Properties

Label 7623.2.a.cw.1.4
Level $7623$
Weight $2$
Character 7623.1
Self dual yes
Analytic conductor $60.870$
Analytic rank $1$
Dimension $8$
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] = [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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.226211\) 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)
 
\(f(q)\) \(=\) \(q+0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} -1.00000 q^{7} -0.893270 q^{8} +O(q^{10})\) \(q+0.226211 q^{2} -1.94883 q^{4} +2.49552 q^{5} -1.00000 q^{7} -0.893270 q^{8} +0.564516 q^{10} -5.13499 q^{13} -0.226211 q^{14} +3.69559 q^{16} -1.43752 q^{17} +6.06848 q^{19} -4.86335 q^{20} -7.08292 q^{23} +1.22764 q^{25} -1.16159 q^{26} +1.94883 q^{28} +6.51769 q^{29} +7.68895 q^{31} +2.62252 q^{32} -0.325184 q^{34} -2.49552 q^{35} -3.98432 q^{37} +1.37276 q^{38} -2.22918 q^{40} +6.74900 q^{41} +0.802299 q^{43} -1.60224 q^{46} -6.75222 q^{47} +1.00000 q^{49} +0.277706 q^{50} +10.0072 q^{52} -6.58167 q^{53} +0.893270 q^{56} +1.47437 q^{58} -2.87625 q^{59} +0.855342 q^{61} +1.73933 q^{62} -6.79793 q^{64} -12.8145 q^{65} -1.64668 q^{67} +2.80149 q^{68} -0.564516 q^{70} -4.52077 q^{71} +14.8479 q^{73} -0.901299 q^{74} -11.8264 q^{76} -2.45291 q^{79} +9.22243 q^{80} +1.52670 q^{82} +2.24780 q^{83} -3.58738 q^{85} +0.181489 q^{86} -1.73566 q^{89} +5.13499 q^{91} +13.8034 q^{92} -1.52743 q^{94} +15.1440 q^{95} -12.0776 q^{97} +0.226211 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + 7 q^{4} - 10 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + 7 q^{4} - 10 q^{5} - 8 q^{7} - 6 q^{10} + 6 q^{13} - q^{14} + q^{16} - 5 q^{17} + 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} - 7 q^{28} + 9 q^{29} + 9 q^{31} + 16 q^{32} - 12 q^{34} + 10 q^{35} + 7 q^{37} + 10 q^{38} - 5 q^{40} - 10 q^{41} + 4 q^{43} - 4 q^{46} - 16 q^{47} + 8 q^{49} + 6 q^{50} + 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} - 19 q^{61} - 18 q^{62} - 4 q^{64} - 4 q^{65} - 19 q^{67} + 9 q^{68} + 6 q^{70} - 13 q^{71} + 25 q^{73} + 33 q^{74} - 26 q^{76} - 4 q^{80} - 13 q^{82} - 25 q^{83} - 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} + 42 q^{94} + 21 q^{95} + 15 q^{97} + q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.226211 0.159956 0.0799778 0.996797i \(-0.474515\pi\)
0.0799778 + 0.996797i \(0.474515\pi\)
\(3\) 0 0
\(4\) −1.94883 −0.974414
\(5\) 2.49552 1.11603 0.558016 0.829830i \(-0.311563\pi\)
0.558016 + 0.829830i \(0.311563\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −0.893270 −0.315819
\(9\) 0 0
\(10\) 0.564516 0.178516
\(11\) 0 0
\(12\) 0 0
\(13\) −5.13499 −1.42419 −0.712094 0.702084i \(-0.752253\pi\)
−0.712094 + 0.702084i \(0.752253\pi\)
\(14\) −0.226211 −0.0604575
\(15\) 0 0
\(16\) 3.69559 0.923897
\(17\) −1.43752 −0.348651 −0.174325 0.984688i \(-0.555774\pi\)
−0.174325 + 0.984688i \(0.555774\pi\)
\(18\) 0 0
\(19\) 6.06848 1.39220 0.696102 0.717943i \(-0.254916\pi\)
0.696102 + 0.717943i \(0.254916\pi\)
\(20\) −4.86335 −1.08748
\(21\) 0 0
\(22\) 0 0
\(23\) −7.08292 −1.47689 −0.738446 0.674313i \(-0.764440\pi\)
−0.738446 + 0.674313i \(0.764440\pi\)
\(24\) 0 0
\(25\) 1.22764 0.245528
\(26\) −1.16159 −0.227807
\(27\) 0 0
\(28\) 1.94883 0.368294
\(29\) 6.51769 1.21030 0.605152 0.796110i \(-0.293112\pi\)
0.605152 + 0.796110i \(0.293112\pi\)
\(30\) 0 0
\(31\) 7.68895 1.38098 0.690488 0.723344i \(-0.257396\pi\)
0.690488 + 0.723344i \(0.257396\pi\)
\(32\) 2.62252 0.463601
\(33\) 0 0
\(34\) −0.325184 −0.0557687
\(35\) −2.49552 −0.421820
\(36\) 0 0
\(37\) −3.98432 −0.655019 −0.327509 0.944848i \(-0.606209\pi\)
−0.327509 + 0.944848i \(0.606209\pi\)
\(38\) 1.37276 0.222691
\(39\) 0 0
\(40\) −2.22918 −0.352464
\(41\) 6.74900 1.05402 0.527008 0.849860i \(-0.323314\pi\)
0.527008 + 0.849860i \(0.323314\pi\)
\(42\) 0 0
\(43\) 0.802299 0.122349 0.0611747 0.998127i \(-0.480515\pi\)
0.0611747 + 0.998127i \(0.480515\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.60224 −0.236237
\(47\) −6.75222 −0.984912 −0.492456 0.870337i \(-0.663901\pi\)
−0.492456 + 0.870337i \(0.663901\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.277706 0.0392735
\(51\) 0 0
\(52\) 10.0072 1.38775
\(53\) −6.58167 −0.904062 −0.452031 0.892002i \(-0.649300\pi\)
−0.452031 + 0.892002i \(0.649300\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.893270 0.119368
\(57\) 0 0
\(58\) 1.47437 0.193595
\(59\) −2.87625 −0.374456 −0.187228 0.982316i \(-0.559950\pi\)
−0.187228 + 0.982316i \(0.559950\pi\)
\(60\) 0 0
\(61\) 0.855342 0.109515 0.0547576 0.998500i \(-0.482561\pi\)
0.0547576 + 0.998500i \(0.482561\pi\)
\(62\) 1.73933 0.220895
\(63\) 0 0
\(64\) −6.79793 −0.849742
