Properties

Label 8752.2.a.s.1.11
Level $8752$
Weight $2$
Character 8752.1
Self dual yes
Analytic conductor $69.885$
Analytic rank $1$
Dimension $18$
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] = [8752,2,Mod(1,8752)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8752, 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("8752.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8752 = 2^{4} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8752.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: \(69.8850718490\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + \cdots + 328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-1.87675\) of defining polynomial
Character \(\chi\) \(=\) 8752.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.13919 q^{3} -1.30620 q^{5} +1.71403 q^{7} -1.70224 q^{9} +O(q^{10})\) \(q+1.13919 q^{3} -1.30620 q^{5} +1.71403 q^{7} -1.70224 q^{9} +5.30835 q^{11} +2.18823 q^{13} -1.48801 q^{15} +0.392924 q^{17} -0.498929 q^{19} +1.95261 q^{21} -8.33591 q^{23} -3.29383 q^{25} -5.35675 q^{27} -4.50948 q^{29} +2.96386 q^{31} +6.04722 q^{33} -2.23887 q^{35} +3.06843 q^{37} +2.49281 q^{39} -10.0885 q^{41} -9.93284 q^{43} +2.22348 q^{45} -0.714605 q^{47} -4.06209 q^{49} +0.447615 q^{51} -10.3693 q^{53} -6.93378 q^{55} -0.568376 q^{57} +6.58293 q^{59} -0.889139 q^{61} -2.91770 q^{63} -2.85827 q^{65} -4.67997 q^{67} -9.49619 q^{69} -11.4239 q^{71} +5.35047 q^{73} -3.75231 q^{75} +9.09868 q^{77} +7.79987 q^{79} -0.995634 q^{81} +17.9975 q^{83} -0.513238 q^{85} -5.13716 q^{87} +13.3683 q^{89} +3.75069 q^{91} +3.37640 q^{93} +0.651703 q^{95} -13.2660 q^{97} -9.03610 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) 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
\(3\) 1.13919 0.657712 0.328856 0.944380i \(-0.393337\pi\)
0.328856 + 0.944380i \(0.393337\pi\)
\(4\) 0 0
\(5\) −1.30620 −0.584152 −0.292076 0.956395i \(-0.594346\pi\)
−0.292076 + 0.956395i \(0.594346\pi\)
\(6\) 0 0
\(7\) 1.71403 0.647843 0.323922 0.946084i \(-0.394999\pi\)
0.323922 + 0.946084i \(0.394999\pi\)
\(8\) 0 0
\(9\) −1.70224 −0.567415
\(10\) 0 0
\(11\) 5.30835 1.60053 0.800263 0.599649i \(-0.204693\pi\)
0.800263 + 0.599649i \(0.204693\pi\)
\(12\) 0 0
\(13\) 2.18823 0.606905 0.303452 0.952847i \(-0.401861\pi\)
0.303452 + 0.952847i \(0.401861\pi\)
\(14\) 0 0
\(15\) −1.48801 −0.384204
\(16\) 0 0
\(17\) 0.392924 0.0952980 0.0476490 0.998864i \(-0.484827\pi\)
0.0476490 + 0.998864i \(0.484827\pi\)
\(18\) 0 0
\(19\) −0.498929 −0.114462 −0.0572311 0.998361i \(-0.518227\pi\)
−0.0572311 + 0.998361i \(0.518227\pi\)
\(20\) 0 0
\(21\) 1.95261 0.426094
\(22\) 0 0
\(23\) −8.33591 −1.73816 −0.869079 0.494674i \(-0.835287\pi\)
−0.869079 + 0.494674i \(0.835287\pi\)
\(24\) 0 0
\(25\) −3.29383 −0.658767
\(26\) 0 0
\(27\) −5.35675 −1.03091
\(28\) 0 0
\(29\) −4.50948 −0.837389 −0.418695 0.908127i \(-0.637512\pi\)
−0.418695 + 0.908127i \(0.637512\pi\)
\(30\) 0 0
\(31\) 2.96386 0.532324 0.266162 0.963928i \(-0.414244\pi\)
0.266162 + 0.963928i \(0.414244\pi\)
\(32\) 0 0
\(33\) 6.04722 1.05269
\(34\) 0 0
\(35\) −2.23887 −0.378439
\(36\) 0 0
\(37\) 3.06843 0.504447 0.252223 0.967669i \(-0.418838\pi\)
0.252223 + 0.967669i \(0.418838\pi\)
\(38\) 0 0
\(39\) 2.49281 0.399169
\(40\) 0 0
\(41\) −10.0885 −1.57556 −0.787780 0.615957i \(-0.788770\pi\)
−0.787780 + 0.615957i \(0.788770\pi\)
\(42\) 0 0
\(43\) −9.93284 −1.51474 −0.757372 0.652984i \(-0.773517\pi\)
−0.757372 + 0.652984i \(0.773517\pi\)
\(44\) 0 0
\(45\) 2.22348 0.331456
\(46\) 0 0
\(47\) −0.714605 −0.104236 −0.0521179 0.998641i \(-0.516597\pi\)
−0.0521179 + 0.998641i \(0.516597\pi\)
\(48\) 0 0
\(49\) −4.06209 −0.580299
\(50\) 0 0
\(51\) 0.447615 0.0626787
\(52\) 0 0
\(53\) −10.3693 −1.42433 −0.712165 0.702012i \(-0.752285\pi\)
−0.712165 + 0.702012i \(0.752285\pi\)
\(54\) 0 0
\(55\) −6.93378 −0.934950
\(56\) 0 0
\(57\) −0.568376 −0.0752832
\(58\) 0 0
\(59\) 6.58293 0.857024 0.428512 0.903536i \(-0.359038\pi\)
0.428512 + 0.903536i \(0.359038\pi\)
\(60\) 0 0
\(61\) −0.889139 −0.113843 −0.0569213 0.998379i \(-0.518128\pi\)
−0.0569213 + 0.998379i \(0.518128\pi\)
\(62\) 0 0
\(63\) −2.91770 −0.367596
\(64\) 0 0
\(65\) −2.85827 −0.354524
