Properties

Label 7623.2.a.cu.1.1
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7623,2,Mod(1,7623)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7623, 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("7623.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7623.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: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.6988960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 693)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.25947\) of defining polynomial
Character \(\chi\) \(=\) 7623.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.25947 q^{2} +3.10522 q^{4} -1.54336 q^{5} -1.00000 q^{7} -2.49721 q^{8} +O(q^{10})\) \(q-2.25947 q^{2} +3.10522 q^{4} -1.54336 q^{5} -1.00000 q^{7} -2.49721 q^{8} +3.48718 q^{10} +2.53716 q^{13} +2.25947 q^{14} -0.568054 q^{16} +6.59570 q^{17} -7.56151 q^{19} -4.79248 q^{20} +5.61973 q^{23} -2.61803 q^{25} -5.73266 q^{26} -3.10522 q^{28} +5.45068 q^{29} -5.32545 q^{31} +6.27793 q^{32} -14.9028 q^{34} +1.54336 q^{35} -6.18609 q^{37} +17.0850 q^{38} +3.85410 q^{40} -9.78690 q^{41} +9.61149 q^{43} -12.6976 q^{46} +3.84418 q^{47} +1.00000 q^{49} +5.91538 q^{50} +7.87845 q^{52} +0.531600 q^{53} +2.49721 q^{56} -12.3157 q^{58} -1.76077 q^{59} -4.74760 q^{61} +12.0327 q^{62} -13.0487 q^{64} -3.91576 q^{65} -9.60824 q^{67} +20.4811 q^{68} -3.48718 q^{70} -9.94228 q^{71} +9.71346 q^{73} +13.9773 q^{74} -23.4802 q^{76} +2.07029 q^{79} +0.876713 q^{80} +22.1132 q^{82} -0.0442404 q^{83} -10.1795 q^{85} -21.7169 q^{86} +6.38297 q^{89} -2.53716 q^{91} +17.4505 q^{92} -8.68582 q^{94} +11.6702 q^{95} -13.2498 q^{97} -2.25947 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{4} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{4} - 8 q^{7} + 14 q^{10} - 2 q^{16} - 6 q^{19} - 12 q^{25} - 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} + 4 q^{40} + 16 q^{43} - 42 q^{46} + 8 q^{49} + 2 q^{52} - 30 q^{58} + 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} + 14 q^{73} - 34 q^{76} + 22 q^{79} + 36 q^{82} - 18 q^{85} + 32 q^{94} - 48 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.25947 −1.59769 −0.798844 0.601538i \(-0.794555\pi\)
−0.798844 + 0.601538i \(0.794555\pi\)
\(3\) 0 0
\(4\) 3.10522 1.55261
\(5\) −1.54336 −0.690212 −0.345106 0.938564i \(-0.612157\pi\)
−0.345106 + 0.938564i \(0.612157\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.49721 −0.882898
\(9\) 0 0
\(10\) 3.48718 1.10274
\(11\) 0 0
\(12\) 0 0
\(13\) 2.53716 0.703683 0.351841 0.936060i \(-0.385556\pi\)
0.351841 + 0.936060i \(0.385556\pi\)
\(14\) 2.25947 0.603870
\(15\) 0 0
\(16\) −0.568054 −0.142014
\(17\) 6.59570 1.59969 0.799846 0.600206i \(-0.204915\pi\)
0.799846 + 0.600206i \(0.204915\pi\)
\(18\) 0 0
\(19\) −7.56151 −1.73473 −0.867365 0.497672i \(-0.834188\pi\)
−0.867365 + 0.497672i \(0.834188\pi\)
\(20\) −4.79248 −1.07163
\(21\) 0 0
\(22\) 0 0
\(23\) 5.61973 1.17179 0.585897 0.810385i \(-0.300742\pi\)
0.585897 + 0.810385i \(0.300742\pi\)
\(24\) 0 0
\(25\) −2.61803 −0.523607
\(26\) −5.73266 −1.12427
\(27\) 0 0
\(28\) −3.10522 −0.586831
\(29\) 5.45068 1.01217 0.506083 0.862485i \(-0.331093\pi\)
0.506083 + 0.862485i \(0.331093\pi\)
\(30\) 0 0
\(31\) −5.32545 −0.956478 −0.478239 0.878230i \(-0.658725\pi\)
−0.478239 + 0.878230i \(0.658725\pi\)
\(32\) 6.27793 1.10979
\(33\) 0 0
\(34\) −14.9028 −2.55581
\(35\) 1.54336 0.260876
\(36\) 0 0
\(37\) −6.18609 −1.01699 −0.508493 0.861066i \(-0.669797\pi\)
−0.508493 + 0.861066i \(0.669797\pi\)
\(38\) 17.0850 2.77156
\(39\) 0 0
\(40\) 3.85410 0.609387
\(41\) −9.78690 −1.52846 −0.764229 0.644946i \(-0.776880\pi\)
−0.764229 + 0.644946i \(0.776880\pi\)
\(42\) 0 0
\(43\) 9.61149 1.46574 0.732870 0.680369i \(-0.238181\pi\)
0.732870 + 0.680369i \(0.238181\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −12.6976 −1.87216
\(47\) 3.84418 0.560731 0.280366 0.959893i \(-0.409544\pi\)
0.280366 + 0.959893i \(0.409544\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.91538 0.836561
\(51\) 0 0
\(52\) 7.87845 1.09254
\(53\) 0.531600 0.0730209 0.0365105 0.999333i \(-0.488376\pi\)
0.0365105 + 0.999333i \(0.488376\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.49721 0.333704
\(57\) 0 0
\(58\) −12.3157 −1.61712
\(59\) −1.76077 −0.229233 −0.114616 0.993410i \(-0.536564\pi\)
−0.114616 + 0.993410i \(0.536564\pi\)
\(60\) 0 0
\(61\) −4.74760 −0.607868 −0.303934 0.952693i \(-0.598300\pi\)
−0.303934 + 0.952693i \(0.598300\pi\)
