Properties

Label 4923.2.a.l.1.3
Level $4923$
Weight $2$
Character 4923.1
Self dual yes
Analytic conductor $39.310$
Analytic rank $0$
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] = [4923,2,Mod(1,4923)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4923, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4923.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4923 = 3^{2} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4923.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: \(39.3103529151\)
Analytic rank: \(0\)
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 1.3
Root \(-1.87675\) of defining polynomial
Character \(\chi\) \(=\) 4923.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.87675 q^{2} +1.52220 q^{4} +1.30620 q^{5} -1.71403 q^{7} +0.896704 q^{8} +O(q^{10})\) \(q-1.87675 q^{2} +1.52220 q^{4} +1.30620 q^{5} -1.71403 q^{7} +0.896704 q^{8} -2.45142 q^{10} +5.30835 q^{11} +2.18823 q^{13} +3.21682 q^{14} -4.72730 q^{16} -0.392924 q^{17} +0.498929 q^{19} +1.98831 q^{20} -9.96246 q^{22} -8.33591 q^{23} -3.29383 q^{25} -4.10676 q^{26} -2.60911 q^{28} +4.50948 q^{29} -2.96386 q^{31} +7.07857 q^{32} +0.737421 q^{34} -2.23887 q^{35} +3.06843 q^{37} -0.936368 q^{38} +1.17128 q^{40} +10.0885 q^{41} +9.93284 q^{43} +8.08039 q^{44} +15.6444 q^{46} -0.714605 q^{47} -4.06209 q^{49} +6.18172 q^{50} +3.33093 q^{52} +10.3693 q^{53} +6.93378 q^{55} -1.53698 q^{56} -8.46318 q^{58} +6.58293 q^{59} -0.889139 q^{61} +5.56243 q^{62} -3.83013 q^{64} +2.85827 q^{65} +4.67997 q^{67} -0.598110 q^{68} +4.20181 q^{70} -11.4239 q^{71} +5.35047 q^{73} -5.75869 q^{74} +0.759473 q^{76} -9.09868 q^{77} -7.79987 q^{79} -6.17481 q^{80} -18.9336 q^{82} +17.9975 q^{83} -0.513238 q^{85} -18.6415 q^{86} +4.76002 q^{88} -13.3683 q^{89} -3.75069 q^{91} -12.6890 q^{92} +1.34114 q^{94} +0.651703 q^{95} -13.2660 q^{97} +7.62355 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.87675 −1.32707 −0.663533 0.748147i \(-0.730944\pi\)
−0.663533 + 0.748147i \(0.730944\pi\)
\(3\) 0 0
\(4\) 1.52220 0.761102
\(5\) 1.30620 0.584152 0.292076 0.956395i \(-0.405654\pi\)
0.292076 + 0.956395i \(0.405654\pi\)
\(6\) 0 0
\(7\) −1.71403 −0.647843 −0.323922 0.946084i \(-0.605001\pi\)
−0.323922 + 0.946084i \(0.605001\pi\)
\(8\) 0.896704 0.317033
\(9\) 0 0
\(10\) −2.45142 −0.775207
\(11\) 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 \(0.401861\pi\)
\(14\) 3.21682 0.859730
\(15\) 0 0
\(16\) −4.72730 −1.18183
\(17\) −0.392924 −0.0952980 −0.0476490 0.998864i \(-0.515173\pi\)
−0.0476490 + 0.998864i \(0.515173\pi\)
\(18\) 0 0
\(19\) 0.498929 0.114462 0.0572311 0.998361i \(-0.481773\pi\)
0.0572311 + 0.998361i \(0.481773\pi\)
\(20\) 1.98831 0.444599
\(21\) 0 0
\(22\) −9.96246 −2.12400
\(23\) −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
\(26\) −4.10676 −0.805402
\(27\) 0 0
\(28\) −2.60911 −0.493075
\(29\) 4.50948 0.837389 0.418695 0.908127i \(-0.362488\pi\)
0.418695 + 0.908127i \(0.362488\pi\)
\(30\) 0 0
\(31\) −2.96386 −0.532324 −0.266162 0.963928i \(-0.585756\pi\)
−0.266162 + 0.963928i \(0.585756\pi\)
\(32\) 7.07857 1.25133
\(33\) 0 0
\(34\) 0.737421 0.126467
\(35\) −2.23887 −0.378439
\(36\) 0 0