\(66\) 0 0
\(67\) −4.67997 −0.571750 −0.285875 0.958267i \(-0.592284\pi\)
−0.285875 + 0.958267i \(0.592284\pi\)
\(68\) 0 0
\(69\) −9.49619 −1.14321
\(70\) 0 0
\(71\) −11.4239 −1.35577 −0.677886 0.735167i \(-0.737104\pi\)
−0.677886 + 0.735167i \(0.737104\pi\)
\(72\) 0 0
\(73\) 5.35047 0.626225 0.313113 0.949716i \(-0.398628\pi\)
0.313113 + 0.949716i \(0.398628\pi\)
\(74\) 0 0
\(75\) −3.75231 −0.433279
\(76\) 0 0
\(77\) 9.09868 1.03689
\(78\) 0 0
\(79\) 7.79987 0.877554 0.438777 0.898596i \(-0.355412\pi\)
0.438777 + 0.898596i \(0.355412\pi\)
\(80\) 0 0
\(81\) −0.995634 −0.110626
\(82\) 0 0
\(83\) 17.9975 1.97548 0.987740 0.156107i \(-0.0498946\pi\)
0.987740 + 0.156107i \(0.0498946\pi\)
\(84\) 0 0
\(85\) −0.513238 −0.0556685
\(86\) 0 0
\(87\) −5.13716 −0.550761
\(88\) 0 0
\(89\) 13.3683 1.41704 0.708519 0.705691i \(-0.249363\pi\)
0.708519 + 0.705691i \(0.249363\pi\)
\(90\) 0 0
\(91\) 3.75069 0.393179
\(92\) 0 0
\(93\) 3.37640 0.350116
\(94\) 0 0
\(95\) 0.651703 0.0668633
\(96\) 0 0
\(97\) −13.2660 −1.34696 −0.673480 0.739206i \(-0.735201\pi\)
−0.673480 + 0.739206i \(0.735201\pi\)
\(98\) 0 0
\(99\) −9.03610 −0.908162
\(100\) 0 0
\(101\) −5.06820 −0.504305 −0.252152 0.967688i \(-0.581138\pi\)
−0.252152 + 0.967688i \(0.581138\pi\)
\(102\) 0 0
\(103\) 1.26789 0.124929 0.0624644 0.998047i \(-0.480104\pi\)
0.0624644 + 0.998047i \(0.480104\pi\)
\(104\) 0 0
\(105\) −2.55050 −0.248904
\(106\) 0 0
\(107\) −11.5228 −1.11395 −0.556974 0.830530i \(-0.688038\pi\)
−0.556974 + 0.830530i \(0.688038\pi\)
\(108\) 0 0
\(109\) 1.84704 0.176914 0.0884570 0.996080i \(-0.471806\pi\)
0.0884570 + 0.996080i \(0.471806\pi\)
\(110\) 0 0
\(111\) 3.49553 0.331781
\(112\) 0 0
\(113\) −10.8186 −1.01773 −0.508865 0.860846i \(-0.669935\pi\)
−0.508865 + 0.860846i \(0.669935\pi\)
\(114\) 0 0
\(115\) 10.8884 1.01535
\(116\) 0 0
\(117\) −3.72489 −0.344367
\(118\) 0 0
\(119\) 0.673484 0.0617382
\(120\) 0 0
\(121\) 17.1786 1.56169
\(122\) 0 0
\(123\) −11.4927 −1.03626
\(124\) 0 0
\(125\) 10.8334 0.968971
\(126\) 0 0
\(127\) −16.9759 −1.50637 −0.753185 0.657809i \(-0.771483\pi\)
−0.753185 + 0.657809i \(0.771483\pi\)
\(128\) 0 0
\(129\) −11.3154 −0.996266
\(130\) 0 0
\(131\) −0.273878 −0.0239289 −0.0119644 0.999928i \(-0.503808\pi\)
−0.0119644 + 0.999928i \(0.503808\pi\)
\(132\) 0 0
\(133\) −0.855181 −0.0741536
\(134\) 0 0
\(135\) 6.99701 0.602206
\(136\) 0 0
\(137\) −5.63213 −0.481185 −0.240593 0.970626i \(-0.577342\pi\)
−0.240593 + 0.970626i \(0.577342\pi\)
\(138\) 0 0
\(139\) 22.7017 1.92553 0.962767 0.270333i \(-0.0871337\pi\)
0.962767 + 0.270333i \(0.0871337\pi\)
\(140\) 0 0
\(141\) −0.814072 −0.0685572
\(142\) 0 0
\(143\) 11.6159 0.971367
\(144\) 0 0
\(145\) 5.89029 0.489162
\(146\) 0 0
\(147\) −4.62750 −0.381670
\(148\) 0 0
\(149\) −3.23862 −0.265318 −0.132659 0.991162i \(-0.542352\pi\)
−0.132659 + 0.991162i \(0.542352\pi\)
\(150\) 0 0
\(151\) −18.3287 −1.49157 −0.745783 0.666189i \(-0.767924\pi\)
−0.745783 + 0.666189i \(0.767924\pi\)
\(152\) 0 0
\(153\) −0.668852 −0.0540735
\(154\) 0 0
\(155\) −3.87140 −0.310958
\(156\) 0 0
\(157\) −2.10134 −0.167705 −0.0838527 0.996478i \(-0.526723\pi\)
−0.0838527 + 0.996478i \(0.526723\pi\)
\(158\) 0 0
\(159\) −11.8126 −0.936799
\(160\) 0 0
\(161\) −14.2880 −1.12605
\(162\) 0 0
\(163\) 23.0706 1.80703 0.903514 0.428558i \(-0.140978\pi\)
0.903514 + 0.428558i \(0.140978\pi\)
\(164\) 0 0
\(165\) −7.89890 −0.614928
\(166\) 0 0
\(167\) 11.3441 0.877832 0.438916 0.898528i \(-0.355363\pi\)
0.438916 + 0.898528i \(0.355363\pi\)
\(168\) 0 0
\(169\) −8.21167 −0.631667
\(170\) 0 0
\(171\) 0.849299 0.0649476
\(172\) 0 0
\(173\) −17.2109 −1.30852 −0.654259 0.756271i \(-0.727019\pi\)
−0.654259 + 0.756271i \(0.727019\pi\)
\(174\) 0 0
\(175\) −5.64574 −0.426778
\(176\) 0 0
\(177\) 7.49921 0.563675
\(178\) 0 0
\(179\) 11.9899 0.896168 0.448084 0.893991i \(-0.352107\pi\)
0.448084 + 0.893991i \(0.352107\pi\)
\(180\) 0 0
\(181\) 1.09690 0.0815319 0.0407660 0.999169i \(-0.487020\pi\)
0.0407660 + 0.999169i \(0.487020\pi\)
\(182\) 0 0
\(183\) −1.01290 −0.0748756
\(184\) 0 0
\(185\) −4.00799 −0.294673
\(186\) 0 0
\(187\) 2.08578 0.152527
\(188\) 0 0
\(189\) −9.18165 −0.667867
\(190\) 0 0