\(62\) 12.0327 1.52815
\(63\) 0 0
\(64\) −13.0487 −1.63109
\(65\) −3.91576 −0.485691
\(66\) 0 0
\(67\) −9.60824 −1.17383 −0.586917 0.809647i \(-0.699658\pi\)
−0.586917 + 0.809647i \(0.699658\pi\)
\(68\) 20.4811 2.48370
\(69\) 0 0
\(70\) −3.48718 −0.416798
\(71\) −9.94228 −1.17993 −0.589966 0.807428i \(-0.700859\pi\)
−0.589966 + 0.807428i \(0.700859\pi\)
\(72\) 0 0
\(73\) 9.71346 1.13687 0.568437 0.822727i \(-0.307548\pi\)
0.568437 + 0.822727i \(0.307548\pi\)
\(74\) 13.9773 1.62483
\(75\) 0 0
\(76\) −23.4802 −2.69336
\(77\) 0 0
\(78\) 0 0
\(79\) 2.07029 0.232926 0.116463 0.993195i \(-0.462844\pi\)
0.116463 + 0.993195i \(0.462844\pi\)
\(80\) 0.876713 0.0980195
\(81\) 0 0
\(82\) 22.1132 2.44200
\(83\) −0.0442404 −0.00485601 −0.00242801 0.999997i \(-0.500773\pi\)
−0.00242801 + 0.999997i \(0.500773\pi\)
\(84\) 0 0
\(85\) −10.1795 −1.10413
\(86\) −21.7169 −2.34179
\(87\) 0 0
\(88\) 0 0
\(89\) 6.38297 0.676594 0.338297 0.941039i \(-0.390149\pi\)
0.338297 + 0.941039i \(0.390149\pi\)
\(90\) 0 0
\(91\) −2.53716 −0.265967
\(92\) 17.4505 1.81934
\(93\) 0 0
\(94\) −8.68582 −0.895874
\(95\) 11.6702 1.19733
\(96\) 0 0
\(97\) −13.2498 −1.34532 −0.672658 0.739953i \(-0.734848\pi\)
−0.672658 + 0.739953i \(0.734848\pi\)
\(98\) −2.25947 −0.228241
\(99\) 0 0
\(100\) −8.12957 −0.812957
\(101\) 5.53071 0.550326 0.275163 0.961398i \(-0.411268\pi\)
0.275163 + 0.961398i \(0.411268\pi\)
\(102\) 0 0
\(103\) 6.70087 0.660257 0.330128 0.943936i \(-0.392908\pi\)
0.330128 + 0.943936i \(0.392908\pi\)
\(104\) −6.33584 −0.621280
\(105\) 0 0
\(106\) −1.20114 −0.116665
\(107\) −5.75478 −0.556335 −0.278168 0.960533i \(-0.589727\pi\)
−0.278168 + 0.960533i \(0.589727\pi\)
\(108\) 0 0
\(109\) 13.7061 1.31281 0.656405 0.754409i \(-0.272076\pi\)
0.656405 + 0.754409i \(0.272076\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.568054 0.0536761
\(113\) −14.6044 −1.37387 −0.686933 0.726721i \(-0.741043\pi\)
−0.686933 + 0.726721i \(0.741043\pi\)
\(114\) 0 0
\(115\) −8.67327 −0.808787
\(116\) 16.9255 1.57150
\(117\) 0 0
\(118\) 3.97841 0.366242
\(119\) −6.59570 −0.604627
\(120\) 0 0
\(121\) 0 0
\(122\) 10.7271 0.971184
\(123\) 0 0
\(124\) −16.5367 −1.48504
\(125\) 11.7574 1.05161
\(126\) 0 0
\(127\) 13.5063 1.19849 0.599244 0.800566i \(-0.295468\pi\)
0.599244 + 0.800566i \(0.295468\pi\)
\(128\) 16.9273 1.49618
\(129\) 0 0
\(130\) 8.84756 0.775983
\(131\) −4.30276 −0.375934 −0.187967 0.982175i \(-0.560190\pi\)
−0.187967 + 0.982175i \(0.560190\pi\)
\(132\) 0 0
\(133\) 7.56151 0.655666
\(134\) 21.7096 1.87542
\(135\) 0 0
\(136\) −16.4709 −1.41236
\(137\) 12.5285 1.07038 0.535191 0.844731i \(-0.320239\pi\)
0.535191 + 0.844731i \(0.320239\pi\)
\(138\) 0 0
\(139\) 11.2263 0.952200 0.476100 0.879391i \(-0.342050\pi\)
0.476100 + 0.879391i \(0.342050\pi\)
\(140\) 4.79248 0.405038
\(141\) 0 0
\(142\) 22.4643 1.88516
\(143\) 0 0
\(144\) 0 0
\(145\) −8.41237 −0.698609
\(146\) −21.9473 −1.81637
\(147\) 0 0
\(148\) −19.2092 −1.57898
\(149\) −12.1151 −0.992505 −0.496252 0.868178i \(-0.665291\pi\)
−0.496252 + 0.868178i \(0.665291\pi\)
\(150\) 0 0
\(151\) 10.7878 0.877898 0.438949 0.898512i \(-0.355351\pi\)
0.438949 + 0.898512i \(0.355351\pi\)
\(152\) 18.8827 1.53159
\(153\) 0 0
\(154\) 0 0
\(155\) 8.21909 0.660173
\(156\) 0 0
\(157\) −3.70295 −0.295527 −0.147764 0.989023i \(-0.547207\pi\)
−0.147764 + 0.989023i \(0.547207\pi\)
\(158\) −4.67776 −0.372143
\(159\) 0 0
\(160\) −9.68911 −0.765992
\(161\) −5.61973 −0.442897
\(162\) 0 0
\(163\) 22.5447 1.76584 0.882918 0.469526i \(-0.155575\pi\)
0.882918 + 0.469526i \(0.155575\pi\)
\(164\) −30.3905 −2.37310
\(165\) 0 0
\(166\) 0.0999599 0.00775839
\(167\) −14.2177 −1.10020 −0.550099 0.835099i \(-0.685410\pi\)
−0.550099 + 0.835099i \(0.685410\pi\)
\(168\) 0 0
\(169\) −6.56280 −0.504830
\(170\) 23.0004 1.76405
\(171\) 0 0
\(172\) 29.8458 2.27572
\(173\) −18.8789 −1.43534 −0.717670 0.696384i \(-0.754791\pi\)
−0.717670 + 0.696384i \(0.754791\pi\)
\(174\) 0 0
\(175\) 2.61803 0.197905
\(176\) 0 0
\(177\) 0 0
\(178\) −14.4222 −1.08099
\(179\) 4.39825 0.328741 0.164370 0.986399i \(-0.447441\pi\)
0.164370 + 0.986399i \(0.447441\pi\)
\(180\) 0 0
\(181\) 4.87924 0.362671 0.181336 0.983421i \(-0.441958\pi\)
0.181336 + 0.983421i \(0.441958\pi\)
\(182\) 5.73266 0.424933
\(183\) 0 0
\(184\) −14.0337 −1.03457
\(185\) 9.54737 0.701937
\(186\) 0 0
\(187\) 0 0
\(188\) 11.9370 0.870597
\(189\) 0 0
\(190\) −26.3684 −1.91296
\(191\) −15.8326 −1.14561 −0.572805 0.819692i \(-0.694145\pi\)