\(37\) 3.06843 0.504447 0.252223 0.967669i \(-0.418838\pi\)
0.252223 + 0.967669i \(0.418838\pi\)
\(38\) −0.936368 −0.151899
\(39\) 0 0
\(40\) 1.17128 0.185195
\(41\) 10.0885 1.57556 0.787780 0.615957i \(-0.211230\pi\)
0.787780 + 0.615957i \(0.211230\pi\)
\(42\) 0 0
\(43\) 9.93284 1.51474 0.757372 0.652984i \(-0.226483\pi\)
0.757372 + 0.652984i \(0.226483\pi\)
\(44\) 8.08039 1.21816
\(45\) 0 0
\(46\) 15.6444 2.30665
\(47\) −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
\(50\) 6.18172 0.874227
\(51\) 0 0
\(52\) 3.33093 0.461916
\(53\) 10.3693 1.42433 0.712165 0.702012i \(-0.247715\pi\)
0.712165 + 0.702012i \(0.247715\pi\)
\(54\) 0 0
\(55\) 6.93378 0.934950
\(56\) −1.53698 −0.205388
\(57\) 0 0
\(58\) −8.46318 −1.11127
\(59\) 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 \(0.518128\pi\)
\(62\) 5.56243 0.706429
\(63\) 0 0
\(64\) −3.83013 −0.478767
\(65\) 2.85827 0.354524
\(66\) 0 0
\(67\) 4.67997 0.571750 0.285875 0.958267i \(-0.407716\pi\)
0.285875 + 0.958267i \(0.407716\pi\)
\(68\) −0.598110 −0.0725315
\(69\) 0 0
\(70\) 4.20181 0.502213
\(71\) −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 \(0.398628\pi\)
\(74\) −5.75869 −0.669434
\(75\) 0 0
\(76\) 0.759473 0.0871175
\(77\) −9.09868 −1.03689
\(78\) 0 0
\(79\) −7.79987 −0.877554 −0.438777 0.898596i \(-0.644588\pi\)
−0.438777 + 0.898596i \(0.644588\pi\)
\(80\) −6.17481 −0.690365
\(81\) 0 0
\(82\) −18.9336 −2.09087
\(83\) 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
\(86\) −18.6415 −2.01016
\(87\) 0 0
\(88\) 4.76002 0.507420
\(89\) −13.3683 −1.41704 −0.708519 0.705691i \(-0.750637\pi\)
−0.708519 + 0.705691i \(0.750637\pi\)
\(90\) 0 0
\(91\) −3.75069 −0.393179
\(92\) −12.6890 −1.32292
\(93\) 0 0
\(94\) 1.34114 0.138328
\(95\) 0.651703 0.0668633
\(96\) 0 0
\(97\) −13.2660 −1.34696 −0.673480 0.739206i \(-0.735201\pi\)
−0.673480 + 0.739206i \(0.735201\pi\)
\(98\) 7.62355 0.770095
\(99\) 0 0
\(100\) −5.01389 −0.501389
\(101\) 5.06820 0.504305 0.252152 0.967688i \(-0.418862\pi\)
0.252152 + 0.967688i \(0.418862\pi\)
\(102\) 0 0
\(103\) −1.26789 −0.124929 −0.0624644 0.998047i \(-0.519896\pi\)
−0.0624644 + 0.998047i \(0.519896\pi\)
\(104\) 1.96219 0.192409
\(105\) 0 0
\(106\) −19.4606 −1.89018
\(107\) −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 \(0.471806\pi\)
\(110\) −13.0130 −1.24074
\(111\) 0 0
\(112\) 8.10275 0.765638
\(113\) 10.8186 1.01773 0.508865 0.860846i \(-0.330065\pi\)
0.508865 + 0.860846i \(0.330065\pi\)
\(114\) 0 0
\(115\) −10.8884 −1.01535
\(116\) 6.86435 0.637339
\(117\) 0 0
\(118\) −12.3545 −1.13733
\(119\) 0.673484 0.0617382
\(120\) 0 0
\(121\) 17.1786 1.56169
\(122\) 1.66869 0.151077
\(123\) 0 0
\(124\) −4.51159 −0.405153
\(125\) −10.8334 −0.968971
\(126\) 0 0
\(127\) 16.9759 1.50637 0.753185 0.657809i \(-0.228517\pi\)
0.753185 + 0.657809i \(0.228517\pi\)
\(128\) −6.96893 −0.615972
\(129\) 0 0
\(130\) −5.36426 −0.470477
\(131\) −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
\(134\) −8.78316 −0.758749
\(135\) 0 0
\(136\) −0.352337 −0.0302126
\(137\) 5.63213 0.481185 0.240593 0.970626i \(-0.422658\pi\)
0.240593 + 0.970626i \(0.422658\pi\)
\(138\) 0 0
\(139\) −22.7017 −1.92553 −0.962767 0.270333i \(-0.912866\pi\)
−0.962767 + 0.270333i \(0.912866\pi\)