\(191\) −1.44468 −0.104533 −0.0522667 0.998633i \(-0.516645\pi\)
−0.0522667 + 0.998633i \(0.516645\pi\)
\(192\) 0 0
\(193\) 1.41359 0.101753 0.0508763 0.998705i \(-0.483799\pi\)
0.0508763 + 0.998705i \(0.483799\pi\)
\(194\) 0 0
\(195\) −3.25611 −0.233175
\(196\) 0 0
\(197\) −5.14237 −0.366379 −0.183189 0.983078i \(-0.558642\pi\)
−0.183189 + 0.983078i \(0.558642\pi\)
\(198\) 0 0
\(199\) 16.7502 1.18739 0.593695 0.804690i \(-0.297668\pi\)
0.593695 + 0.804690i \(0.297668\pi\)
\(200\) 0 0
\(201\) −5.33139 −0.376047
\(202\) 0 0
\(203\) −7.72939 −0.542497
\(204\) 0 0
\(205\) 13.1776 0.920366
\(206\) 0 0
\(207\) 14.1897 0.986256
\(208\) 0 0
\(209\) −2.64849 −0.183200
\(210\) 0 0
\(211\) 11.7475 0.808734 0.404367 0.914597i \(-0.367492\pi\)
0.404367 + 0.914597i \(0.367492\pi\)
\(212\) 0 0
\(213\) −13.0141 −0.891708
\(214\) 0 0
\(215\) 12.9743 0.884840
\(216\) 0 0
\(217\) 5.08014 0.344863
\(218\) 0 0
\(219\) 6.09521 0.411876
\(220\) 0 0
\(221\) 0.859806 0.0578368
\(222\) 0 0
\(223\) −26.9958 −1.80778 −0.903888 0.427770i \(-0.859299\pi\)
−0.903888 + 0.427770i \(0.859299\pi\)
\(224\) 0 0
\(225\) 5.60691 0.373794
\(226\) 0 0
\(227\) 6.16279 0.409039 0.204519 0.978863i \(-0.434437\pi\)
0.204519 + 0.978863i \(0.434437\pi\)
\(228\) 0 0
\(229\) 5.04203 0.333187 0.166593 0.986026i \(-0.446723\pi\)
0.166593 + 0.986026i \(0.446723\pi\)
\(230\) 0 0
\(231\) 10.3651 0.681975
\(232\) 0 0
\(233\) −8.21008 −0.537860 −0.268930 0.963160i \(-0.586670\pi\)
−0.268930 + 0.963160i \(0.586670\pi\)
\(234\) 0 0
\(235\) 0.933419 0.0608896
\(236\) 0 0
\(237\) 8.88554 0.577178
\(238\) 0 0
\(239\) −25.4229 −1.64447 −0.822235 0.569148i \(-0.807273\pi\)
−0.822235 + 0.569148i \(0.807273\pi\)
\(240\) 0 0
\(241\) 26.7739 1.72466 0.862328 0.506350i \(-0.169006\pi\)
0.862328 + 0.506350i \(0.169006\pi\)
\(242\) 0 0
\(243\) 14.9360 0.958148
\(244\) 0 0
\(245\) 5.30592 0.338983
\(246\) 0 0
\(247\) −1.09177 −0.0694677
\(248\) 0 0
\(249\) 20.5026 1.29930
\(250\) 0 0
\(251\) −23.7572 −1.49954 −0.749770 0.661699i \(-0.769836\pi\)
−0.749770 + 0.661699i \(0.769836\pi\)
\(252\) 0 0
\(253\) −44.2499 −2.78197
\(254\) 0 0
\(255\) −0.584676 −0.0366138
\(256\) 0 0
\(257\) −6.92184 −0.431773 −0.215886 0.976418i \(-0.569264\pi\)
−0.215886 + 0.976418i \(0.569264\pi\)
\(258\) 0 0
\(259\) 5.25939 0.326802
\(260\) 0 0
\(261\) 7.67623 0.475147
\(262\) 0 0
\(263\) 9.02135 0.556281 0.278140 0.960540i \(-0.410282\pi\)
0.278140 + 0.960540i \(0.410282\pi\)
\(264\) 0 0
\(265\) 13.5444 0.832025
\(266\) 0 0
\(267\) 15.2291 0.932004
\(268\) 0 0
\(269\) 7.43191 0.453131 0.226566 0.973996i \(-0.427250\pi\)
0.226566 + 0.973996i \(0.427250\pi\)
\(270\) 0 0
\(271\) −24.3754 −1.48070 −0.740351 0.672220i \(-0.765341\pi\)
−0.740351 + 0.672220i \(0.765341\pi\)
\(272\) 0 0
\(273\) 4.27275 0.258599
\(274\) 0 0
\(275\) −17.4848 −1.05437
\(276\) 0 0
\(277\) 15.8453 0.952052 0.476026 0.879431i \(-0.342077\pi\)
0.476026 + 0.879431i \(0.342077\pi\)
\(278\) 0 0
\(279\) −5.04520 −0.302049
\(280\) 0 0
\(281\) 5.55329 0.331281 0.165641 0.986186i \(-0.447031\pi\)
0.165641 + 0.986186i \(0.447031\pi\)
\(282\) 0 0
\(283\) 21.5322 1.27996 0.639978 0.768394i \(-0.278944\pi\)
0.639978 + 0.768394i \(0.278944\pi\)
\(284\) 0 0
\(285\) 0.742414 0.0439768
\(286\) 0 0
\(287\) −17.2920 −1.02072
\(288\) 0 0
\(289\) −16.8456 −0.990918
\(290\) 0 0
\(291\) −15.1125 −0.885912
\(292\) 0 0
\(293\) −23.9442 −1.39884 −0.699418 0.714713i \(-0.746557\pi\)
−0.699418 + 0.714713i \(0.746557\pi\)
\(294\) 0 0
\(295\) −8.59863 −0.500632
\(296\) 0 0
\(297\) −28.4355 −1.65000
\(298\) 0 0
\(299\) −18.2408 −1.05490
\(300\) 0 0
\(301\) −17.0252 −0.981316
\(302\) 0 0
\(303\) −5.77365 −0.331687
\(304\) 0 0
\(305\) 1.16140 0.0665013
\(306\) 0 0
\(307\) −1.31760 −0.0751996 −0.0375998 0.999293i \(-0.511971\pi\)
−0.0375998 + 0.999293i \(0.511971\pi\)
\(308\) 0 0
\(309\) 1.44437 0.0821672
\(310\) 0 0
\(311\) 21.9974 1.24736 0.623679 0.781680i \(-0.285637\pi\)
0.623679 + 0.781680i \(0.285637\pi\)
\(312\) 0 0
\(313\) −0.891821 −0.0504087 −0.0252044 0.999682i \(-0.508024\pi\)
−0.0252044 + 0.999682i \(0.508024\pi\)
\(314\) 0 0
\(315\) 3.81111 0.214732
\(316\) 0 0
\(317\) −24.3281 −1.36640 −0.683202 0.730229i \(-0.739413\pi\)