−0.572805 + 0.819692i \(0.694145\pi\)
\(192\) 0 0
\(193\) −8.33475 −0.599948 −0.299974 0.953947i \(-0.596978\pi\)
−0.299974 + 0.953947i \(0.596978\pi\)
\(194\) 29.9376 2.14940
\(195\) 0 0
\(196\) 3.10522 0.221801
\(197\) 3.63957 0.259309 0.129654 0.991559i \(-0.458613\pi\)
0.129654 + 0.991559i \(0.458613\pi\)
\(198\) 0 0
\(199\) −4.58189 −0.324801 −0.162401 0.986725i \(-0.551924\pi\)
−0.162401 + 0.986725i \(0.551924\pi\)
\(200\) 6.53779 0.462291
\(201\) 0 0
\(202\) −12.4965 −0.879250
\(203\) −5.45068 −0.382562
\(204\) 0 0
\(205\) 15.1047 1.05496
\(206\) −15.1404 −1.05488
\(207\) 0 0
\(208\) −1.44125 −0.0999325
\(209\) 0 0
\(210\) 0 0
\(211\) 1.16946 0.0805087 0.0402543 0.999189i \(-0.487183\pi\)
0.0402543 + 0.999189i \(0.487183\pi\)
\(212\) 1.65074 0.113373
\(213\) 0 0
\(214\) 13.0028 0.888850
\(215\) −14.8340 −1.01167
\(216\) 0 0
\(217\) 5.32545 0.361515
\(218\) −30.9686 −2.09746
\(219\) 0 0
\(220\) 0 0
\(221\) 16.7344 1.12568
\(222\) 0 0
\(223\) −3.93300 −0.263373 −0.131687 0.991291i \(-0.542039\pi\)
−0.131687 + 0.991291i \(0.542039\pi\)
\(224\) −6.27793 −0.419462
\(225\) 0 0
\(226\) 32.9982 2.19501
\(227\) 24.3591 1.61677 0.808384 0.588655i \(-0.200342\pi\)
0.808384 + 0.588655i \(0.200342\pi\)
\(228\) 0 0
\(229\) −24.2275 −1.60100 −0.800498 0.599336i \(-0.795431\pi\)
−0.800498 + 0.599336i \(0.795431\pi\)
\(230\) 19.5970 1.29219
\(231\) 0 0
\(232\) −13.6115 −0.893638
\(233\) 28.1961 1.84719 0.923594 0.383372i \(-0.125237\pi\)
0.923594 + 0.383372i \(0.125237\pi\)
\(234\) 0 0
\(235\) −5.93296 −0.387024
\(236\) −5.46757 −0.355909
\(237\) 0 0
\(238\) 14.9028 0.966005
\(239\) 18.6392 1.20567 0.602834 0.797867i \(-0.294038\pi\)
0.602834 + 0.797867i \(0.294038\pi\)
\(240\) 0 0
\(241\) −17.4814 −1.12608 −0.563039 0.826430i \(-0.690368\pi\)
−0.563039 + 0.826430i \(0.690368\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −14.7423 −0.943782
\(245\) −1.54336 −0.0986018
\(246\) 0 0
\(247\) −19.1848 −1.22070
\(248\) 13.2988 0.844473
\(249\) 0 0
\(250\) −26.5655 −1.68015
\(251\) −4.04114 −0.255075 −0.127537 0.991834i \(-0.540707\pi\)
−0.127537 + 0.991834i \(0.540707\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −30.5171 −1.91481
\(255\) 0 0
\(256\) −12.1495 −0.759341
\(257\) 9.68475 0.604118 0.302059 0.953289i \(-0.402326\pi\)
0.302059 + 0.953289i \(0.402326\pi\)
\(258\) 0 0
\(259\) 6.18609 0.384385
\(260\) −12.1593 −0.754088
\(261\) 0 0
\(262\) 9.72197 0.600625
\(263\) 5.08831 0.313759 0.156879 0.987618i \(-0.449857\pi\)
0.156879 + 0.987618i \(0.449857\pi\)
\(264\) 0 0
\(265\) −0.820452 −0.0504000
\(266\) −17.0850 −1.04755
\(267\) 0 0
\(268\) −29.8357 −1.82251
\(269\) −0.224607 −0.0136945 −0.00684725 0.999977i \(-0.502180\pi\)
−0.00684725 + 0.999977i \(0.502180\pi\)
\(270\) 0 0
\(271\) −24.3590 −1.47971 −0.739853 0.672769i \(-0.765105\pi\)
−0.739853 + 0.672769i \(0.765105\pi\)
\(272\) −3.74671 −0.227178
\(273\) 0 0
\(274\) −28.3078 −1.71014
\(275\) 0 0
\(276\) 0 0
\(277\) −17.4997 −1.05146 −0.525729 0.850652i \(-0.676207\pi\)
−0.525729 + 0.850652i \(0.676207\pi\)
\(278\) −25.3655 −1.52132
\(279\) 0 0
\(280\) −3.85410 −0.230327
\(281\) −14.3550 −0.856345 −0.428173 0.903697i \(-0.640842\pi\)
−0.428173 + 0.903697i \(0.640842\pi\)
\(282\) 0 0
\(283\) 28.0055 1.66475 0.832376 0.554212i \(-0.186980\pi\)
0.832376 + 0.554212i \(0.186980\pi\)
\(284\) −30.8730 −1.83197
\(285\) 0 0
\(286\) 0 0
\(287\) 9.78690 0.577702
\(288\) 0 0
\(289\) 26.5032 1.55901
\(290\) 19.0075 1.11616
\(291\) 0 0
\(292\) 30.1624 1.76512
\(293\) −13.3587 −0.780421 −0.390211 0.920726i \(-0.627598\pi\)
−0.390211 + 0.920726i \(0.627598\pi\)
\(294\) 0 0
\(295\) 2.71750 0.158219
\(296\) 15.4480 0.897895
\(297\) 0 0
\(298\) 27.3737 1.58571
\(299\) 14.2582 0.824571
\(300\) 0 0
\(301\) −9.61149 −0.553997
\(302\) −24.3747 −1.40261
\(303\) 0 0
\(304\) 4.29535 0.246355
\(305\) 7.32727 0.419558
\(306\) 0 0
\(307\) −3.14669 −0.179591 −0.0897955 0.995960i \(-0.528621\pi\)
−0.0897955 + 0.995960i \(0.528621\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −18.5708 −1.05475
\(311\) 24.5598 1.39266 0.696329 0.717722i \(-0.254815\pi\)
0.696329 + 0.717722i \(0.254815\pi\)
\(312\) 0 0
\(313\) −21.5445 −1.21776 −0.608882 0.793261i \(-0.708382\pi\)
−0.608882 + 0.793261i \(0.708382\pi\)
\(314\) 8.36670 0.472160
\(315\) 0 0
\(316\) 6.42870 0.361642
\(317\) −7.38572 −0.414823 −0.207412 0.978254i \(-0.566504\pi\)
−0.207412 + 0.978254i \(0.566504\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 20.1389 1.12580
\(321\) 0 0