\(65\) −12.8145 −1.58944
\(66\) 0 0
\(67\) −1.64668 −0.201174 −0.100587 0.994928i \(-0.532072\pi\)
−0.100587 + 0.994928i \(0.532072\pi\)
\(68\) 2.80149 0.339730
\(69\) 0 0
\(70\) −0.564516 −0.0674725
\(71\) −4.52077 −0.536517 −0.268258 0.963347i \(-0.586448\pi\)
−0.268258 + 0.963347i \(0.586448\pi\)
\(72\) 0 0
\(73\) 14.8479 1.73782 0.868910 0.494970i \(-0.164821\pi\)
0.868910 + 0.494970i \(0.164821\pi\)
\(74\) −0.901299 −0.104774
\(75\) 0 0
\(76\) −11.8264 −1.35658
\(77\) 0 0
\(78\) 0 0
\(79\) −2.45291 −0.275973 −0.137987 0.990434i \(-0.544063\pi\)
−0.137987 + 0.990434i \(0.544063\pi\)
\(80\) 9.22243 1.03110
\(81\) 0 0
\(82\) 1.52670 0.168596
\(83\) 2.24780 0.246728 0.123364 0.992361i \(-0.460632\pi\)
0.123364 + 0.992361i \(0.460632\pi\)
\(84\) 0 0
\(85\) −3.58738 −0.389106
\(86\) 0.181489 0.0195705
\(87\) 0 0
\(88\) 0 0
\(89\) −1.73566 −0.183980 −0.0919898 0.995760i \(-0.529323\pi\)
−0.0919898 + 0.995760i \(0.529323\pi\)
\(90\) 0 0
\(91\) 5.13499 0.538293
\(92\) 13.8034 1.43910
\(93\) 0 0
\(94\) −1.52743 −0.157542
\(95\) 15.1440 1.55374
\(96\) 0 0
\(97\) −12.0776 −1.22629 −0.613145 0.789970i \(-0.710096\pi\)
−0.613145 + 0.789970i \(0.710096\pi\)
\(98\) 0.226211 0.0228508
\(99\) 0 0
\(100\) −2.39246 −0.239246
\(101\) −3.69338 −0.367505 −0.183753 0.982973i \(-0.558825\pi\)
−0.183753 + 0.982973i \(0.558825\pi\)
\(102\) 0 0
\(103\) −1.15156 −0.113467 −0.0567334 0.998389i \(-0.518069\pi\)
−0.0567334 + 0.998389i \(0.518069\pi\)
\(104\) 4.58693 0.449785
\(105\) 0 0
\(106\) −1.48885 −0.144610
\(107\) 1.16714 0.112831 0.0564157 0.998407i \(-0.482033\pi\)
0.0564157 + 0.998407i \(0.482033\pi\)
\(108\) 0 0
\(109\) −9.30234 −0.891003 −0.445501 0.895281i \(-0.646975\pi\)
−0.445501 + 0.895281i \(0.646975\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.69559 −0.349200
\(113\) −3.29733 −0.310187 −0.155093 0.987900i \(-0.549568\pi\)
−0.155093 + 0.987900i \(0.549568\pi\)
\(114\) 0 0
\(115\) −17.6756 −1.64826
\(116\) −12.7019 −1.17934
\(117\) 0 0
\(118\) −0.650640 −0.0598963
\(119\) 1.43752 0.131778
\(120\) 0 0
\(121\) 0 0
\(122\) 0.193488 0.0175176
\(123\) 0 0
\(124\) −14.9844 −1.34564
\(125\) −9.41402 −0.842015
\(126\) 0 0
\(127\) 0.289205 0.0256628 0.0128314 0.999918i \(-0.495916\pi\)
0.0128314 + 0.999918i \(0.495916\pi\)
\(128\) −6.78282 −0.599522
\(129\) 0 0
\(130\) −2.89878 −0.254240
\(131\) −16.5059 −1.44212 −0.721062 0.692871i \(-0.756346\pi\)
−0.721062 + 0.692871i \(0.756346\pi\)
\(132\) 0 0
\(133\) −6.06848 −0.526204
\(134\) −0.372498 −0.0321789
\(135\) 0 0
\(136\) 1.28410 0.110110
\(137\) −9.32588 −0.796763 −0.398382 0.917220i \(-0.630428\pi\)
−0.398382 + 0.917220i \(0.630428\pi\)
\(138\) 0 0
\(139\) −4.82649 −0.409377 −0.204689 0.978827i \(-0.565618\pi\)
−0.204689 + 0.978827i \(0.565618\pi\)
\(140\) 4.86335 0.411028
\(141\) 0 0
\(142\) −1.02265 −0.0858189
\(143\) 0 0
\(144\) 0 0
\(145\) 16.2650 1.35074
\(146\) 3.35877 0.277974
\(147\) 0 0
\(148\) 7.76476 0.638260
\(149\) 0.921915 0.0755262 0.0377631 0.999287i \(-0.487977\pi\)
0.0377631 + 0.999287i \(0.487977\pi\)
\(150\) 0 0
\(151\) −18.0964 −1.47267 −0.736333 0.676620i \(-0.763444\pi\)
−0.736333 + 0.676620i \(0.763444\pi\)
\(152\) −5.42079 −0.439684
\(153\) 0 0
\(154\) 0 0
\(155\) 19.1880 1.54121
\(156\) 0 0
\(157\) −12.2733 −0.979518 −0.489759 0.871858i \(-0.662915\pi\)
−0.489759 + 0.871858i \(0.662915\pi\)
\(158\) −0.554875 −0.0441435
\(159\) 0 0
\(160\) 6.54457 0.517394
\(161\) 7.08292 0.558212
\(162\) 0 0
\(163\) 8.09913 0.634373 0.317186 0.948363i \(-0.397262\pi\)
0.317186 + 0.948363i \(0.397262\pi\)
\(164\) −13.1526 −1.02705
\(165\) 0 0
\(166\) 0.508478 0.0394655
\(167\) −13.0516 −1.00996 −0.504982 0.863130i \(-0.668501\pi\)
−0.504982 + 0.863130i \(0.668501\pi\)
\(168\) 0 0
\(169\) 13.3681 1.02831
\(170\) −0.811505 −0.0622396
\(171\) 0 0
\(172\) −1.56354 −0.119219
\(173\) 5.91219 0.449495 0.224748 0.974417i \(-0.427844\pi\)
0.224748 + 0.974417i \(0.427844\pi\)
\(174\) 0 0
\(175\) −1.22764 −0.0928007
\(176\) 0 0
\(177\) 0 0
\(178\) −0.392626 −0.0294286
\(179\) −4.33508 −0.324019 −0.162009 0.986789i \(-0.551798\pi\)
−0.162009 + 0.986789i \(0.551798\pi\)
\(180\) 0 0
\(181\) 10.8307 0.805040 0.402520 0.915411i \(-0.368134\pi\)
0.402520 + 0.915411i \(0.368134\pi\)
\(182\) 1.16159 0.0861029
\(183\) 0 0
\(184\) 6.32696 0.466430
\(185\) −9.94297 −0.731022
\(186\) 0 0
\(187\) 0 0
\(188\) 13.1589 0.959713
\(189\) 0 0
\(190\) 3.42575 0.248530
\(191\) −11.6556 −0.843370 −0.421685 0.906742i \(-0.638561\pi\)
−0.421685 + 0.906742i \(0.638561\pi\)
\(192\) 0 0
\(193\) −22.4454 −1.61566 −0.807829 0.589418i \(-0.799357\pi\)
−0.807829 + 0.589418i \(0.799357\pi\)
\(194\) −2.73208 −0.196152
\(195\) 0 0