−0.683202 + 0.730229i \(0.739413\pi\)
\(318\) 0 0
\(319\) −23.9379 −1.34026
\(320\) 0 0
\(321\) −13.1266 −0.732657
\(322\) 0 0
\(323\) −0.196041 −0.0109080
\(324\) 0 0
\(325\) −7.20765 −0.399809
\(326\) 0 0
\(327\) 2.10413 0.116359
\(328\) 0 0
\(329\) −1.22486 −0.0675285
\(330\) 0 0
\(331\) −27.7538 −1.52549 −0.762745 0.646700i \(-0.776149\pi\)
−0.762745 + 0.646700i \(0.776149\pi\)
\(332\) 0 0
\(333\) −5.22322 −0.286231
\(334\) 0 0
\(335\) 6.11300 0.333989
\(336\) 0 0
\(337\) 6.10740 0.332691 0.166346 0.986068i \(-0.446803\pi\)
0.166346 + 0.986068i \(0.446803\pi\)
\(338\) 0 0
\(339\) −12.3245 −0.669374
\(340\) 0 0
\(341\) 15.7332 0.851999
\(342\) 0 0
\(343\) −18.9608 −1.02379
\(344\) 0 0
\(345\) 12.4040 0.667806
\(346\) 0 0
\(347\) −27.7441 −1.48938 −0.744691 0.667409i \(-0.767403\pi\)
−0.744691 + 0.667409i \(0.767403\pi\)
\(348\) 0 0
\(349\) −14.3589 −0.768616 −0.384308 0.923205i \(-0.625560\pi\)
−0.384308 + 0.923205i \(0.625560\pi\)
\(350\) 0 0
\(351\) −11.7218 −0.625663
\(352\) 0 0
\(353\) 1.89459 0.100839 0.0504194 0.998728i \(-0.483944\pi\)
0.0504194 + 0.998728i \(0.483944\pi\)
\(354\) 0 0
\(355\) 14.9220 0.791977
\(356\) 0 0
\(357\) 0.767227 0.0406060
\(358\) 0 0
\(359\) −10.0371 −0.529740 −0.264870 0.964284i \(-0.585329\pi\)
−0.264870 + 0.964284i \(0.585329\pi\)
\(360\) 0 0
\(361\) −18.7511 −0.986898
\(362\) 0 0
\(363\) 19.5697 1.02714
\(364\) 0 0
\(365\) −6.98880 −0.365810
\(366\) 0 0
\(367\) −5.02023 −0.262054 −0.131027 0.991379i \(-0.541827\pi\)
−0.131027 + 0.991379i \(0.541827\pi\)
\(368\) 0 0
\(369\) 17.1731 0.893996
\(370\) 0 0
\(371\) −17.7733 −0.922742
\(372\) 0 0
\(373\) −3.12699 −0.161910 −0.0809548 0.996718i \(-0.525797\pi\)
−0.0809548 + 0.996718i \(0.525797\pi\)
\(374\) 0 0
\(375\) 12.3413 0.637304
\(376\) 0 0
\(377\) −9.86776 −0.508215
\(378\) 0 0
\(379\) −16.8372 −0.864867 −0.432433 0.901666i \(-0.642345\pi\)
−0.432433 + 0.901666i \(0.642345\pi\)
\(380\) 0 0
\(381\) −19.3388 −0.990757
\(382\) 0 0
\(383\) 3.31954 0.169621 0.0848104 0.996397i \(-0.472972\pi\)
0.0848104 + 0.996397i \(0.472972\pi\)
\(384\) 0 0
\(385\) −11.8847 −0.605701
\(386\) 0 0
\(387\) 16.9081 0.859488
\(388\) 0 0
\(389\) −13.8035 −0.699867 −0.349933 0.936775i \(-0.613796\pi\)
−0.349933 + 0.936775i \(0.613796\pi\)
\(390\) 0 0
\(391\) −3.27538 −0.165643
\(392\) 0 0
\(393\) −0.312000 −0.0157383
\(394\) 0 0
\(395\) −10.1882 −0.512625
\(396\) 0 0
\(397\) 13.6460 0.684875 0.342438 0.939541i \(-0.388747\pi\)
0.342438 + 0.939541i \(0.388747\pi\)
\(398\) 0 0
\(399\) −0.974214 −0.0487717
\(400\) 0 0
\(401\) −4.21927 −0.210700 −0.105350 0.994435i \(-0.533596\pi\)
−0.105350 + 0.994435i \(0.533596\pi\)
\(402\) 0 0
\(403\) 6.48558 0.323070
\(404\) 0 0
\(405\) 1.30050 0.0646224
\(406\) 0 0
\(407\) 16.2883 0.807381
\(408\) 0 0
\(409\) 3.32926 0.164621 0.0823105 0.996607i \(-0.473770\pi\)
0.0823105 + 0.996607i \(0.473770\pi\)
\(410\) 0 0
\(411\) −6.41607 −0.316481
\(412\) 0 0
\(413\) 11.2833 0.555217
\(414\) 0 0
\(415\) −23.5084 −1.15398
\(416\) 0 0
\(417\) 25.8616 1.26645
\(418\) 0 0
\(419\) 17.2091 0.840718 0.420359 0.907358i \(-0.361904\pi\)
0.420359 + 0.907358i \(0.361904\pi\)
\(420\) 0 0
\(421\) 5.90690 0.287884 0.143942 0.989586i \(-0.454022\pi\)
0.143942 + 0.989586i \(0.454022\pi\)
\(422\) 0 0
\(423\) 1.21643 0.0591450
\(424\) 0 0
\(425\) −1.29423 −0.0627792
\(426\) 0 0
\(427\) −1.52401 −0.0737521
\(428\) 0 0
\(429\) 13.2327 0.638880
\(430\) 0 0
\(431\) 27.0646 1.30366 0.651829 0.758366i \(-0.274002\pi\)
0.651829 + 0.758366i \(0.274002\pi\)
\(432\) 0 0
\(433\) −5.65104 −0.271571 −0.135786 0.990738i \(-0.543356\pi\)
−0.135786 + 0.990738i \(0.543356\pi\)
\(434\) 0 0
\(435\) 6.71017 0.321728
\(436\) 0 0
\(437\) 4.15903 0.198953
\(438\) 0 0
\(439\) 26.7713 1.27772 0.638861 0.769322i \(-0.279406\pi\)
0.638861 + 0.769322i \(0.279406\pi\)
\(440\) 0 0
\(441\) 6.91468 0.329270
\(442\) 0 0
\(443\) −34.1995 −1.62487 −0.812435 0.583052i \(-0.801858\pi\)
−0.812435 + 0.583052i \(0.801858\pi\)
\(444\) 0 0
\(445\) −17.4617 −0.827765
\(446\) 0 0
\(447\) −3.68940 −0.174503
\(448\) 0 0
\(449\) −7.11797 −0.335918 −0.167959 0.985794i \(-0.553718\pi\)
−0.167959 + 0.985794i \(0.553718\pi\)
\(450\) 0 0
\(451\) −53.5533 −2.52173