\(322\) 12.6976 0.707611
\(323\) −49.8735 −2.77503
\(324\) 0 0
\(325\) −6.64238 −0.368453
\(326\) −50.9391 −2.82126
\(327\) 0 0
\(328\) 24.4400 1.34947
\(329\) −3.84418 −0.211937
\(330\) 0 0
\(331\) −21.0897 −1.15919 −0.579597 0.814903i \(-0.696790\pi\)
−0.579597 + 0.814903i \(0.696790\pi\)
\(332\) −0.137376 −0.00753949
\(333\) 0 0
\(334\) 32.1245 1.75777
\(335\) 14.8290 0.810195
\(336\) 0 0
\(337\) −20.2973 −1.10566 −0.552831 0.833293i \(-0.686452\pi\)
−0.552831 + 0.833293i \(0.686452\pi\)
\(338\) 14.8285 0.806562
\(339\) 0 0
\(340\) −31.6097 −1.71428
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −24.0019 −1.29410
\(345\) 0 0
\(346\) 42.6565 2.29323
\(347\) 16.4180 0.881366 0.440683 0.897663i \(-0.354736\pi\)
0.440683 + 0.897663i \(0.354736\pi\)
\(348\) 0 0
\(349\) −28.3977 −1.52009 −0.760047 0.649868i \(-0.774824\pi\)
−0.760047 + 0.649868i \(0.774824\pi\)
\(350\) −5.91538 −0.316190
\(351\) 0 0
\(352\) 0 0
\(353\) 6.54122 0.348154 0.174077 0.984732i \(-0.444306\pi\)
0.174077 + 0.984732i \(0.444306\pi\)
\(354\) 0 0
\(355\) 15.3445 0.814403
\(356\) 19.8205 1.05049
\(357\) 0 0
\(358\) −9.93773 −0.525225
\(359\) 3.56688 0.188253 0.0941265 0.995560i \(-0.469994\pi\)
0.0941265 + 0.995560i \(0.469994\pi\)
\(360\) 0 0
\(361\) 38.1765 2.00929
\(362\) −11.0245 −0.579436
\(363\) 0 0
\(364\) −7.87845 −0.412943
\(365\) −14.9914 −0.784685
\(366\) 0 0
\(367\) 6.38020 0.333044 0.166522 0.986038i \(-0.446746\pi\)
0.166522 + 0.986038i \(0.446746\pi\)
\(368\) −3.19231 −0.166411
\(369\) 0 0
\(370\) −21.5720 −1.12148
\(371\) −0.531600 −0.0275993
\(372\) 0 0
\(373\) −24.6135 −1.27444 −0.637218 0.770683i \(-0.719915\pi\)
−0.637218 + 0.770683i \(0.719915\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −9.59974 −0.495069
\(377\) 13.8293 0.712243
\(378\) 0 0
\(379\) 31.5946 1.62291 0.811453 0.584418i \(-0.198677\pi\)
0.811453 + 0.584418i \(0.198677\pi\)
\(380\) 36.2384 1.85899
\(381\) 0 0
\(382\) 35.7734 1.83033
\(383\) −17.0187 −0.869614 −0.434807 0.900524i \(-0.643183\pi\)
−0.434807 + 0.900524i \(0.643183\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 18.8321 0.958531
\(387\) 0 0
\(388\) −41.1436 −2.08875
\(389\) 2.70796 0.137299 0.0686495 0.997641i \(-0.478131\pi\)
0.0686495 + 0.997641i \(0.478131\pi\)
\(390\) 0 0
\(391\) 37.0660 1.87451
\(392\) −2.49721 −0.126128
\(393\) 0 0
\(394\) −8.22352 −0.414295
\(395\) −3.19520 −0.160768
\(396\) 0 0
\(397\) −20.9423 −1.05106 −0.525531 0.850774i \(-0.676133\pi\)
−0.525531 + 0.850774i \(0.676133\pi\)
\(398\) 10.3526 0.518931
\(399\) 0 0
\(400\) 1.48718 0.0743592
\(401\) 12.7290 0.635658 0.317829 0.948148i \(-0.397046\pi\)
0.317829 + 0.948148i \(0.397046\pi\)
\(402\) 0 0
\(403\) −13.5115 −0.673057
\(404\) 17.1741 0.854441
\(405\) 0 0
\(406\) 12.3157 0.611216
\(407\) 0 0
\(408\) 0 0
\(409\) 8.25063 0.407967 0.203984 0.978974i \(-0.434611\pi\)
0.203984 + 0.978974i \(0.434611\pi\)
\(410\) −34.1287 −1.68550
\(411\) 0 0
\(412\) 20.8077 1.02512
\(413\) 1.76077 0.0866418
\(414\) 0 0
\(415\) 0.0682789 0.00335168
\(416\) 15.9281 0.780941
\(417\) 0 0
\(418\) 0 0
\(419\) 5.81767 0.284212 0.142106 0.989851i \(-0.454613\pi\)
0.142106 + 0.989851i \(0.454613\pi\)
\(420\) 0 0
\(421\) 7.60495 0.370643 0.185321 0.982678i \(-0.440667\pi\)
0.185321 + 0.982678i \(0.440667\pi\)
\(422\) −2.64235 −0.128628
\(423\) 0 0
\(424\) −1.32752 −0.0644700
\(425\) −17.2678 −0.837609
\(426\) 0 0
\(427\) 4.74760 0.229753
\(428\) −17.8698 −0.863771
\(429\) 0 0
\(430\) 33.5171 1.61634
\(431\) −23.3450 −1.12449 −0.562244 0.826971i \(-0.690062\pi\)
−0.562244 + 0.826971i \(0.690062\pi\)
\(432\) 0 0
\(433\) −13.5912 −0.653153 −0.326576 0.945171i \(-0.605895\pi\)
−0.326576 + 0.945171i \(0.605895\pi\)
\(434\) −12.0327 −0.577588
\(435\) 0 0
\(436\) 42.5605 2.03828
\(437\) −42.4936 −2.03275
\(438\) 0 0
\(439\) −37.4203 −1.78597 −0.892986 0.450084i \(-0.851394\pi\)
−0.892986 + 0.450084i \(0.851394\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −37.8109 −1.79848
\(443\) 29.2812 1.39119 0.695597 0.718432i \(-0.255140\pi\)
0.695597 + 0.718432i \(0.255140\pi\)
\(444\) 0 0
\(445\) −9.85124 −0.466993
\(446\) 8.88651 0.420789
\(447\) 0 0
\(448\) 13.0487 0.616493
\(449\) 4.90362 0.231416 0.115708 0.993283i \(-0.463086\pi\)
0.115708 + 0.993283i \(0.463086\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −45.3498 −2.13308
\(453\) 0 0
\(454\) −55.0387 −2.58309
\(455\) 3.91576 0.183574
\(456\) 0 0
\(457\) 1.55754 0.0728585 0.0364292 0.999336i \(-0.488402\pi\)
