pi\)
−0.808085 + 0.589067i \(0.799496\pi\)
\(968\) 0 0
\(969\) 13.1265 0.421683
\(970\) 0 0
\(971\) 32.3492 1.03814 0.519068 0.854733i \(-0.326279\pi\)
0.519068 + 0.854733i \(0.326279\pi\)
\(972\) 0 0
\(973\) 68.9812 2.21144
\(974\) 0 0
\(975\) −46.6909 −1.49531
\(976\) 0 0
\(977\) −20.4938 −0.655655 −0.327827 0.944738i \(-0.606316\pi\)
−0.327827 + 0.944738i \(0.606316\pi\)
\(978\) 0 0
\(979\) −41.6415 −1.33087
\(980\) 0 0
\(981\) −27.0617 −0.864013
\(982\) 0 0
\(983\) −13.1227 −0.418551 −0.209275 0.977857i \(-0.567110\pi\)
−0.209275 + 0.977857i \(0.567110\pi\)
\(984\) 0 0
\(985\) 7.75101 0.246968
\(986\) 0 0
\(987\) −35.1268 −1.11810
\(988\) 0 0
\(989\) 31.1746 0.991294
\(990\) 0 0
\(991\) 5.85840 0.186098 0.0930491 0.995662i \(-0.470339\pi\)
0.0930491 + 0.995662i \(0.470339\pi\)
\(992\) 0 0
\(993\) −59.0935 −1.87528
\(994\) 0 0
\(995\) 5.40988 0.171505
\(996\) 0 0
\(997\) −43.3653 −1.37339 −0.686697 0.726944i \(-0.740940\pi\)
−0.686697 + 0.726944i \(0.740940\pi\)
\(998\) 0 0
\(999\) 0.778000 0.0246148
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 6044.2.a.b.1.8 63
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6044.2.a.b.1.8 63 1.1 even 1 trivial