\(452\) 0 0
\(453\) −20.8799 −0.981021
\(454\) 0 0
\(455\) −4.89916 −0.229676
\(456\) 0 0
\(457\) 25.8336 1.20845 0.604223 0.796815i \(-0.293484\pi\)
0.604223 + 0.796815i \(0.293484\pi\)
\(458\) 0 0
\(459\) −2.10480 −0.0982435
\(460\) 0 0
\(461\) −38.8134 −1.80772 −0.903860 0.427829i \(-0.859279\pi\)
−0.903860 + 0.427829i \(0.859279\pi\)
\(462\) 0 0
\(463\) −22.8908 −1.06383 −0.531913 0.846799i \(-0.678527\pi\)
−0.531913 + 0.846799i \(0.678527\pi\)
\(464\) 0 0
\(465\) −4.41026 −0.204521
\(466\) 0 0
\(467\) 12.1901 0.564089 0.282045 0.959401i \(-0.408987\pi\)
0.282045 + 0.959401i \(0.408987\pi\)
\(468\) 0 0
\(469\) −8.02163 −0.370404
\(470\) 0 0
\(471\) −2.39383 −0.110302
\(472\) 0 0
\(473\) −52.7270 −2.42439
\(474\) 0 0
\(475\) 1.64339 0.0754039
\(476\) 0 0
\(477\) 17.6510 0.808186
\(478\) 0 0
\(479\) 35.1228 1.60480 0.802400 0.596786i \(-0.203556\pi\)
0.802400 + 0.596786i \(0.203556\pi\)
\(480\) 0 0
\(481\) 6.71442 0.306151
\(482\) 0 0
\(483\) −16.2768 −0.740619
\(484\) 0 0
\(485\) 17.3281 0.786828
\(486\) 0 0
\(487\) −7.59211 −0.344031 −0.172016 0.985094i \(-0.555028\pi\)
−0.172016 + 0.985094i \(0.555028\pi\)
\(488\) 0 0
\(489\) 26.2818 1.18850
\(490\) 0 0
\(491\) −18.2927 −0.825537 −0.412768 0.910836i \(-0.635438\pi\)
−0.412768 + 0.910836i \(0.635438\pi\)
\(492\) 0 0
\(493\) −1.77188 −0.0798015
\(494\) 0 0
\(495\) 11.8030 0.530504
\(496\) 0 0
\(497\) −19.5810 −0.878328
\(498\) 0 0
\(499\) −5.76535 −0.258093 −0.129046 0.991639i \(-0.541192\pi\)
−0.129046 + 0.991639i \(0.541192\pi\)
\(500\) 0 0
\(501\) 12.9231 0.577361
\(502\) 0 0
\(503\) 21.1805 0.944394 0.472197 0.881493i \(-0.343461\pi\)
0.472197 + 0.881493i \(0.343461\pi\)
\(504\) 0 0
\(505\) 6.62010 0.294590
\(506\) 0 0
\(507\) −9.35466 −0.415455
\(508\) 0 0
\(509\) 18.1907 0.806289 0.403145 0.915136i \(-0.367917\pi\)
0.403145 + 0.915136i \(0.367917\pi\)
\(510\) 0 0
\(511\) 9.17088 0.405696
\(512\) 0 0
\(513\) 2.67264 0.118000
\(514\) 0 0
\(515\) −1.65612 −0.0729773
\(516\) 0 0
\(517\) −3.79337 −0.166832
\(518\) 0 0
\(519\) −19.6065 −0.860628
\(520\) 0 0
\(521\) 21.4553 0.939976 0.469988 0.882673i \(-0.344258\pi\)
0.469988 + 0.882673i \(0.344258\pi\)
\(522\) 0 0
\(523\) −11.6184 −0.508038 −0.254019 0.967199i \(-0.581753\pi\)
−0.254019 + 0.967199i \(0.581753\pi\)
\(524\) 0 0
\(525\) −6.43157 −0.280697
\(526\) 0 0
\(527\) 1.16457 0.0507294
\(528\) 0 0
\(529\) 46.4874 2.02119
\(530\) 0 0
\(531\) −11.2057 −0.486288
\(532\) 0 0
\(533\) −22.0759 −0.956214
\(534\) 0 0
\(535\) 15.0511 0.650714
\(536\) 0 0
\(537\) 13.6588 0.589421
\(538\) 0 0
\(539\) −21.5630 −0.928785
\(540\) 0 0
\(541\) 15.8443 0.681198 0.340599 0.940209i \(-0.389370\pi\)
0.340599 + 0.940209i \(0.389370\pi\)
\(542\) 0 0
\(543\) 1.24958 0.0536245
\(544\) 0 0
\(545\) −2.41260 −0.103345
\(546\) 0 0
\(547\) 1.00000 0.0427569
\(548\) 0 0
\(549\) 1.51353 0.0645959
\(550\) 0 0
\(551\) 2.24991 0.0958495
\(552\) 0 0
\(553\) 13.3692 0.568517
\(554\) 0 0
\(555\) −4.56587 −0.193810
\(556\) 0 0
\(557\) 11.0799 0.469471 0.234735 0.972059i \(-0.424578\pi\)
0.234735 + 0.972059i \(0.424578\pi\)
\(558\) 0 0
\(559\) −21.7353 −0.919305
\(560\) 0 0
\(561\) 2.37610 0.100319
\(562\) 0 0
\(563\) −41.1603 −1.73470 −0.867351 0.497697i \(-0.834179\pi\)
−0.867351 + 0.497697i \(0.834179\pi\)
\(564\) 0 0
\(565\) 14.1313 0.594509
\(566\) 0 0
\(567\) −1.70655 −0.0716683
\(568\) 0 0
\(569\) −9.52628 −0.399363 −0.199681 0.979861i \(-0.563991\pi\)
−0.199681 + 0.979861i \(0.563991\pi\)
\(570\) 0 0
\(571\) −4.76393 −0.199364 −0.0996822 0.995019i \(-0.531783\pi\)
−0.0996822 + 0.995019i \(0.531783\pi\)
\(572\) 0 0
\(573\) −1.64577 −0.0687530
\(574\) 0 0
\(575\) 27.4571 1.14504
\(576\) 0 0
\(577\) 33.8345 1.40855 0.704274 0.709929i \(-0.251273\pi\)
0.704274 + 0.709929i \(0.251273\pi\)
\(578\) 0 0
\(579\) 1.61035 0.0669240
\(580\) 0 0
\(581\) 30.8483 1.27980
\(582\) 0 0
\(583\) −55.0437 −2.27968
\(584\) 0 0
\(585\) 4.86547 0.201162
\(586\) 0 0
\(587\) 12.2101 0.503964 0.251982 0.967732i \(-0.418918\pi\)
0.251982 + 0.967732i \(0.418918\pi\)
\(588\) 0 0
\(589\) −1.47875 −0.0609310
\(590\) 0 0
\(591\) −5.85814 −0.240972
\(592\) 0 0
\(593\) 32.7830 1.34624 0.673118 0.739535i \(-0.264954\pi\)