0.0364292 + 0.999336i \(0.488402\pi\)
\(458\) 54.7413 2.55789
\(459\) 0 0
\(460\) −26.9324 −1.25573
\(461\) 17.6580 0.822414 0.411207 0.911542i \(-0.365107\pi\)
0.411207 + 0.911542i \(0.365107\pi\)
\(462\) 0 0
\(463\) −4.54726 −0.211329 −0.105664 0.994402i \(-0.533697\pi\)
−0.105664 + 0.994402i \(0.533697\pi\)
\(464\) −3.09628 −0.143741
\(465\) 0 0
\(466\) −63.7083 −2.95123
\(467\) 2.04275 0.0945271 0.0472636 0.998882i \(-0.484950\pi\)
0.0472636 + 0.998882i \(0.484950\pi\)
\(468\) 0 0
\(469\) 9.60824 0.443667
\(470\) 13.4054 0.618344
\(471\) 0 0
\(472\) 4.39701 0.202389
\(473\) 0 0
\(474\) 0 0
\(475\) 19.7963 0.908317
\(476\) −20.4811 −0.938749
\(477\) 0 0
\(478\) −42.1147 −1.92628
\(479\) 8.13148 0.371537 0.185768 0.982594i \(-0.440523\pi\)
0.185768 + 0.982594i \(0.440523\pi\)
\(480\) 0 0
\(481\) −15.6951 −0.715636
\(482\) 39.4988 1.79912
\(483\) 0 0
\(484\) 0 0
\(485\) 20.4493 0.928555
\(486\) 0 0
\(487\) −28.7773 −1.30402 −0.652011 0.758209i \(-0.726075\pi\)
−0.652011 + 0.758209i \(0.726075\pi\)
\(488\) 11.8558 0.536685
\(489\) 0 0
\(490\) 3.48718 0.157535
\(491\) 5.29273 0.238858 0.119429 0.992843i \(-0.461894\pi\)
0.119429 + 0.992843i \(0.461894\pi\)
\(492\) 0 0
\(493\) 35.9510 1.61915
\(494\) 43.3476 1.95030
\(495\) 0 0
\(496\) 3.02514 0.135833
\(497\) 9.94228 0.445972
\(498\) 0 0
\(499\) −25.4668 −1.14005 −0.570025 0.821628i \(-0.693066\pi\)
−0.570025 + 0.821628i \(0.693066\pi\)
\(500\) 36.5092 1.63274
\(501\) 0 0
\(502\) 9.13085 0.407530
\(503\) 42.0133 1.87328 0.936640 0.350293i \(-0.113918\pi\)
0.936640 + 0.350293i \(0.113918\pi\)
\(504\) 0 0
\(505\) −8.53588 −0.379842
\(506\) 0 0
\(507\) 0 0
\(508\) 41.9399 1.86078
\(509\) 31.9838 1.41766 0.708829 0.705380i \(-0.249224\pi\)
0.708829 + 0.705380i \(0.249224\pi\)
\(510\) 0 0
\(511\) −9.71346 −0.429698
\(512\) −6.40330 −0.282989
\(513\) 0 0
\(514\) −21.8824 −0.965193
\(515\) −10.3419 −0.455717
\(516\) 0 0
\(517\) 0 0
\(518\) −13.9773 −0.614127
\(519\) 0 0
\(520\) 9.77849 0.428815
\(521\) −33.9429 −1.48707 −0.743533 0.668699i \(-0.766852\pi\)
−0.743533 + 0.668699i \(0.766852\pi\)
\(522\) 0 0
\(523\) 17.1926 0.751779 0.375890 0.926664i \(-0.377337\pi\)
0.375890 + 0.926664i \(0.377337\pi\)
\(524\) −13.3610 −0.583679
\(525\) 0 0
\(526\) −11.4969 −0.501289
\(527\) −35.1250 −1.53007
\(528\) 0 0
\(529\) 8.58133 0.373101
\(530\) 1.85379 0.0805234
\(531\) 0 0
\(532\) 23.4802 1.01799
\(533\) −24.8310 −1.07555
\(534\) 0 0
\(535\) 8.88170 0.383989
\(536\) 23.9938 1.03638
\(537\) 0 0
\(538\) 0.507492 0.0218796
\(539\) 0 0
\(540\) 0 0
\(541\) 7.65418 0.329079 0.164539 0.986370i \(-0.447386\pi\)
0.164539 + 0.986370i \(0.447386\pi\)
\(542\) 55.0386 2.36411
\(543\) 0 0
\(544\) 41.4073 1.77532
\(545\) −21.1535 −0.906117
\(546\) 0 0
\(547\) −39.9128 −1.70655 −0.853275 0.521462i \(-0.825387\pi\)
−0.853275 + 0.521462i \(0.825387\pi\)
\(548\) 38.9038 1.66189
\(549\) 0 0
\(550\) 0 0
\(551\) −41.2154 −1.75583
\(552\) 0 0
\(553\) −2.07029 −0.0880376
\(554\) 39.5402 1.67990
\(555\) 0 0
\(556\) 34.8600 1.47840
\(557\) −17.3280 −0.734211 −0.367105 0.930179i \(-0.619651\pi\)
−0.367105 + 0.930179i \(0.619651\pi\)
\(558\) 0 0
\(559\) 24.3859 1.03142
\(560\) −0.876713 −0.0370479
\(561\) 0 0
\(562\) 32.4346 1.36817
\(563\) −36.7312 −1.54803 −0.774017 0.633164i \(-0.781756\pi\)
−0.774017 + 0.633164i \(0.781756\pi\)
\(564\) 0 0
\(565\) 22.5399 0.948259
\(566\) −63.2776 −2.65976
\(567\) 0 0
\(568\) 24.8280 1.04176
\(569\) −30.9746 −1.29852 −0.649261 0.760566i \(-0.724922\pi\)
−0.649261 + 0.760566i \(0.724922\pi\)
\(570\) 0 0
\(571\) 26.7450 1.11925 0.559623 0.828748i \(-0.310946\pi\)
0.559623 + 0.828748i \(0.310946\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −22.1132 −0.922989
\(575\) −14.7126 −0.613559
\(576\) 0 0
\(577\) −30.6529 −1.27610 −0.638048 0.769996i \(-0.720258\pi\)
−0.638048 + 0.769996i \(0.720258\pi\)
\(578\) −59.8833 −2.49082
\(579\) 0 0
\(580\) −26.1222 −1.08467
\(581\) 0.0442404 0.00183540
\(582\) 0 0
\(583\) 0 0
\(584\) −24.2566 −1.00374
\(585\) 0 0
\(586\) 30.1835 1.24687
\(587\) 16.6367 0.686670 0.343335 0.939213i \(-0.388443\pi\)
0.343335 + 0.939213i \(0.388443\pi\)
\(588\) 0 0
\(589\) 40.2684 1.65923
\(590\) −6.14013 −0.252785
\(591\) 0 0
\(592\) 3.51403 0.144426
\(593\) 37.7740 1.55119 0.775596 0.631230i \(-0.217450\pi\)
0.775596 + 0.631230i \(0.217450\pi\)
\(594\) 0 0
\(595\) 10.1795 0.417321
\(596\) −37.6199 −1.54097
\(597\) 0 0
\(598\) −32.2160 −1.31741
\(599\) −25.0536 −1.02366 −0.511832 0.859086i \(-0.671033\pi\)