\(196\) −1.94883 −0.139202
\(197\) 24.1022 1.71721 0.858604 0.512639i \(-0.171332\pi\)
0.858604 + 0.512639i \(0.171332\pi\)
\(198\) 0 0
\(199\) 18.7205 1.32706 0.663531 0.748148i \(-0.269057\pi\)
0.663531 + 0.748148i \(0.269057\pi\)
\(200\) −1.09661 −0.0775422
\(201\) 0 0
\(202\) −0.835485 −0.0587845
\(203\) −6.51769 −0.457452
\(204\) 0 0
\(205\) 16.8423 1.17632
\(206\) −0.260497 −0.0181497
\(207\) 0 0
\(208\) −18.9768 −1.31580
\(209\) 0 0
\(210\) 0 0
\(211\) 7.56636 0.520890 0.260445 0.965489i \(-0.416131\pi\)
0.260445 + 0.965489i \(0.416131\pi\)
\(212\) 12.8266 0.880931
\(213\) 0 0
\(214\) 0.264019 0.0180480
\(215\) 2.00216 0.136546
\(216\) 0 0
\(217\) −7.68895 −0.521960
\(218\) −2.10430 −0.142521
\(219\) 0 0
\(220\) 0 0
\(221\) 7.38167 0.496545
\(222\) 0 0
\(223\) 17.5244 1.17352 0.586760 0.809761i \(-0.300403\pi\)
0.586760 + 0.809761i \(0.300403\pi\)
\(224\) −2.62252 −0.175225
\(225\) 0 0
\(226\) −0.745893 −0.0496161
\(227\) −25.6773 −1.70426 −0.852130 0.523330i \(-0.824689\pi\)
−0.852130 + 0.523330i \(0.824689\pi\)
\(228\) 0 0
\(229\) −19.8369 −1.31086 −0.655429 0.755257i \(-0.727512\pi\)
−0.655429 + 0.755257i \(0.727512\pi\)
\(230\) −3.99842 −0.263648
\(231\) 0 0
\(232\) −5.82205 −0.382236
\(233\) −20.2146 −1.32430 −0.662151 0.749371i \(-0.730356\pi\)
−0.662151 + 0.749371i \(0.730356\pi\)
\(234\) 0 0
\(235\) −16.8503 −1.09919
\(236\) 5.60532 0.364875
\(237\) 0 0
\(238\) 0.325184 0.0210786
\(239\) 17.1004 1.10613 0.553066 0.833137i \(-0.313458\pi\)
0.553066 + 0.833137i \(0.313458\pi\)
\(240\) 0 0
\(241\) 24.1529 1.55582 0.777912 0.628373i \(-0.216279\pi\)
0.777912 + 0.628373i \(0.216279\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.66691 −0.106713
\(245\) 2.49552 0.159433
\(246\) 0 0
\(247\) −31.1615 −1.98276
\(248\) −6.86831 −0.436138
\(249\) 0 0
\(250\) −2.12956 −0.134685
\(251\) −11.8947 −0.750790 −0.375395 0.926865i \(-0.622493\pi\)
−0.375395 + 0.926865i \(0.622493\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0.0654215 0.00410491
\(255\) 0 0
\(256\) 12.0615 0.753845
\(257\) −22.9609 −1.43226 −0.716131 0.697966i \(-0.754089\pi\)
−0.716131 + 0.697966i \(0.754089\pi\)
\(258\) 0 0
\(259\) 3.98432 0.247574
\(260\) 24.9732 1.54877
\(261\) 0 0
\(262\) −3.73381 −0.230676
\(263\) 1.93774 0.119486 0.0597432 0.998214i \(-0.480972\pi\)
0.0597432 + 0.998214i \(0.480972\pi\)
\(264\) 0 0
\(265\) −16.4247 −1.00896
\(266\) −1.37276 −0.0841692
\(267\) 0 0
\(268\) 3.20910 0.196027
\(269\) −7.18676 −0.438184 −0.219092 0.975704i \(-0.570310\pi\)
−0.219092 + 0.975704i \(0.570310\pi\)
\(270\) 0 0
\(271\) −1.19110 −0.0723543 −0.0361772 0.999345i \(-0.511518\pi\)
−0.0361772 + 0.999345i \(0.511518\pi\)
\(272\) −5.31250 −0.322118
\(273\) 0 0
\(274\) −2.10962 −0.127447
\(275\) 0 0
\(276\) 0 0
\(277\) −10.3402 −0.621285 −0.310642 0.950527i \(-0.600544\pi\)
−0.310642 + 0.950527i \(0.600544\pi\)
\(278\) −1.09181 −0.0654822
\(279\) 0 0
\(280\) 2.22918 0.133219
\(281\) −13.1513 −0.784541 −0.392271 0.919850i \(-0.628310\pi\)
−0.392271 + 0.919850i \(0.628310\pi\)
\(282\) 0 0
\(283\) −0.300031 −0.0178350 −0.00891748 0.999960i \(-0.502839\pi\)
−0.00891748 + 0.999960i \(0.502839\pi\)
\(284\) 8.81021 0.522790
\(285\) 0 0
\(286\) 0 0
\(287\) −6.74900 −0.398381
\(288\) 0 0
\(289\) −14.9335 −0.878443
\(290\) 3.67934 0.216058
\(291\) 0 0
\(292\) −28.9361 −1.69336
\(293\) 16.1441 0.943148 0.471574 0.881826i \(-0.343686\pi\)
0.471574 + 0.881826i \(0.343686\pi\)
\(294\) 0 0
\(295\) −7.17775 −0.417905
\(296\) 3.55908 0.206867
\(297\) 0 0
\(298\) 0.208548 0.0120808
\(299\) 36.3707 2.10337
\(300\) 0 0
\(301\) −0.802299 −0.0462437
\(302\) −4.09361 −0.235561
\(303\) 0 0
\(304\) 22.4266 1.28625
\(305\) 2.13453 0.122223
\(306\) 0 0
\(307\) 28.6376 1.63443 0.817217 0.576330i \(-0.195516\pi\)
0.817217 + 0.576330i \(0.195516\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.34053 0.246526
\(311\) 31.8228 1.80450 0.902252 0.431210i \(-0.141913\pi\)
0.902252 + 0.431210i \(0.141913\pi\)
\(312\) 0 0
\(313\) 0.0342232 0.00193441 0.000967206 1.00000i \(-0.499692\pi\)
0.000967206 1.00000i \(0.499692\pi\)
\(314\) −2.77636 −0.156679
\(315\) 0 0
\(316\) 4.78029 0.268912
\(317\) −22.3894 −1.25752 −0.628758 0.777601i \(-0.716436\pi\)
−0.628758 + 0.777601i \(0.716436\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −16.9644 −0.948339
\(321\) 0 0
\(322\) 1.60224 0.0892892
\(323\) −8.72359 −0.485393
\(324\) 0 0
\(325\) −6.30391 −0.349678
\(326\) 1.83211 0.101471
\(327\) 0 0
\(328\) −6.02868 −0.332878
\(329\) 6.75222 0.372262
\(330\) 0 0
\(331\) 10.7577 0.591297 0.295648 0.955297i \(-0.404464\pi\)
0.295648 + 0.955297i \(0.404464\pi\)
\(332\) −4.38057 −0.240415
\(333\) 0 0