0.673118 + 0.739535i \(0.264954\pi\)
\(594\) 0 0
\(595\) −0.879707 −0.0360645
\(596\) 0 0
\(597\) 19.0817 0.780961
\(598\) 0 0
\(599\) 26.1846 1.06987 0.534936 0.844892i \(-0.320336\pi\)
0.534936 + 0.844892i \(0.320336\pi\)
\(600\) 0 0
\(601\) −14.1011 −0.575196 −0.287598 0.957751i \(-0.592857\pi\)
−0.287598 + 0.957751i \(0.592857\pi\)
\(602\) 0 0
\(603\) 7.96646 0.324419
\(604\) 0 0
\(605\) −22.4387 −0.912262
\(606\) 0 0
\(607\) −16.3651 −0.664239 −0.332120 0.943237i \(-0.607764\pi\)
−0.332120 + 0.943237i \(0.607764\pi\)
\(608\) 0 0
\(609\) −8.80525 −0.356807
\(610\) 0 0
\(611\) −1.56372 −0.0632612
\(612\) 0 0
\(613\) −13.3441 −0.538961 −0.269481 0.963006i \(-0.586852\pi\)
−0.269481 + 0.963006i \(0.586852\pi\)
\(614\) 0 0
\(615\) 15.0118 0.605336
\(616\) 0 0
\(617\) 34.5802 1.39215 0.696074 0.717971i \(-0.254929\pi\)
0.696074 + 0.717971i \(0.254929\pi\)
\(618\) 0 0
\(619\) 20.0240 0.804832 0.402416 0.915457i \(-0.368171\pi\)
0.402416 + 0.915457i \(0.368171\pi\)
\(620\) 0 0
\(621\) 44.6534 1.79188
\(622\) 0 0
\(623\) 22.9137 0.918019
\(624\) 0 0
\(625\) 2.31852 0.0927408
\(626\) 0 0
\(627\) −3.01714 −0.120493
\(628\) 0 0
\(629\) 1.20566 0.0480728
\(630\) 0 0
\(631\) 1.64737 0.0655808 0.0327904 0.999462i \(-0.489561\pi\)
0.0327904 + 0.999462i \(0.489561\pi\)
\(632\) 0 0
\(633\) 13.3827 0.531914
\(634\) 0 0
\(635\) 22.1740 0.879948
\(636\) 0 0
\(637\) −8.88878 −0.352186
\(638\) 0 0
\(639\) 19.4463 0.769285
\(640\) 0 0
\(641\) 30.9733 1.22337 0.611686 0.791100i \(-0.290491\pi\)
0.611686 + 0.791100i \(0.290491\pi\)
\(642\) 0 0
\(643\) −24.8226 −0.978907 −0.489454 0.872029i \(-0.662804\pi\)
−0.489454 + 0.872029i \(0.662804\pi\)
\(644\) 0 0
\(645\) 14.7802 0.581970
\(646\) 0 0
\(647\) −36.5122 −1.43544 −0.717721 0.696331i \(-0.754815\pi\)
−0.717721 + 0.696331i \(0.754815\pi\)
\(648\) 0 0
\(649\) 34.9445 1.37169
\(650\) 0 0
\(651\) 5.78725 0.226820
\(652\) 0 0
\(653\) −26.5164 −1.03767 −0.518834 0.854875i \(-0.673634\pi\)
−0.518834 + 0.854875i \(0.673634\pi\)
\(654\) 0 0
\(655\) 0.357740 0.0139781
\(656\) 0 0
\(657\) −9.10781 −0.355329
\(658\) 0 0
\(659\) 25.7938 1.00478 0.502391 0.864641i \(-0.332454\pi\)
0.502391 + 0.864641i \(0.332454\pi\)
\(660\) 0 0
\(661\) −41.9540 −1.63182 −0.815910 0.578179i \(-0.803764\pi\)
−0.815910 + 0.578179i \(0.803764\pi\)
\(662\) 0 0
\(663\) 0.979483 0.0380400
\(664\) 0 0
\(665\) 1.11704 0.0433169
\(666\) 0 0
\(667\) 37.5906 1.45551
\(668\) 0 0
\(669\) −30.7534 −1.18900
\(670\) 0 0
\(671\) −4.71986 −0.182208
\(672\) 0 0
\(673\) −3.20648 −0.123601 −0.0618003 0.998089i \(-0.519684\pi\)
−0.0618003 + 0.998089i \(0.519684\pi\)
\(674\) 0 0
\(675\) 17.6443 0.679128
\(676\) 0 0
\(677\) 4.84844 0.186341 0.0931704 0.995650i \(-0.470300\pi\)
0.0931704 + 0.995650i \(0.470300\pi\)
\(678\) 0 0
\(679\) −22.7384 −0.872618
\(680\) 0 0
\(681\) 7.02059 0.269030
\(682\) 0 0
\(683\) 32.3263 1.23693 0.618466 0.785812i \(-0.287755\pi\)
0.618466 + 0.785812i \(0.287755\pi\)
\(684\) 0 0
\(685\) 7.35670 0.281085
\(686\) 0 0
\(687\) 5.74384 0.219141
\(688\) 0 0
\(689\) −22.6903 −0.864432
\(690\) 0 0
\(691\) 9.59431 0.364985 0.182492 0.983207i \(-0.441584\pi\)
0.182492 + 0.983207i \(0.441584\pi\)
\(692\) 0 0
\(693\) −15.4882 −0.588347
\(694\) 0 0
\(695\) −29.6530 −1.12480
\(696\) 0 0
\(697\) −3.96401 −0.150148
\(698\) 0 0
\(699\) −9.35285 −0.353757
\(700\) 0 0
\(701\) −48.0675 −1.81549 −0.907743 0.419527i \(-0.862196\pi\)
−0.907743 + 0.419527i \(0.862196\pi\)
\(702\) 0 0
\(703\) −1.53093 −0.0577401
\(704\) 0 0
\(705\) 1.06334 0.0400478
\(706\) 0 0
\(707\) −8.68706 −0.326710
\(708\) 0 0
\(709\) 12.6771 0.476099 0.238049 0.971253i \(-0.423492\pi\)
0.238049 + 0.971253i \(0.423492\pi\)
\(710\) 0 0
\(711\) −13.2773 −0.497937
\(712\) 0 0
\(713\) −24.7064 −0.925263
\(714\) 0 0
\(715\) −15.1727 −0.567426
\(716\) 0 0
\(717\) −28.9615 −1.08159
\(718\) 0 0
\(719\) −14.9369 −0.557053 −0.278526 0.960429i \(-0.589846\pi\)
−0.278526 + 0.960429i \(0.589846\pi\)
\(720\) 0 0
\(721\) 2.17320 0.0809343
\(722\) 0 0
\(723\) 30.5005 1.13433
\(724\) 0 0
\(725\) 14.8535 0.551644
\(726\) 0 0
\(727\) 16.9039 0.626932 0.313466 0.949599i \(-0.398510\pi\)
0.313466 + 0.949599i \(0.398510\pi\)
\(728\) 0 0
\(729\) 20.0019 0.740811