\(140\) −3.40802 −0.288030
\(141\) 0 0
\(142\) 21.4399 1.79920
\(143\) 11.6159 0.971367
\(144\) 0 0
\(145\) 5.89029 0.489162
\(146\) −10.0415 −0.831042
\(147\) 0 0
\(148\) 4.67078 0.383936
\(149\) 3.23862 0.265318 0.132659 0.991162i \(-0.457648\pi\)
0.132659 + 0.991162i \(0.457648\pi\)
\(150\) 0 0
\(151\) 18.3287 1.49157 0.745783 0.666189i \(-0.232076\pi\)
0.745783 + 0.666189i \(0.232076\pi\)
\(152\) 0.447392 0.0362883
\(153\) 0 0
\(154\) 17.0760 1.37602
\(155\) −3.87140 −0.310958
\(156\) 0 0
\(157\) −2.10134 −0.167705 −0.0838527 0.996478i \(-0.526723\pi\)
−0.0838527 + 0.996478i \(0.526723\pi\)
\(158\) 14.6384 1.16457
\(159\) 0 0
\(160\) 9.24605 0.730965
\(161\) 14.2880 1.12605
\(162\) 0 0
\(163\) −23.0706 −1.80703 −0.903514 0.428558i \(-0.859022\pi\)
−0.903514 + 0.428558i \(0.859022\pi\)
\(164\) 15.3568 1.19916
\(165\) 0 0
\(166\) −33.7768 −2.62159
\(167\) 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
\(170\) 0.963222 0.0738757
\(171\) 0 0
\(172\) 15.1198 1.15287
\(173\) 17.2109 1.30852 0.654259 0.756271i \(-0.272981\pi\)
0.654259 + 0.756271i \(0.272981\pi\)
\(174\) 0 0
\(175\) 5.64574 0.426778
\(176\) −25.0942 −1.89154
\(177\) 0 0
\(178\) 25.0890 1.88050
\(179\) 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 \(0.487020\pi\)
\(182\) 7.03912 0.521774
\(183\) 0 0
\(184\) −7.47485 −0.551053
\(185\) 4.00799 0.294673
\(186\) 0 0
\(187\) −2.08578 −0.152527
\(188\) −1.08778 −0.0793342
\(189\) 0 0
\(190\) −1.22309 −0.0887320
\(191\) −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 \(0.483799\pi\)
\(194\) 24.8970 1.78750
\(195\) 0 0
\(196\) −6.18334 −0.441667
\(197\) 5.14237 0.366379 0.183189 0.983078i \(-0.441358\pi\)
0.183189 + 0.983078i \(0.441358\pi\)
\(198\) 0 0
\(199\) −16.7502 −1.18739 −0.593695 0.804690i \(-0.702332\pi\)
−0.593695 + 0.804690i \(0.702332\pi\)
\(200\) −2.95360 −0.208851
\(201\) 0 0
\(202\) −9.51176 −0.669245
\(203\) −7.72939 −0.542497
\(204\) 0 0
\(205\) 13.1776 0.920366
\(206\) 2.37951 0.165789
\(207\) 0 0
\(208\) −10.3444 −0.717255
\(209\) 2.64849 0.183200
\(210\) 0 0
\(211\) −11.7475 −0.808734 −0.404367 0.914597i \(-0.632508\pi\)
−0.404367 + 0.914597i \(0.632508\pi\)
\(212\) 15.7842 1.08406
\(213\) 0 0
\(214\) 21.6254 1.47828
\(215\) 12.9743 0.884840
\(216\) 0 0
\(217\) 5.08014 0.344863
\(218\) −3.46643 −0.234777
\(219\) 0 0
\(220\) 10.5546 0.711593
\(221\) −0.859806 −0.0578368
\(222\) 0 0
\(223\) 26.9958 1.80778 0.903888 0.427770i \(-0.140701\pi\)
0.903888 + 0.427770i \(0.140701\pi\)
\(224\) −12.1329 −0.810664
\(225\) 0 0
\(226\) −20.3039 −1.35059
\(227\) 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 \(0.446723\pi\)
\(230\) 20.4348 1.34743
\(231\) 0 0
\(232\) 4.04367 0.265480
\(233\) 8.21008 0.537860 0.268930 0.963160i \(-0.413330\pi\)
0.268930 + 0.963160i \(0.413330\pi\)
\(234\) 0 0
\(235\) −0.933419 −0.0608896
\(236\) 10.0206 0.652283
\(237\) 0 0
\(238\) −1.26396 −0.0819306
\(239\) −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 \(0.169006\pi\)
\(242\) −32.2399 −2.07246
\(243\) 0 0
\(244\) −1.35345 −0.0866458
\(245\) −5.30592 −0.338983
\(246\) 0 0
\(247\) 1.09177 0.0694677
\(248\) −2.65770 −0.168764
\(249\) 0 0
\(250\) 20.3317 1.28589