\(334\) −2.95242 −0.161549
\(335\) −4.10933 −0.224517
\(336\) 0 0
\(337\) −7.50492 −0.408819 −0.204410 0.978885i \(-0.565527\pi\)
−0.204410 + 0.978885i \(0.565527\pi\)
\(338\) 3.02401 0.164485
\(339\) 0 0
\(340\) 6.99118 0.379150
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −0.716669 −0.0386402
\(345\) 0 0
\(346\) 1.33740 0.0718993
\(347\) −27.6818 −1.48604 −0.743019 0.669271i \(-0.766607\pi\)
−0.743019 + 0.669271i \(0.766607\pi\)
\(348\) 0 0
\(349\) 11.0605 0.592055 0.296028 0.955179i \(-0.404338\pi\)
0.296028 + 0.955179i \(0.404338\pi\)
\(350\) −0.277706 −0.0148440
\(351\) 0 0
\(352\) 0 0
\(353\) −31.9202 −1.69894 −0.849469 0.527638i \(-0.823078\pi\)
−0.849469 + 0.527638i \(0.823078\pi\)
\(354\) 0 0
\(355\) −11.2817 −0.598770
\(356\) 3.38251 0.179272
\(357\) 0 0
\(358\) −0.980644 −0.0518286
\(359\) 3.57826 0.188854 0.0944268 0.995532i \(-0.469898\pi\)
0.0944268 + 0.995532i \(0.469898\pi\)
\(360\) 0 0
\(361\) 17.8264 0.938233
\(362\) 2.45003 0.128771
\(363\) 0 0
\(364\) −10.0072 −0.524520
\(365\) 37.0534 1.93946
\(366\) 0 0
\(367\) 2.29397 0.119744 0.0598720 0.998206i \(-0.480931\pi\)
0.0598720 + 0.998206i \(0.480931\pi\)
\(368\) −26.1756 −1.36450
\(369\) 0 0
\(370\) −2.24921 −0.116931
\(371\) 6.58167 0.341703
\(372\) 0 0
\(373\) −7.96856 −0.412596 −0.206298 0.978489i \(-0.566142\pi\)
−0.206298 + 0.978489i \(0.566142\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 6.03155 0.311054
\(377\) −33.4682 −1.72370
\(378\) 0 0
\(379\) −11.6212 −0.596941 −0.298470 0.954419i \(-0.596476\pi\)
−0.298470 + 0.954419i \(0.596476\pi\)
\(380\) −29.5131 −1.51399
\(381\) 0 0
\(382\) −2.63663 −0.134902
\(383\) −12.5785 −0.642729 −0.321364 0.946956i \(-0.604141\pi\)
−0.321364 + 0.946956i \(0.604141\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.07741 −0.258433
\(387\) 0 0
\(388\) 23.5371 1.19491
\(389\) 0.438294 0.0222224 0.0111112 0.999938i \(-0.496463\pi\)
0.0111112 + 0.999938i \(0.496463\pi\)
\(390\) 0 0
\(391\) 10.1819 0.514920
\(392\) −0.893270 −0.0451169
\(393\) 0 0
\(394\) 5.45218 0.274677
\(395\) −6.12128 −0.307995
\(396\) 0 0
\(397\) 16.8147 0.843905 0.421952 0.906618i \(-0.361345\pi\)
0.421952 + 0.906618i \(0.361345\pi\)
\(398\) 4.23480 0.212271
\(399\) 0 0
\(400\) 4.53685 0.226842
\(401\) −36.8609 −1.84074 −0.920372 0.391043i \(-0.872114\pi\)
−0.920372 + 0.391043i \(0.872114\pi\)
\(402\) 0 0
\(403\) −39.4827 −1.96677
\(404\) 7.19776 0.358102
\(405\) 0 0
\(406\) −1.47437 −0.0731720
\(407\) 0 0
\(408\) 0 0
\(409\) −16.2739 −0.804692 −0.402346 0.915488i \(-0.631805\pi\)
−0.402346 + 0.915488i \(0.631805\pi\)
\(410\) 3.80992 0.188158
\(411\) 0 0
\(412\) 2.24420 0.110564
\(413\) 2.87625 0.141531
\(414\) 0 0
\(415\) 5.60944 0.275356
\(416\) −13.4666 −0.660255
\(417\) 0 0
\(418\) 0 0
\(419\) −5.56352 −0.271796 −0.135898 0.990723i \(-0.543392\pi\)
−0.135898 + 0.990723i \(0.543392\pi\)
\(420\) 0 0
\(421\) 21.4914 1.04743 0.523713 0.851895i \(-0.324546\pi\)
0.523713 + 0.851895i \(0.324546\pi\)
\(422\) 1.71160 0.0833192
\(423\) 0 0
\(424\) 5.87921 0.285520
\(425\) −1.76476 −0.0856035
\(426\) 0 0
\(427\) −0.855342 −0.0413929
\(428\) −2.27455 −0.109944
\(429\) 0 0
\(430\) 0.452910 0.0218413
\(431\) −27.7188 −1.33517 −0.667583 0.744536i \(-0.732671\pi\)
−0.667583 + 0.744536i \(0.732671\pi\)
\(432\) 0 0
\(433\) 9.28812 0.446358 0.223179 0.974777i \(-0.428356\pi\)
0.223179 + 0.974777i \(0.428356\pi\)
\(434\) −1.73933 −0.0834904
\(435\) 0 0
\(436\) 18.1287 0.868206
\(437\) −42.9826 −2.05613
\(438\) 0 0
\(439\) 14.7118 0.702156 0.351078 0.936346i \(-0.385815\pi\)
0.351078 + 0.936346i \(0.385815\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.66982 0.0794251
\(443\) −2.44345 −0.116092 −0.0580459 0.998314i \(-0.518487\pi\)
−0.0580459 + 0.998314i \(0.518487\pi\)
\(444\) 0 0
\(445\) −4.33138 −0.205327
\(446\) 3.96421 0.187711
\(447\) 0 0
\(448\) 6.79793 0.321172
\(449\) −4.76935 −0.225080 −0.112540 0.993647i \(-0.535899\pi\)
−0.112540 + 0.993647i \(0.535899\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 6.42593 0.302250
\(453\) 0 0
\(454\) −5.80849 −0.272606
\(455\) 12.8145 0.600752
\(456\) 0 0
\(457\) −31.0875 −1.45421 −0.727106 0.686525i \(-0.759135\pi\)
−0.727106 + 0.686525i \(0.759135\pi\)
\(458\) −4.48733 −0.209679
\(459\) 0 0
\(460\) 34.4467 1.60609
\(461\) 29.7215 1.38427 0.692134 0.721769i \(-0.256671\pi\)
0.692134 + 0.721769i \(0.256671\pi\)
\(462\) 0 0
\(463\) −25.4553 −1.18301 −0.591505 0.806302i \(-0.701466\pi\)
−0.591505 + 0.806302i \(0.701466\pi\)
\(464\) 24.0867 1.11820
\(465\) 0 0
\(466\) −4.57277 −0.211829
\(467\) 3.18491 0.147380 0.0736901 0.997281i \(-0.476522\pi\)
0.0736901 + 0.997281i \(0.476522\pi\)
\(468\) 0 0