\(730\) 0 0
\(731\) −3.90285 −0.144352
\(732\) 0 0
\(733\) 17.0731 0.630609 0.315305 0.948991i \(-0.397893\pi\)
0.315305 + 0.948991i \(0.397893\pi\)
\(734\) 0 0
\(735\) 6.04445 0.222953
\(736\) 0 0
\(737\) −24.8429 −0.915101
\(738\) 0 0
\(739\) 0.313121 0.0115183 0.00575917 0.999983i \(-0.498167\pi\)
0.00575917 + 0.999983i \(0.498167\pi\)
\(740\) 0 0
\(741\) −1.24373 −0.0456897
\(742\) 0 0
\(743\) 19.0657 0.699451 0.349726 0.936852i \(-0.386275\pi\)
0.349726 + 0.936852i \(0.386275\pi\)
\(744\) 0 0
\(745\) 4.23029 0.154986
\(746\) 0 0
\(747\) −30.6361 −1.12092
\(748\) 0 0
\(749\) −19.7504 −0.721663
\(750\) 0 0
\(751\) 14.1901 0.517803 0.258901 0.965904i \(-0.416640\pi\)
0.258901 + 0.965904i \(0.416640\pi\)
\(752\) 0 0
\(753\) −27.0640 −0.986266
\(754\) 0 0
\(755\) 23.9410 0.871301
\(756\) 0 0
\(757\) −32.0979 −1.16662 −0.583309 0.812250i \(-0.698242\pi\)
−0.583309 + 0.812250i \(0.698242\pi\)
\(758\) 0 0
\(759\) −50.4091 −1.82973
\(760\) 0 0
\(761\) 27.3107 0.990013 0.495006 0.868889i \(-0.335166\pi\)
0.495006 + 0.868889i \(0.335166\pi\)
\(762\) 0 0
\(763\) 3.16588 0.114613
\(764\) 0 0
\(765\) 0.873656 0.0315871
\(766\) 0 0
\(767\) 14.4049 0.520132
\(768\) 0 0
\(769\) 15.5549 0.560923 0.280461 0.959865i \(-0.409513\pi\)
0.280461 + 0.959865i \(0.409513\pi\)
\(770\) 0 0
\(771\) −7.88530 −0.283982
\(772\) 0 0
\(773\) −49.6932 −1.78734 −0.893670 0.448724i \(-0.851878\pi\)
−0.893670 + 0.448724i \(0.851878\pi\)
\(774\) 0 0
\(775\) −9.76245 −0.350678
\(776\) 0 0
\(777\) 5.99145 0.214942
\(778\) 0 0
\(779\) 5.03345 0.180342
\(780\) 0 0
\(781\) −60.6423 −2.16995
\(782\) 0 0
\(783\) 24.1562 0.863271
\(784\) 0 0
\(785\) 2.74478 0.0979654
\(786\) 0 0
\(787\) 14.9449 0.532729 0.266365 0.963872i \(-0.414177\pi\)
0.266365 + 0.963872i \(0.414177\pi\)
\(788\) 0 0
\(789\) 10.2770 0.365873
\(790\) 0 0
\(791\) −18.5435 −0.659330
\(792\) 0 0
\(793\) −1.94564 −0.0690916
\(794\) 0 0
\(795\) 15.4296 0.547233
\(796\) 0 0
\(797\) −13.1154 −0.464573 −0.232286 0.972647i \(-0.574621\pi\)
−0.232286 + 0.972647i \(0.574621\pi\)
\(798\) 0 0
\(799\) −0.280785 −0.00993347
\(800\) 0 0
\(801\) −22.7561 −0.804049
\(802\) 0 0
\(803\) 28.4022 1.00229
\(804\) 0 0
\(805\) 18.6630 0.657786
\(806\) 0 0
\(807\) 8.46636 0.298030
\(808\) 0 0
\(809\) −19.4909 −0.685262 −0.342631 0.939470i \(-0.611318\pi\)
−0.342631 + 0.939470i \(0.611318\pi\)
\(810\) 0 0
\(811\) −18.5367 −0.650912 −0.325456 0.945557i \(-0.605518\pi\)
−0.325456 + 0.945557i \(0.605518\pi\)
\(812\) 0 0
\(813\) −27.7683 −0.973876
\(814\) 0 0
\(815\) −30.1349 −1.05558
\(816\) 0 0
\(817\) 4.95579 0.173381
\(818\) 0 0
\(819\) −6.38459 −0.223095
\(820\) 0 0
\(821\) −12.1526 −0.424130 −0.212065 0.977256i \(-0.568019\pi\)
−0.212065 + 0.977256i \(0.568019\pi\)
\(822\) 0 0
\(823\) −15.5837 −0.543214 −0.271607 0.962408i \(-0.587555\pi\)
−0.271607 + 0.962408i \(0.587555\pi\)
\(824\) 0 0
\(825\) −19.9185 −0.693475
\(826\) 0 0
\(827\) −14.2481 −0.495456 −0.247728 0.968830i \(-0.579684\pi\)
−0.247728 + 0.968830i \(0.579684\pi\)
\(828\) 0 0
\(829\) 52.8914 1.83700 0.918498 0.395426i \(-0.129403\pi\)
0.918498 + 0.395426i \(0.129403\pi\)
\(830\) 0 0
\(831\) 18.0508 0.626176
\(832\) 0 0
\(833\) −1.59609 −0.0553014
\(834\) 0 0
\(835\) −14.8177 −0.512787
\(836\) 0 0
\(837\) −15.8766 −0.548777
\(838\) 0 0
\(839\) 25.6169 0.884395 0.442198 0.896918i \(-0.354199\pi\)
0.442198 + 0.896918i \(0.354199\pi\)
\(840\) 0 0
\(841\) −8.66460 −0.298779
\(842\) 0 0
\(843\) 6.32625 0.217888
\(844\) 0 0
\(845\) 10.7261 0.368989
\(846\) 0 0
\(847\) 29.4446 1.01173
\(848\) 0 0
\(849\) 24.5293 0.841842
\(850\) 0 0
\(851\) −25.5782 −0.876808
\(852\) 0 0
\(853\) 23.7883 0.814498 0.407249 0.913317i \(-0.366488\pi\)
0.407249 + 0.913317i \(0.366488\pi\)
\(854\) 0 0
\(855\) −1.10936 −0.0379392
\(856\) 0 0
\(857\) −1.27557 −0.0435727 −0.0217863 0.999763i \(-0.506935\pi\)
−0.0217863 + 0.999763i \(0.506935\pi\)
\(858\) 0 0
\(859\) 38.6456 1.31857 0.659285 0.751893i \(-0.270859\pi\)
0.659285 + 0.751893i \(0.270859\pi\)
\(860\) 0 0
\(861\) −19.6989 −0.671337
\(862\) 0 0
\(863\) −13.3344 −0.453908 −0.226954 0.973905i \(-0.572877\pi\)
−0.226954 + 0.973905i \(0.572877\pi\)
\(864\) 0 0
\(865\) 22.4809 0.764373