−0.511832 + 0.859086i \(0.671033\pi\)
\(600\) 0 0
\(601\) −42.9247 −1.75093 −0.875467 0.483277i \(-0.839446\pi\)
−0.875467 + 0.483277i \(0.839446\pi\)
\(602\) 21.7169 0.885115
\(603\) 0 0
\(604\) 33.4985 1.36303
\(605\) 0 0
\(606\) 0 0
\(607\) 8.72896 0.354298 0.177149 0.984184i \(-0.443313\pi\)
0.177149 + 0.984184i \(0.443313\pi\)
\(608\) −47.4706 −1.92519
\(609\) 0 0
\(610\) −16.5558 −0.670323
\(611\) 9.75332 0.394577
\(612\) 0 0
\(613\) 8.79301 0.355146 0.177573 0.984108i \(-0.443175\pi\)
0.177573 + 0.984108i \(0.443175\pi\)
\(614\) 7.10986 0.286931
\(615\) 0 0
\(616\) 0 0
\(617\) −8.31423 −0.334718 −0.167359 0.985896i \(-0.553524\pi\)
−0.167359 + 0.985896i \(0.553524\pi\)
\(618\) 0 0
\(619\) 2.35834 0.0947899 0.0473949 0.998876i \(-0.484908\pi\)
0.0473949 + 0.998876i \(0.484908\pi\)
\(620\) 25.5221 1.02499
\(621\) 0 0
\(622\) −55.4922 −2.22504
\(623\) −6.38297 −0.255728
\(624\) 0 0
\(625\) −5.05573 −0.202229
\(626\) 48.6791 1.94561
\(627\) 0 0
\(628\) −11.4985 −0.458838
\(629\) −40.8016 −1.62686
\(630\) 0 0
\(631\) 40.3297 1.60550 0.802750 0.596316i \(-0.203369\pi\)
0.802750 + 0.596316i \(0.203369\pi\)
\(632\) −5.16995 −0.205649
\(633\) 0 0
\(634\) 16.6878 0.662758
\(635\) −20.8451 −0.827211
\(636\) 0 0
\(637\) 2.53716 0.100526
\(638\) 0 0
\(639\) 0 0
\(640\) −26.1250 −1.03268
\(641\) 12.6982 0.501551 0.250775 0.968045i \(-0.419314\pi\)
0.250775 + 0.968045i \(0.419314\pi\)
\(642\) 0 0
\(643\) 14.7309 0.580929 0.290465 0.956886i \(-0.406190\pi\)
0.290465 + 0.956886i \(0.406190\pi\)
\(644\) −17.4505 −0.687645
\(645\) 0 0
\(646\) 112.688 4.43364
\(647\) −4.64105 −0.182459 −0.0912293 0.995830i \(-0.529080\pi\)
−0.0912293 + 0.995830i \(0.529080\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 15.0083 0.588673
\(651\) 0 0
\(652\) 70.0062 2.74165
\(653\) −22.8707 −0.895001 −0.447500 0.894284i \(-0.647686\pi\)
−0.447500 + 0.894284i \(0.647686\pi\)
\(654\) 0 0
\(655\) 6.64072 0.259474
\(656\) 5.55949 0.217062
\(657\) 0 0
\(658\) 8.68582 0.338609
\(659\) 17.8597 0.695717 0.347858 0.937547i \(-0.386909\pi\)
0.347858 + 0.937547i \(0.386909\pi\)
\(660\) 0 0
\(661\) −18.7110 −0.727774 −0.363887 0.931443i \(-0.618551\pi\)
−0.363887 + 0.931443i \(0.618551\pi\)
\(662\) 47.6516 1.85203
\(663\) 0 0
\(664\) 0.110478 0.00428736
\(665\) −11.6702 −0.452549
\(666\) 0 0
\(667\) 30.6313 1.18605
\(668\) −44.1491 −1.70818
\(669\) 0 0
\(670\) −33.5057 −1.29444
\(671\) 0 0
\(672\) 0 0
\(673\) 7.15894 0.275957 0.137978 0.990435i \(-0.455940\pi\)
0.137978 + 0.990435i \(0.455940\pi\)
\(674\) 45.8611 1.76650
\(675\) 0 0
\(676\) −20.3789 −0.783804
\(677\) −23.0053 −0.884166 −0.442083 0.896974i \(-0.645760\pi\)
−0.442083 + 0.896974i \(0.645760\pi\)
\(678\) 0 0
\(679\) 13.2498 0.508482
\(680\) 25.4205 0.974831
\(681\) 0 0
\(682\) 0 0
\(683\) −18.0889 −0.692152 −0.346076 0.938206i \(-0.612486\pi\)
−0.346076 + 0.938206i \(0.612486\pi\)
\(684\) 0 0
\(685\) −19.3360 −0.738792
\(686\) 2.25947 0.0862671
\(687\) 0 0
\(688\) −5.45985 −0.208155
\(689\) 1.34876 0.0513836
\(690\) 0 0
\(691\) −26.6850 −1.01514 −0.507572 0.861609i \(-0.669457\pi\)
−0.507572 + 0.861609i \(0.669457\pi\)
\(692\) −58.6232 −2.22852
\(693\) 0 0
\(694\) −37.0961 −1.40815
\(695\) −17.3262 −0.657221
\(696\) 0 0
\(697\) −64.5514 −2.44506
\(698\) 64.1638 2.42864
\(699\) 0 0
\(700\) 8.12957 0.307269
\(701\) −14.2569 −0.538477 −0.269238 0.963074i \(-0.586772\pi\)
−0.269238 + 0.963074i \(0.586772\pi\)
\(702\) 0 0
\(703\) 46.7762 1.76420
\(704\) 0 0
\(705\) 0 0
\(706\) −14.7797 −0.556242
\(707\) −5.53071 −0.208004
\(708\) 0 0
\(709\) 16.0584 0.603085 0.301543 0.953453i \(-0.402498\pi\)
0.301543 + 0.953453i \(0.402498\pi\)
\(710\) −34.6706 −1.30116
\(711\) 0 0
\(712\) −15.9396 −0.597363
\(713\) −29.9276 −1.12080
\(714\) 0 0
\(715\) 0 0
\(716\) 13.6575 0.510406
\(717\) 0 0
\(718\) −8.05928 −0.300770
\(719\) −37.0621 −1.38218 −0.691091 0.722768i \(-0.742869\pi\)
−0.691091 + 0.722768i \(0.742869\pi\)
\(720\) 0 0
\(721\) −6.70087 −0.249554
\(722\) −86.2588 −3.21022
\(723\) 0 0
\(724\) 15.1511 0.563087
\(725\) −14.2701 −0.529977
\(726\) 0 0
\(727\) 14.6597 0.543698 0.271849 0.962340i \(-0.412365\pi\)
0.271849 + 0.962340i \(0.412365\pi\)
\(728\) 6.33584 0.234822
\(729\) 0 0
\(730\) 33.8726 1.25368
\(731\) 63.3945 2.34473
\(732\) 0 0
\(733\) −28.9555 −1.06949 −0.534747 0.845012i \(-0.679593\pi\)
−0.534747 + 0.845012i \(0.679593\pi\)
\(734\) −14.4159 −0.532100
\(735\) 0 0
\(736\) 35.2802 1.30045
\(737\) 0 0
\(738\) 0 0