\(469\) 1.64668 0.0760366
\(470\) −3.81173 −0.175822
\(471\) 0 0
\(472\) 2.56927 0.118260
\(473\) 0 0
\(474\) 0 0
\(475\) 7.44990 0.341825
\(476\) −2.80149 −0.128406
\(477\) 0 0
\(478\) 3.86830 0.176932
\(479\) 3.23354 0.147744 0.0738720 0.997268i \(-0.476464\pi\)
0.0738720 + 0.997268i \(0.476464\pi\)
\(480\) 0 0
\(481\) 20.4594 0.932870
\(482\) 5.46366 0.248863
\(483\) 0 0
\(484\) 0 0
\(485\) −30.1398 −1.36858
\(486\) 0 0
\(487\) 9.87096 0.447296 0.223648 0.974670i \(-0.428203\pi\)
0.223648 + 0.974670i \(0.428203\pi\)
\(488\) −0.764051 −0.0345870
\(489\) 0 0
\(490\) 0.564516 0.0255022
\(491\) 4.22600 0.190717 0.0953584 0.995443i \(-0.469600\pi\)
0.0953584 + 0.995443i \(0.469600\pi\)
\(492\) 0 0
\(493\) −9.36933 −0.421974
\(494\) −7.04910 −0.317154
\(495\) 0 0
\(496\) 28.4152 1.27588
\(497\) 4.52077 0.202784
\(498\) 0 0
\(499\) −20.3953 −0.913018 −0.456509 0.889719i \(-0.650900\pi\)
−0.456509 + 0.889719i \(0.650900\pi\)
\(500\) 18.3463 0.820472
\(501\) 0 0
\(502\) −2.69073 −0.120093
\(503\) −23.9593 −1.06829 −0.534145 0.845393i \(-0.679366\pi\)
−0.534145 + 0.845393i \(0.679366\pi\)
\(504\) 0 0
\(505\) −9.21692 −0.410147
\(506\) 0 0
\(507\) 0 0
\(508\) −0.563611 −0.0250062
\(509\) −3.69341 −0.163708 −0.0818538 0.996644i \(-0.526084\pi\)
−0.0818538 + 0.996644i \(0.526084\pi\)
\(510\) 0 0
\(511\) −14.8479 −0.656834
\(512\) 16.2941 0.720104
\(513\) 0 0
\(514\) −5.19402 −0.229098
\(515\) −2.87375 −0.126633
\(516\) 0 0
\(517\) 0 0
\(518\) 0.901299 0.0396008
\(519\) 0 0
\(520\) 11.4468 0.501975
\(521\) 0.238270 0.0104388 0.00521939 0.999986i \(-0.498339\pi\)
0.00521939 + 0.999986i \(0.498339\pi\)
\(522\) 0 0
\(523\) 21.9914 0.961618 0.480809 0.876825i \(-0.340343\pi\)
0.480809 + 0.876825i \(0.340343\pi\)
\(524\) 32.1671 1.40523
\(525\) 0 0
\(526\) 0.438339 0.0191125
\(527\) −11.0531 −0.481479
\(528\) 0 0
\(529\) 27.1678 1.18121
\(530\) −3.71546 −0.161389
\(531\) 0 0
\(532\) 11.8264 0.512740
\(533\) −34.6560 −1.50112
\(534\) 0 0
\(535\) 2.91262 0.125923
\(536\) 1.47093 0.0635345
\(537\) 0 0
\(538\) −1.62573 −0.0700900
\(539\) 0 0
\(540\) 0 0
\(541\) −28.6309 −1.23094 −0.615470 0.788161i \(-0.711034\pi\)
−0.615470 + 0.788161i \(0.711034\pi\)
\(542\) −0.269441 −0.0115735
\(543\) 0 0
\(544\) −3.76994 −0.161635
\(545\) −23.2142 −0.994388
\(546\) 0 0
\(547\) −6.40847 −0.274007 −0.137003 0.990571i \(-0.543747\pi\)
−0.137003 + 0.990571i \(0.543747\pi\)
\(548\) 18.1745 0.776378
\(549\) 0 0
\(550\) 0 0
\(551\) 39.5524 1.68499
\(552\) 0 0
\(553\) 2.45291 0.104308
\(554\) −2.33908 −0.0993780
\(555\) 0 0
\(556\) 9.40600 0.398903
\(557\) −22.5193 −0.954172 −0.477086 0.878857i \(-0.658307\pi\)
−0.477086 + 0.878857i \(0.658307\pi\)
\(558\) 0 0
\(559\) −4.11979 −0.174249
\(560\) −9.22243 −0.389719
\(561\) 0 0
\(562\) −2.97498 −0.125492
\(563\) −29.8267 −1.25705 −0.628523 0.777791i \(-0.716340\pi\)
−0.628523 + 0.777791i \(0.716340\pi\)
\(564\) 0 0
\(565\) −8.22856 −0.346178
\(566\) −0.0678703 −0.00285280
\(567\) 0 0
\(568\) 4.03827 0.169442
\(569\) 29.2764 1.22733 0.613665 0.789567i \(-0.289695\pi\)
0.613665 + 0.789567i \(0.289695\pi\)
\(570\) 0 0
\(571\) −37.9252 −1.58712 −0.793559 0.608493i \(-0.791774\pi\)
−0.793559 + 0.608493i \(0.791774\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.52670 −0.0637232
\(575\) −8.69527 −0.362618
\(576\) 0 0
\(577\) 8.77263 0.365209 0.182605 0.983186i \(-0.441547\pi\)
0.182605 + 0.983186i \(0.441547\pi\)
\(578\) −3.37813 −0.140512
\(579\) 0 0
\(580\) −31.6978 −1.31618
\(581\) −2.24780 −0.0932544
\(582\) 0 0
\(583\) 0 0
\(584\) −13.2632 −0.548836
\(585\) 0 0
\(586\) 3.65198 0.150862
\(587\) −9.17011 −0.378491 −0.189246 0.981930i \(-0.560604\pi\)
−0.189246 + 0.981930i \(0.560604\pi\)
\(588\) 0 0
\(589\) 46.6602 1.92260
\(590\) −1.62369 −0.0668462
\(591\) 0 0
\(592\) −14.7244 −0.605170
\(593\) 7.25596 0.297967 0.148983 0.988840i \(-0.452400\pi\)
0.148983 + 0.988840i \(0.452400\pi\)
\(594\) 0 0
\(595\) 3.58738 0.147068
\(596\) −1.79665 −0.0735938
\(597\) 0 0
\(598\) 8.22747 0.336446
\(599\) 20.0181 0.817916 0.408958 0.912553i \(-0.365892\pi\)
0.408958 + 0.912553i \(0.365892\pi\)
\(600\) 0 0
\(601\) 11.2706 0.459736 0.229868 0.973222i \(-0.426171\pi\)
0.229868 + 0.973222i \(0.426171\pi\)
\(602\) −0.181489 −0.00739694
\(603\) 0 0
\(604\) 35.2668 1.43499
\(605\) 0 0
\(606\) 0 0
\(607\) 13.5260 0.549005 0.274502 0.961586i \(-0.411487\pi\)
0.274502 + 0.961586i \(0.411487\pi\)
\(608\) 15.9147 0.645427
\(609\) 0 0
\(610\) 0.482854 0.0195502
\(611\) 34.6725 1.40270
\(612\) 0 0
\(613\) 30.7968 1.24387 0.621935 0.783069i \(-0.286347\pi\)
0.621935 + 0.783069i \(0.286347\pi\)
\(614\) 6.47815 0.261437
\(615\) 0 0
\(616\) 0 0