\(866\) 0 0
\(867\) −19.1904 −0.651739
\(868\) 0 0
\(869\) 41.4044 1.40455
\(870\) 0 0
\(871\) −10.2408 −0.346998
\(872\) 0 0
\(873\) 22.5820 0.764284
\(874\) 0 0
\(875\) 18.5688 0.627741
\(876\) 0 0
\(877\) −5.39699 −0.182244 −0.0911218 0.995840i \(-0.529045\pi\)
−0.0911218 + 0.995840i \(0.529045\pi\)
\(878\) 0 0
\(879\) −27.2770 −0.920031
\(880\) 0 0
\(881\) −5.04786 −0.170067 −0.0850333 0.996378i \(-0.527100\pi\)
−0.0850333 + 0.996378i \(0.527100\pi\)
\(882\) 0 0
\(883\) 36.4145 1.22545 0.612723 0.790298i \(-0.290074\pi\)
0.612723 + 0.790298i \(0.290074\pi\)
\(884\) 0 0
\(885\) −9.79549 −0.329272
\(886\) 0 0
\(887\) 24.9346 0.837224 0.418612 0.908165i \(-0.362517\pi\)
0.418612 + 0.908165i \(0.362517\pi\)
\(888\) 0 0
\(889\) −29.0973 −0.975891
\(890\) 0 0
\(891\) −5.28517 −0.177060
\(892\) 0 0
\(893\) 0.356538 0.0119311
\(894\) 0 0
\(895\) −15.6613 −0.523498
\(896\) 0 0
\(897\) −20.7798 −0.693818
\(898\) 0 0
\(899\) −13.3654 −0.445763
\(900\) 0 0
\(901\) −4.07434 −0.135736
\(902\) 0 0
\(903\) −19.3950 −0.645424
\(904\) 0 0
\(905\) −1.43277 −0.0476270
\(906\) 0 0
\(907\) 27.1951 0.903000 0.451500 0.892271i \(-0.350889\pi\)
0.451500 + 0.892271i \(0.350889\pi\)
\(908\) 0 0
\(909\) 8.62731 0.286150
\(910\) 0 0
\(911\) 47.9588 1.58895 0.794473 0.607299i \(-0.207747\pi\)
0.794473 + 0.607299i \(0.207747\pi\)
\(912\) 0 0
\(913\) 95.5369 3.16181
\(914\) 0 0
\(915\) 1.32305 0.0437387
\(916\) 0 0
\(917\) −0.469436 −0.0155021
\(918\) 0 0
\(919\) 20.3674 0.671857 0.335928 0.941887i \(-0.390950\pi\)
0.335928 + 0.941887i \(0.390950\pi\)
\(920\) 0 0
\(921\) −1.50100 −0.0494597
\(922\) 0 0
\(923\) −24.9982 −0.822825
\(924\) 0 0
\(925\) −10.1069 −0.332313
\(926\) 0 0
\(927\) −2.15826 −0.0708864
\(928\) 0 0
\(929\) 3.43001 0.112535 0.0562675 0.998416i \(-0.482080\pi\)
0.0562675 + 0.998416i \(0.482080\pi\)
\(930\) 0 0
\(931\) 2.02670 0.0664224
\(932\) 0 0
\(933\) 25.0593 0.820403
\(934\) 0 0
\(935\) −2.72445 −0.0890989
\(936\) 0 0
\(937\) −55.8441 −1.82435 −0.912173 0.409805i \(-0.865597\pi\)
−0.912173 + 0.409805i \(0.865597\pi\)
\(938\) 0 0
\(939\) −1.01595 −0.0331544
\(940\) 0 0
\(941\) −35.2757 −1.14996 −0.574978 0.818169i \(-0.694989\pi\)
−0.574978 + 0.818169i \(0.694989\pi\)
\(942\) 0 0
\(943\) 84.0969 2.73857
\(944\) 0 0
\(945\) 11.9931 0.390135
\(946\) 0 0
\(947\) 45.2546 1.47058 0.735289 0.677754i \(-0.237046\pi\)
0.735289 + 0.677754i \(0.237046\pi\)
\(948\) 0 0
\(949\) 11.7080 0.380059
\(950\) 0 0
\(951\) −27.7144 −0.898701
\(952\) 0 0
\(953\) 39.0690 1.26557 0.632784 0.774328i \(-0.281912\pi\)
0.632784 + 0.774328i \(0.281912\pi\)
\(954\) 0 0
\(955\) 1.88705 0.0610634
\(956\) 0 0
\(957\) −27.2698 −0.881508
\(958\) 0 0
\(959\) −9.65365 −0.311733
\(960\) 0 0
\(961\) −22.2156 −0.716631
\(962\) 0 0
\(963\) 19.6146 0.632070
\(964\) 0 0
\(965\) −1.84644 −0.0594390
\(966\) 0 0
\(967\) −11.8771 −0.381942 −0.190971 0.981596i \(-0.561164\pi\)
−0.190971 + 0.981596i \(0.561164\pi\)
\(968\) 0 0
\(969\) −0.223328 −0.00717434
\(970\) 0 0
\(971\) 43.3430 1.39094 0.695471 0.718554i \(-0.255196\pi\)
0.695471 + 0.718554i \(0.255196\pi\)
\(972\) 0 0
\(973\) 38.9115 1.24744
\(974\) 0 0
\(975\) −8.21089 −0.262959
\(976\) 0 0
\(977\) 30.9084 0.988846 0.494423 0.869221i \(-0.335379\pi\)
0.494423 + 0.869221i \(0.335379\pi\)
\(978\) 0 0
\(979\) 70.9637 2.26801
\(980\) 0 0
\(981\) −3.14411 −0.100384
\(982\) 0 0
\(983\) −35.0128 −1.11673 −0.558367 0.829594i \(-0.688572\pi\)
−0.558367 + 0.829594i \(0.688572\pi\)
\(984\) 0 0
\(985\) 6.71698 0.214021
\(986\) 0 0
\(987\) −1.39535 −0.0444143
\(988\) 0 0
\(989\) 82.7992 2.63286
\(990\) 0 0
\(991\) 22.7329 0.722135 0.361067 0.932540i \(-0.382412\pi\)
0.361067 + 0.932540i \(0.382412\pi\)
\(992\) 0 0
\(993\) −31.6169 −1.00333
\(994\) 0 0
\(995\) −21.8792 −0.693616
\(996\) 0 0
\(997\) −10.6754 −0.338095 −0.169047 0.985608i \(-0.554069\pi\)
−0.169047 + 0.985608i \(0.554069\pi\)
\(998\) 0 0
\(999\) −16.4368 −0.520038
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 8752.2.a.s.1.11 18
4.3 odd 2 547.2.a.b.1.16 18
12.11 even 2 4923.2.a.l.1.3 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.16 18 4.3 odd 2
4923.2.a.l.1.3 18 12.11 even 2
8752.2.a.s.1.11 18 1.1 even 1 trivial