\(739\) 8.43921 0.310441 0.155221 0.987880i \(-0.450391\pi\)
0.155221 + 0.987880i \(0.450391\pi\)
\(740\) 29.6467 1.08983
\(741\) 0 0
\(742\) 1.20114 0.0440951
\(743\) −47.7420 −1.75149 −0.875743 0.482778i \(-0.839628\pi\)
−0.875743 + 0.482778i \(0.839628\pi\)
\(744\) 0 0
\(745\) 18.6979 0.685039
\(746\) 55.6135 2.03615
\(747\) 0 0
\(748\) 0 0
\(749\) 5.75478 0.210275
\(750\) 0 0
\(751\) −0.931618 −0.0339952 −0.0169976 0.999856i \(-0.505411\pi\)
−0.0169976 + 0.999856i \(0.505411\pi\)
\(752\) −2.18370 −0.0796314
\(753\) 0 0
\(754\) −31.2468 −1.13794
\(755\) −16.6495 −0.605936
\(756\) 0 0
\(757\) −15.9127 −0.578355 −0.289178 0.957275i \(-0.593382\pi\)
−0.289178 + 0.957275i \(0.593382\pi\)
\(758\) −71.3872 −2.59290
\(759\) 0 0
\(760\) −29.1428 −1.05712
\(761\) −0.221975 −0.00804657 −0.00402328 0.999992i \(-0.501281\pi\)
−0.00402328 + 0.999992i \(0.501281\pi\)
\(762\) 0 0
\(763\) −13.7061 −0.496195
\(764\) −49.1638 −1.77868
\(765\) 0 0
\(766\) 38.4532 1.38937
\(767\) −4.46736 −0.161307
\(768\) 0 0
\(769\) −44.0307 −1.58779 −0.793894 0.608056i \(-0.791949\pi\)
−0.793894 + 0.608056i \(0.791949\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −25.8812 −0.931485
\(773\) 44.3759 1.59609 0.798045 0.602598i \(-0.205868\pi\)
0.798045 + 0.602598i \(0.205868\pi\)
\(774\) 0 0
\(775\) 13.9422 0.500819
\(776\) 33.0876 1.18778
\(777\) 0 0
\(778\) −6.11856 −0.219361
\(779\) 74.0038 2.65146
\(780\) 0 0
\(781\) 0 0
\(782\) −83.7497 −2.99488
\(783\) 0 0
\(784\) −0.568054 −0.0202876
\(785\) 5.71498 0.203977
\(786\) 0 0
\(787\) −17.2307 −0.614210 −0.307105 0.951676i \(-0.599360\pi\)
−0.307105 + 0.951676i \(0.599360\pi\)
\(788\) 11.3017 0.402605
\(789\) 0 0
\(790\) 7.21948 0.256857
\(791\) 14.6044 0.519272
\(792\) 0 0
\(793\) −12.0454 −0.427746
\(794\) 47.3185 1.67927
\(795\) 0 0
\(796\) −14.2278 −0.504290
\(797\) −32.7314 −1.15941 −0.579703 0.814828i \(-0.696832\pi\)
−0.579703 + 0.814828i \(0.696832\pi\)
\(798\) 0 0
\(799\) 25.3551 0.896997
\(800\) −16.4358 −0.581094
\(801\) 0 0
\(802\) −28.7609 −1.01558
\(803\) 0 0
\(804\) 0 0
\(805\) 8.67327 0.305693
\(806\) 30.5289 1.07534
\(807\) 0 0
\(808\) −13.8113 −0.485882
\(809\) −36.9725 −1.29988 −0.649942 0.759984i \(-0.725207\pi\)
−0.649942 + 0.759984i \(0.725207\pi\)
\(810\) 0 0
\(811\) −16.4600 −0.577990 −0.288995 0.957331i \(-0.593321\pi\)
−0.288995 + 0.957331i \(0.593321\pi\)
\(812\) −16.9255 −0.593970
\(813\) 0 0
\(814\) 0 0
\(815\) −34.7946 −1.21880
\(816\) 0 0
\(817\) −72.6774 −2.54266
\(818\) −18.6421 −0.651805
\(819\) 0 0
\(820\) 46.9035 1.63794
\(821\) 7.48406 0.261195 0.130598 0.991435i \(-0.458310\pi\)
0.130598 + 0.991435i \(0.458310\pi\)
\(822\) 0 0
\(823\) −47.6217 −1.65999 −0.829994 0.557772i \(-0.811656\pi\)
−0.829994 + 0.557772i \(0.811656\pi\)
\(824\) −16.7335 −0.582939
\(825\) 0 0
\(826\) −3.97841 −0.138427
\(827\) −37.8258 −1.31533 −0.657666 0.753310i \(-0.728456\pi\)
−0.657666 + 0.753310i \(0.728456\pi\)
\(828\) 0 0
\(829\) −6.27021 −0.217773 −0.108887 0.994054i \(-0.534729\pi\)
−0.108887 + 0.994054i \(0.534729\pi\)
\(830\) −0.154274 −0.00535494
\(831\) 0 0
\(832\) −33.1067 −1.14777
\(833\) 6.59570 0.228527
\(834\) 0 0
\(835\) 21.9431 0.759371
\(836\) 0 0
\(837\) 0 0
\(838\) −13.1449 −0.454082
\(839\) 13.8793 0.479166 0.239583 0.970876i \(-0.422989\pi\)
0.239583 + 0.970876i \(0.422989\pi\)
\(840\) 0 0
\(841\) 0.709871 0.0244783
\(842\) −17.1832 −0.592172
\(843\) 0 0
\(844\) 3.63142 0.124998
\(845\) 10.1288 0.348440
\(846\) 0 0
\(847\) 0 0
\(848\) −0.301978 −0.0103700
\(849\) 0 0
\(850\) 39.0160 1.33824
\(851\) −34.7641 −1.19170
\(852\) 0 0
\(853\) −25.4608 −0.871762 −0.435881 0.900004i \(-0.643563\pi\)
−0.435881 + 0.900004i \(0.643563\pi\)
\(854\) −10.7271 −0.367073
\(855\) 0 0
\(856\) 14.3709 0.491187
\(857\) −50.8757 −1.73788 −0.868941 0.494916i \(-0.835199\pi\)
−0.868941 + 0.494916i \(0.835199\pi\)
\(858\) 0 0
\(859\) −37.2976 −1.27258 −0.636288 0.771452i \(-0.719531\pi\)
−0.636288 + 0.771452i \(0.719531\pi\)
\(860\) −46.0629 −1.57073
\(861\) 0 0
\(862\) 52.7474 1.79658
\(863\) −12.8773 −0.438348 −0.219174 0.975686i \(-0.570336\pi\)
−0.219174 + 0.975686i \(0.570336\pi\)
\(864\) 0 0
\(865\) 29.1370 0.990689
\(866\) 30.7090 1.04353
\(867\) 0 0
\(868\) 16.5367 0.561291
\(869\) 0 0
\(870\) 0 0
\(871\) −24.3777 −0.826007
\(872\) −34.2271 −1.15908
\(873\) 0 0
\(874\) 96.0133 3.24770
\(875\) −11.7574 −0.397472
\(876\) 0 0
\(877\) −10.8105 −0.365046 −0.182523 0.983202i \(-0.558426\pi\)
−0.182523 + 0.983202i \(0.558426\pi\)