\(617\) −23.6896 −0.953707 −0.476853 0.878983i \(-0.658223\pi\)
−0.476853 + 0.878983i \(0.658223\pi\)
\(618\) 0 0
\(619\) −32.4878 −1.30579 −0.652897 0.757446i \(-0.726447\pi\)
−0.652897 + 0.757446i \(0.726447\pi\)
\(620\) −37.3940 −1.50178
\(621\) 0 0
\(622\) 7.19867 0.288640
\(623\) 1.73566 0.0695378
\(624\) 0 0
\(625\) −29.6311 −1.18524
\(626\) 0.00774169 0.000309420 0
\(627\) 0 0
\(628\) 23.9186 0.954456
\(629\) 5.72756 0.228373
\(630\) 0 0
\(631\) −15.1333 −0.602448 −0.301224 0.953553i \(-0.597395\pi\)
−0.301224 + 0.953553i \(0.597395\pi\)
\(632\) 2.19111 0.0871575
\(633\) 0 0
\(634\) −5.06474 −0.201147
\(635\) 0.721718 0.0286405
\(636\) 0 0
\(637\) −5.13499 −0.203456
\(638\) 0 0
\(639\) 0 0
\(640\) −16.9267 −0.669086
\(641\) 16.5502 0.653695 0.326848 0.945077i \(-0.394014\pi\)
0.326848 + 0.945077i \(0.394014\pi\)
\(642\) 0 0
\(643\) 1.99506 0.0786773 0.0393387 0.999226i \(-0.487475\pi\)
0.0393387 + 0.999226i \(0.487475\pi\)
\(644\) −13.8034 −0.543930
\(645\) 0 0
\(646\) −1.97337 −0.0776414
\(647\) −40.4517 −1.59032 −0.795160 0.606399i \(-0.792613\pi\)
−0.795160 + 0.606399i \(0.792613\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −1.42602 −0.0559329
\(651\) 0 0
\(652\) −15.7838 −0.618142
\(653\) 41.4856 1.62346 0.811729 0.584034i \(-0.198527\pi\)
0.811729 + 0.584034i \(0.198527\pi\)
\(654\) 0 0
\(655\) −41.1908 −1.60946
\(656\) 24.9415 0.973803
\(657\) 0 0
\(658\) 1.52743 0.0595454
\(659\) −51.1359 −1.99197 −0.995985 0.0895158i \(-0.971468\pi\)
−0.995985 + 0.0895158i \(0.971468\pi\)
\(660\) 0 0
\(661\) −42.8840 −1.66800 −0.833998 0.551768i \(-0.813954\pi\)
−0.833998 + 0.551768i \(0.813954\pi\)
\(662\) 2.43351 0.0945812
\(663\) 0 0
\(664\) −2.00789 −0.0779213
\(665\) −15.1440 −0.587260
\(666\) 0 0
\(667\) −46.1643 −1.78749
\(668\) 25.4353 0.984122
\(669\) 0 0
\(670\) −0.929577 −0.0359127
\(671\) 0 0
\(672\) 0 0
\(673\) 25.0072 0.963958 0.481979 0.876183i \(-0.339918\pi\)
0.481979 + 0.876183i \(0.339918\pi\)
\(674\) −1.69770 −0.0653929
\(675\) 0 0
\(676\) −26.0521 −1.00200
\(677\) −9.91890 −0.381214 −0.190607 0.981666i \(-0.561046\pi\)
−0.190607 + 0.981666i \(0.561046\pi\)
\(678\) 0 0
\(679\) 12.0776 0.463494
\(680\) 3.20450 0.122887
\(681\) 0 0
\(682\) 0 0
\(683\) 39.8980 1.52666 0.763328 0.646011i \(-0.223564\pi\)
0.763328 + 0.646011i \(0.223564\pi\)
\(684\) 0 0
\(685\) −23.2729 −0.889214
\(686\) −0.226211 −0.00863679
\(687\) 0 0
\(688\) 2.96497 0.113038
\(689\) 33.7968 1.28756
\(690\) 0 0
\(691\) −6.61401 −0.251609 −0.125804 0.992055i \(-0.540151\pi\)
−0.125804 + 0.992055i \(0.540151\pi\)
\(692\) −11.5218 −0.437995
\(693\) 0 0
\(694\) −6.26194 −0.237700
\(695\) −12.0446 −0.456878
\(696\) 0 0
\(697\) −9.70185 −0.367484
\(698\) 2.50201 0.0947025
\(699\) 0 0
\(700\) 2.39246 0.0904264
\(701\) 14.6016 0.551495 0.275748 0.961230i \(-0.411075\pi\)
0.275748 + 0.961230i \(0.411075\pi\)
\(702\) 0 0
\(703\) −24.1788 −0.911920
\(704\) 0 0
\(705\) 0 0
\(706\) −7.22070 −0.271755
\(707\) 3.69338 0.138904
\(708\) 0 0
\(709\) −4.14406 −0.155633 −0.0778167 0.996968i \(-0.524795\pi\)
−0.0778167 + 0.996968i \(0.524795\pi\)
\(710\) −2.55205 −0.0957766
\(711\) 0 0
\(712\) 1.55041 0.0581042
\(713\) −54.4602 −2.03955
\(714\) 0 0
\(715\) 0 0
\(716\) 8.44832 0.315729
\(717\) 0 0
\(718\) 0.809444 0.0302082
\(719\) −17.0223 −0.634823 −0.317412 0.948288i \(-0.602814\pi\)
−0.317412 + 0.948288i \(0.602814\pi\)
\(720\) 0 0
\(721\) 1.15156 0.0428864
\(722\) 4.03254 0.150076
\(723\) 0 0
\(724\) −21.1072 −0.784442
\(725\) 8.00136 0.297163
\(726\) 0 0
\(727\) −21.6199 −0.801837 −0.400918 0.916114i \(-0.631309\pi\)
−0.400918 + 0.916114i \(0.631309\pi\)
\(728\) −4.58693 −0.170003
\(729\) 0 0
\(730\) 8.38190 0.310228
\(731\) −1.15332 −0.0426572
\(732\) 0 0
\(733\) 48.2326 1.78151 0.890755 0.454484i \(-0.150176\pi\)
0.890755 + 0.454484i \(0.150176\pi\)
\(734\) 0.518921 0.0191537
\(735\) 0 0
\(736\) −18.5751 −0.684688
\(737\) 0 0
\(738\) 0 0
\(739\) 8.16347 0.300298 0.150149 0.988663i \(-0.452025\pi\)
0.150149 + 0.988663i \(0.452025\pi\)
\(740\) 19.3772 0.712318
\(741\) 0 0
\(742\) 1.48885 0.0546574
\(743\) 19.5612 0.717633 0.358816 0.933408i \(-0.383180\pi\)
0.358816 + 0.933408i \(0.383180\pi\)
\(744\) 0 0
\(745\) 2.30066 0.0842897
\(746\) −1.80258 −0.0659971
\(747\) 0 0
\(748\) 0 0
\(749\) −1.16714 −0.0426462
\(750\) 0 0
\(751\) 1.11630 0.0407343 0.0203671 0.999793i \(-0.493516\pi\)
0.0203671 + 0.999793i \(0.493516\pi\)
\(752\) −24.9534 −0.909958
\(753\) 0 0
\(754\) −7.57089 −0.275716
\(755\) −45.1600 −1.64354
\(756\) 0 0
\(757\) −26.5773 −0.965969 −0.482984 0.875629i \(-0.660447\pi\)
−0.482984 + 0.875629i \(0.660447\pi\)
\(758\) −2.62885 −0.0954840
\(759\) 0 0
\(760\) −13.5277 −0.490701
\(761\) 6.30003 0.228376 0.114188 0.993459i \(-0.463573\pi\)
0.114188 + 0.993459i \(0.463573\pi\)
\(762\) 0 0
\(763\) 9.30234 0.336767
\(764\) 22.7148 0.821792
\(765\) 0 0
\(766\) −2.84539 −0.102808
\(767\) 14.7695 0.533296
\(768\) 0 0
\(769\) −13.1916 −0.475700 −0.237850 0.971302i \(-0.576443\pi\)
−0.237850 + 0.971302i \(0.576443\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 43.7423 1.57432
\(773\) −45.8178 −1.64795 −0.823976 0.566625i \(-0.808249\pi\)
−0.823976 + 0.566625i \(0.808249\pi\)
\(774\) 0 0
\(775\) 9.43925 0.339068
\(776\) 10.7885 0.387285
\(777\) 0 0
\(778\) 0.0991470 0.00355459
\(779\) 40.9562 1.46741
\(780\) 0 0
\(781\) 0 0
\(782\) 2.30326 0.0823643
\(783\) 0 0
\(784\) 3.69559 0.131985
\(785\) −30.6284 −1.09317
\(786\) 0 0
\(787\) 18.8479 0.671856 0.335928 0.941888i \(-0.390950\pi\)
0.335928 + 0.941888i \(0.390950\pi\)
\(788\) −46.9710 −1.67327
\(789\) 0 0
\(790\) −1.38470 −0.0492656
\(791\) 3.29733 0.117240
\(792\) 0 0
\(793\) −4.39217 −0.155970
\(794\) 3.80367 0.134987
\(795\) 0 0
\(796\) −36.4831 −1.29311
\(797\) 28.1613 0.997526 0.498763 0.866738i \(-0.333788\pi\)
0.498763 + 0.866738i \(0.333788\pi\)
\(798\) 0 0
\(799\) 9.70648 0.343391
\(800\) 3.21951 0.113827
\(801\) 0 0
\(802\) −8.33835 −0.294437
\(803\) 0 0
\(804\) 0 0
\(805\) 17.6756 0.622983
\(806\) −8.93143 −0.314596
\(807\) 0 0
\(808\) 3.29919 0.116065
\(809\) −6.54552 −0.230128 −0.115064 0.993358i \(-0.536707\pi\)
−0.115064 + 0.993358i \(0.536707\pi\)
\(810\) 0 0
\(811\) −18.8046 −0.660320 −0.330160 0.943925i \(-0.607103\pi\)
−0.330160 + 0.943925i \(0.607103\pi\)
\(812\) 12.7019 0.445748
\(813\) 0 0
\(814\) 0 0
\(815\) 20.2116 0.707980
\(816\) 0 0
\(817\) 4.86873 0.170335
\(818\) −3.68134 −0.128715
\(819\) 0 0
\(820\) −32.8227 −1.14622
\(821\) −11.5880 −0.404424 −0.202212 0.979342i \(-0.564813\pi\)
−0.202212 + 0.979342i \(0.564813\pi\)
\(822\) 0 0
\(823\) 12.3464 0.430368 0.215184 0.976574i \(-0.430965\pi\)
0.215184 + 0.976574i \(0.430965\pi\)
\(824\) 1.02866 0.0358349
\(825\) 0 0
\(826\) 0.650640 0.0226387
\(827\) −4.49579 −0.156334 −0.0781669 0.996940i \(-0.524907\pi\)
−0.0781669 + 0.996940i \(0.524907\pi\)
\(828\) 0 0
\(829\) −19.8629 −0.689866 −0.344933 0.938627i \(-0.612098\pi\)
−0.344933 + 0.938627i \(0.612098\pi\)
\(830\) 1.26892 0.0440448
\(831\) 0 0
\(832\) 34.9073 1.21019
\(833\) −1.43752 −0.0498073
\(834\) 0 0
\(835\) −32.5706 −1.12715
\(836\) 0 0
\(837\) 0 0
\(838\) −1.25853 −0.0434753
\(839\) 43.8528 1.51397 0.756984 0.653434i \(-0.226672\pi\)
0.756984 + 0.653434i \(0.226672\pi\)
\(840\) 0 0
\(841\) 13.4802 0.464836
\(842\) 4.86160 0.167542
\(843\) 0 0
\(844\) −14.7455 −0.507562
\(845\) 33.3604 1.14763
\(846\) 0 0
\(847\) 0 0
\(848\) −24.3232 −0.835261
\(849\) 0 0
\(850\) −0.399209 −0.0136928
\(851\) 28.2207 0.967392
\(852\) 0 0
\(853\) −39.1407 −1.34015 −0.670076 0.742293i \(-0.733738\pi\)
−0.670076 + 0.742293i \(0.733738\pi\)
\(854\) −0.193488 −0.00662102
\(855\) 0 0
\(856\) −1.04257 −0.0356342
\(857\) 35.0524 1.19737 0.598684 0.800986i \(-0.295691\pi\)
0.598684 + 0.800986i \(0.295691\pi\)
\(858\) 0 0
\(859\) −32.5206 −1.10959 −0.554794 0.831988i \(-0.687203\pi\)
−0.554794 + 0.831988i \(0.687203\pi\)
\(860\) −3.90186 −0.133052
\(861\) 0 0
\(862\) −6.27030 −0.213567
\(863\) −12.6940 −0.432107 −0.216054 0.976381i \(-0.569319\pi\)
−0.216054 + 0.976381i \(0.569319\pi\)
\(864\) 0 0
\(865\) 14.7540 0.501651
\(866\) 2.10108 0.0713975
\(867\) 0 0
\(868\) 14.9844 0.508605
\(869\) 0 0
\(870\) 0 0
\(871\) 8.45568 0.286510
\(872\) 8.30950 0.281395
\(873\) 0 0
\(874\) −9.72314 −0.328890
\(875\) 9.41402 0.318252
\(876\) 0 0
\(877\) 28.1340 0.950017 0.475008 0.879981i \(-0.342445\pi\)
0.475008 + 0.879981i \(0.342445\pi\)
\(878\) 3.32798 0.112314
\(879\) 0 0
\(880\) 0 0
\(881\) 36.7964 1.23970 0.619850 0.784720i \(-0.287193\pi\)
0.619850 + 0.784720i \(0.287193\pi\)
\(882\) 0 0
\(883\) −2.28419 −0.0768692 −0.0384346 0.999261i \(-0.512237\pi\)
−0.0384346 + 0.999261i \(0.512237\pi\)
\(884\) −14.3856 −0.483840
\(885\) 0 0
\(886\) −0.552736 −0.0185695
\(887\) 42.2676 1.41921 0.709604 0.704600i \(-0.248874\pi\)
0.709604 + 0.704600i \(0.248874\pi\)
\(888\) 0 0
\(889\) −0.289205 −0.00969963
\(890\) −0.979808 −0.0328432
\(891\) 0 0
\(892\) −34.1520 −1.14349
\(893\) −40.9757 −1.37120
\(894\) 0 0
\(895\) −10.8183 −0.361616
\(896\) 6.78282 0.226598
\(897\) 0 0
\(898\) −1.07888 −0.0360027
\(899\) 50.1142 1.67140
\(900\) 0 0
\(901\) 9.46132 0.315202
\(902\) 0 0
\(903\) 0 0
\(904\) 2.94540 0.0979627
\(905\) 27.0283 0.898451
\(906\) 0 0
\(907\) 39.5286 1.31252 0.656262 0.754533i \(-0.272136\pi\)
0.656262 + 0.754533i \(0.272136\pi\)