\(878\) 84.5501 2.85343
\(879\) 0 0
\(880\) 0 0
\(881\) −19.9303 −0.671467 −0.335734 0.941957i \(-0.608984\pi\)
−0.335734 + 0.941957i \(0.608984\pi\)
\(882\) 0 0
\(883\) −27.1591 −0.913976 −0.456988 0.889473i \(-0.651072\pi\)
−0.456988 + 0.889473i \(0.651072\pi\)
\(884\) 51.9639 1.74773
\(885\) 0 0
\(886\) −66.1602 −2.22269
\(887\) −6.37071 −0.213908 −0.106954 0.994264i \(-0.534110\pi\)
−0.106954 + 0.994264i \(0.534110\pi\)
\(888\) 0 0
\(889\) −13.5063 −0.452986
\(890\) 22.2586 0.746110
\(891\) 0 0
\(892\) −12.2128 −0.408916
\(893\) −29.0678 −0.972718
\(894\) 0 0
\(895\) −6.78809 −0.226901
\(896\) −16.9273 −0.565502
\(897\) 0 0
\(898\) −11.0796 −0.369731
\(899\) −29.0273 −0.968114
\(900\) 0 0
\(901\) 3.50628 0.116811
\(902\) 0 0
\(903\) 0 0
\(904\) 36.4703 1.21298
\(905\) −7.53044 −0.250320
\(906\) 0 0
\(907\) −11.2413 −0.373262 −0.186631 0.982430i \(-0.559757\pi\)
−0.186631 + 0.982430i \(0.559757\pi\)
\(908\) 75.6403 2.51021
\(909\) 0 0
\(910\) −8.84756 −0.293294
\(911\) −32.3301 −1.07114 −0.535572 0.844490i \(-0.679904\pi\)
−0.535572 + 0.844490i \(0.679904\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −3.51921 −0.116405
\(915\) 0 0
\(916\) −75.2315 −2.48572
\(917\) 4.30276 0.142090
\(918\) 0 0
\(919\) −16.9783 −0.560063 −0.280031 0.959991i \(-0.590345\pi\)
−0.280031 + 0.959991i \(0.590345\pi\)
\(920\) 21.6590 0.714076
\(921\) 0 0
\(922\) −39.8977 −1.31396
\(923\) −25.2252 −0.830298
\(924\) 0 0
\(925\) 16.1954 0.532501
\(926\) 10.2744 0.337638
\(927\) 0 0
\(928\) 34.2189 1.12329
\(929\) 7.11309 0.233373 0.116686 0.993169i \(-0.462773\pi\)
0.116686 + 0.993169i \(0.462773\pi\)
\(930\) 0 0
\(931\) −7.56151 −0.247819
\(932\) 87.5551 2.86796
\(933\) 0 0
\(934\) −4.61554 −0.151025
\(935\) 0 0
\(936\) 0 0
\(937\) 15.6317 0.510665 0.255332 0.966853i \(-0.417815\pi\)
0.255332 + 0.966853i \(0.417815\pi\)
\(938\) −21.7096 −0.708843
\(939\) 0 0
\(940\) −18.4231 −0.600897
\(941\) −5.05065 −0.164646 −0.0823232 0.996606i \(-0.526234\pi\)
−0.0823232 + 0.996606i \(0.526234\pi\)
\(942\) 0 0
\(943\) −54.9997 −1.79104
\(944\) 1.00021 0.0325541
\(945\) 0 0
\(946\) 0 0
\(947\) 20.3081 0.659925 0.329963 0.943994i \(-0.392964\pi\)
0.329963 + 0.943994i \(0.392964\pi\)
\(948\) 0 0
\(949\) 24.6447 0.799999
\(950\) −44.7292 −1.45121
\(951\) 0 0
\(952\) 16.4709 0.533823
\(953\) −29.3431 −0.950515 −0.475257 0.879847i \(-0.657645\pi\)
−0.475257 + 0.879847i \(0.657645\pi\)
\(954\) 0 0
\(955\) 24.4355 0.790714
\(956\) 57.8787 1.87193
\(957\) 0 0
\(958\) −18.3729 −0.593600
\(959\) −12.5285 −0.404567
\(960\) 0 0
\(961\) −2.63962 −0.0851491
\(962\) 35.4627 1.14336
\(963\) 0 0
\(964\) −54.2837 −1.74836
\(965\) 12.8635 0.414092
\(966\) 0 0
\(967\) 35.2155 1.13245 0.566227 0.824249i \(-0.308402\pi\)
0.566227 + 0.824249i \(0.308402\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −46.2046 −1.48354
\(971\) −4.93270 −0.158298 −0.0791490 0.996863i \(-0.525220\pi\)
−0.0791490 + 0.996863i \(0.525220\pi\)
\(972\) 0 0
\(973\) −11.2263 −0.359898
\(974\) 65.0215 2.08342
\(975\) 0 0
\(976\) 2.69689 0.0863255
\(977\) −30.3307 −0.970366 −0.485183 0.874413i \(-0.661247\pi\)
−0.485183 + 0.874413i \(0.661247\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.79248 −0.153090
\(981\) 0 0
\(982\) −11.9588 −0.381620
\(983\) 40.7718 1.30042 0.650209 0.759756i \(-0.274681\pi\)
0.650209 + 0.759756i \(0.274681\pi\)
\(984\) 0 0
\(985\) −5.61718 −0.178978
\(986\) −81.2303 −2.58690
\(987\) 0 0
\(988\) −59.5730 −1.89527
\(989\) 54.0140 1.71754
\(990\) 0 0
\(991\) 11.0927 0.352370 0.176185 0.984357i \(-0.443624\pi\)
0.176185 + 0.984357i \(0.443624\pi\)
\(992\) −33.4328 −1.06149
\(993\) 0 0
\(994\) −22.4643 −0.712525
\(995\) 7.07151 0.224182
\(996\) 0 0
\(997\) −26.3185 −0.833515 −0.416757 0.909018i \(-0.636834\pi\)
−0.416757 + 0.909018i \(0.636834\pi\)
\(998\) 57.5415 1.82144
\(999\) 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 7623.2.a.cu.1.1 8
3.2 odd 2 inner 7623.2.a.cu.1.8 8
11.3 even 5 693.2.m.h.64.1 16
11.4 even 5 693.2.m.h.379.1 yes 16
11.10 odd 2 7623.2.a.cv.1.8 8
33.14 odd 10 693.2.m.h.64.4 yes 16
33.26 odd 10 693.2.m.h.379.4 yes 16
33.32 even 2 7623.2.a.cv.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
693.2.m.h.64.1 16 11.3 even 5
693.2.m.h.64.4 yes 16 33.14 odd 10
693.2.m.h.379.1 yes 16 11.4 even 5
693.2.m.h.379.4 yes 16 33.26 odd 10
7623.2.a.cu.1.1 8 1.1 even 1 trivial
7623.2.a.cu.1.8 8 3.2 odd 2 inner
7623.2.a.cv.1.1 8 33.32 even 2
7623.2.a.cv.1.8 8 11.10 odd 2