\(908\) 50.0406 1.66065
\(909\) 0 0
\(910\) 2.89878 0.0960936
\(911\) 35.2296 1.16721 0.583604 0.812039i \(-0.301642\pi\)
0.583604 + 0.812039i \(0.301642\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −7.03234 −0.232609
\(915\) 0 0
\(916\) 38.6587 1.27732
\(917\) 16.5059 0.545072
\(918\) 0 0
\(919\) 39.6105 1.30663 0.653314 0.757087i \(-0.273378\pi\)
0.653314 + 0.757087i \(0.273378\pi\)
\(920\) 15.7891 0.520551
\(921\) 0 0
\(922\) 6.72334 0.221421
\(923\) 23.2141 0.764101
\(924\) 0 0
\(925\) −4.89131 −0.160825
\(926\) −5.75828 −0.189229
\(927\) 0 0
\(928\) 17.0928 0.561098
\(929\) −2.61657 −0.0858469 −0.0429234 0.999078i \(-0.513667\pi\)
−0.0429234 + 0.999078i \(0.513667\pi\)
\(930\) 0 0
\(931\) 6.06848 0.198886
\(932\) 39.3947 1.29042
\(933\) 0 0
\(934\) 0.720463 0.0235743
\(935\) 0 0
\(936\) 0 0
\(937\) −53.8401 −1.75888 −0.879439 0.476011i \(-0.842082\pi\)
−0.879439 + 0.476011i \(0.842082\pi\)
\(938\) 0.372498 0.0121625
\(939\) 0 0
\(940\) 32.8384 1.07107
\(941\) 29.6476 0.966484 0.483242 0.875487i \(-0.339459\pi\)
0.483242 + 0.875487i \(0.339459\pi\)
\(942\) 0 0
\(943\) −47.8026 −1.55667
\(944\) −10.6294 −0.345959
\(945\) 0 0
\(946\) 0 0
\(947\) 15.7861 0.512980 0.256490 0.966547i \(-0.417434\pi\)
0.256490 + 0.966547i \(0.417434\pi\)
\(948\) 0 0
\(949\) −76.2440 −2.47498
\(950\) 1.68525 0.0546768
\(951\) 0 0
\(952\) −1.28410 −0.0416178
\(953\) −35.1119 −1.13739 −0.568693 0.822550i \(-0.692551\pi\)
−0.568693 + 0.822550i \(0.692551\pi\)
\(954\) 0 0
\(955\) −29.0869 −0.941229
\(956\) −33.3257 −1.07783
\(957\) 0 0
\(958\) 0.731463 0.0236325
\(959\) 9.32588 0.301148
\(960\) 0 0
\(961\) 28.1200 0.907096
\(962\) 4.62816 0.149218
\(963\) 0 0
\(964\) −47.0698 −1.51602
\(965\) −56.0131 −1.80313
\(966\) 0 0
\(967\) 49.2820 1.58480 0.792401 0.610001i \(-0.208831\pi\)
0.792401 + 0.610001i \(0.208831\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −6.81797 −0.218912
\(971\) −37.3424 −1.19838 −0.599188 0.800608i \(-0.704510\pi\)
−0.599188 + 0.800608i \(0.704510\pi\)
\(972\) 0 0
\(973\) 4.82649 0.154730
\(974\) 2.23292 0.0715475
\(975\) 0 0
\(976\) 3.16099 0.101181
\(977\) −40.0485 −1.28126 −0.640632 0.767848i \(-0.721328\pi\)
−0.640632 + 0.767848i \(0.721328\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −4.86335 −0.155354
\(981\) 0 0
\(982\) 0.955970 0.0305062
\(983\) 52.0414 1.65986 0.829932 0.557864i \(-0.188379\pi\)
0.829932 + 0.557864i \(0.188379\pi\)
\(984\) 0 0
\(985\) 60.1475 1.91646
\(986\) −2.11945 −0.0674970
\(987\) 0 0
\(988\) 60.7285 1.93203
\(989\) −5.68262 −0.180697
\(990\) 0 0
\(991\) 45.4828 1.44481 0.722404 0.691471i \(-0.243037\pi\)
0.722404 + 0.691471i \(0.243037\pi\)
\(992\) 20.1645 0.640222
\(993\) 0 0
\(994\) 1.02265 0.0324365
\(995\) 46.7175 1.48104
\(996\) 0 0
\(997\) −27.9594 −0.885482 −0.442741 0.896650i \(-0.645994\pi\)
−0.442741 + 0.896650i \(0.645994\pi\)
\(998\) −4.61364 −0.146042
\(999\) 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.cw.1.4 8
3.2 odd 2 847.2.a.o.1.5 8
11.2 odd 10 693.2.m.i.631.2 16
11.6 odd 10 693.2.m.i.190.2 16
11.10 odd 2 7623.2.a.ct.1.5 8
21.20 even 2 5929.2.a.bs.1.5 8
33.2 even 10 77.2.f.b.15.3 16
33.5 odd 10 847.2.f.x.729.2 16
33.8 even 10 847.2.f.w.372.2 16
33.14 odd 10 847.2.f.v.372.3 16
33.17 even 10 77.2.f.b.36.3 yes 16
33.20 odd 10 847.2.f.x.323.2 16
33.26 odd 10 847.2.f.v.148.3 16
33.29 even 10 847.2.f.w.148.2 16
33.32 even 2 847.2.a.p.1.4 8
231.2 even 30 539.2.q.g.312.3 32
231.17 odd 30 539.2.q.f.520.3 32
231.68 odd 30 539.2.q.f.312.3 32
231.83 odd 10 539.2.f.e.344.3 16
231.101 odd 30 539.2.q.f.422.2 32
231.116 even 30 539.2.q.g.520.3 32
231.149 even 30 539.2.q.g.410.2 32
231.167 odd 10 539.2.f.e.246.3 16
231.200 even 30 539.2.q.g.422.2 32
231.215 odd 30 539.2.q.f.410.2 32
231.230 odd 2 5929.2.a.bt.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.f.b.15.3 16 33.2 even 10
77.2.f.b.36.3 yes 16 33.17 even 10
539.2.f.e.246.3 16 231.167 odd 10
539.2.f.e.344.3 16 231.83 odd 10
539.2.q.f.312.3 32 231.68 odd 30
539.2.q.f.410.2 32 231.215 odd 30
539.2.q.f.422.2 32 231.101 odd 30
539.2.q.f.520.3 32 231.17 odd 30
539.2.q.g.312.3 32 231.2 even 30
539.2.q.g.410.2 32 231.149 even 30
539.2.q.g.422.2 32 231.200 even 30
539.2.q.g.520.3 32 231.116 even 30
693.2.m.i.190.2 16 11.6 odd 10
693.2.m.i.631.2 16 11.2 odd 10
847.2.a.o.1.5 8 3.2 odd 2
847.2.a.p.1.4 8 33.32 even 2
847.2.f.v.148.3 16 33.26 odd 10
847.2.f.v.372.3 16 33.14 odd 10
847.2.f.w.148.2 16 33.29 even 10
847.2.f.w.372.2 16 33.8 even 10
847.2.f.x.323.2 16 33.20 odd 10
847.2.f.x.729.2 16 33.5 odd 10
5929.2.a.bs.1.5 8 21.20 even 2
5929.2.a.bt.1.4 8 231.230 odd 2
7623.2.a.ct.1.5 8 11.10 odd 2
7623.2.a.cw.1.4 8 1.1 even 1 trivial