\(251\) −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
\(254\) −31.8596 −1.99905
\(255\) 0 0
\(256\) 20.7392 1.29620
\(257\) 6.92184 0.431773 0.215886 0.976418i \(-0.430736\pi\)
0.215886 + 0.976418i \(0.430736\pi\)
\(258\) 0 0
\(259\) −5.25939 −0.326802
\(260\) 4.35087 0.269829
\(261\) 0 0
\(262\) 0.514002 0.0317551
\(263\) 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
\(266\) 1.60496 0.0984066
\(267\) 0 0
\(268\) 7.12388 0.435160
\(269\) −7.43191 −0.453131 −0.226566 0.973996i \(-0.572750\pi\)
−0.226566 + 0.973996i \(0.572750\pi\)
\(270\) 0 0
\(271\) 24.3754 1.48070 0.740351 0.672220i \(-0.234659\pi\)
0.740351 + 0.672220i \(0.234659\pi\)
\(272\) 1.85747 0.112626
\(273\) 0 0
\(274\) −10.5701 −0.638564
\(275\) −17.4848 −1.05437
\(276\) 0 0
\(277\) 15.8453 0.952052 0.476026 0.879431i \(-0.342077\pi\)
0.476026 + 0.879431i \(0.342077\pi\)
\(278\) 42.6055 2.55531
\(279\) 0 0
\(280\) −2.00761 −0.119977
\(281\) −5.55329 −0.331281 −0.165641 0.986186i \(-0.552969\pi\)
−0.165641 + 0.986186i \(0.552969\pi\)
\(282\) 0 0
\(283\) −21.5322 −1.27996 −0.639978 0.768394i \(-0.721056\pi\)
−0.639978 + 0.768394i \(0.721056\pi\)
\(284\) −17.3896 −1.03188
\(285\) 0 0
\(286\) −21.8001 −1.28907
\(287\) −17.2920 −1.02072
\(288\) 0 0
\(289\) −16.8456 −0.990918
\(290\) −11.0546 −0.649150
\(291\) 0 0
\(292\) 8.14451 0.476621
\(293\) 23.9442 1.39884 0.699418 0.714713i \(-0.253443\pi\)
0.699418 + 0.714713i \(0.253443\pi\)
\(294\) 0 0
\(295\) 8.59863 0.500632
\(296\) 2.75148 0.159926
\(297\) 0 0
\(298\) −6.07809 −0.352094
\(299\) −18.2408 −1.05490
\(300\) 0 0
\(301\) −17.0252 −0.981316
\(302\) −34.3984 −1.97941
\(303\) 0 0
\(304\) −2.35859 −0.135274
\(305\) −1.16140 −0.0665013
\(306\) 0 0
\(307\) 1.31760 0.0751996 0.0375998 0.999293i \(-0.488029\pi\)
0.0375998 + 0.999293i \(0.488029\pi\)
\(308\) −13.8500 −0.789180
\(309\) 0 0
\(310\) 7.26566 0.412662
\(311\) 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 \(0.508024\pi\)
\(314\) 3.94370 0.222556
\(315\) 0 0
\(316\) −11.8730 −0.667908
\(317\) 24.3281 1.36640 0.683202 0.730229i \(-0.260587\pi\)
0.683202 + 0.730229i \(0.260587\pi\)
\(318\) 0 0
\(319\) 23.9379 1.34026
\(320\) −5.00293 −0.279672
\(321\) 0 0
\(322\) −26.8151 −1.49435
\(323\) −0.196041 −0.0109080
\(324\) 0 0
\(325\) −7.20765 −0.399809
\(326\) 43.2978 2.39804
\(327\) 0 0
\(328\) 9.04641 0.499504
\(329\) 1.22486 0.0675285
\(330\) 0 0
\(331\) 27.7538 1.52549 0.762745 0.646700i \(-0.223851\pi\)
0.762745 + 0.646700i \(0.223851\pi\)
\(332\) 27.3958 1.50354
\(333\) 0 0
\(334\) −21.2901 −1.16494
\(335\) 6.11300 0.333989
\(336\) 0 0
\(337\) 6.10740 0.332691 0.166346 0.986068i \(-0.446803\pi\)
0.166346 + 0.986068i \(0.446803\pi\)
\(338\) 15.4113 0.838263
\(339\) 0 0
\(340\) −0.781253 −0.0423694
\(341\) −15.7332 −0.851999
\(342\) 0 0
\(343\) 18.9608 1.02379
\(344\) 8.90682 0.480224
\(345\) 0 0
\(346\) −32.3005 −1.73649
\(347\) −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 \(0.625560\pi\)
\(350\) −10.5957 −0.566362
\(351\) 0 0
\(352\) 37.5755 2.00278
\(353\) −1.89459 −0.100839 −0.0504194 0.998728i \(-0.516056\pi\)
−0.0504194 + 0.998728i \(0.516056\pi\)
\(354\) 0 0
\(355\) −14.9220 −0.791977
\(356\) −20.3493 −1.07851
\(357\) 0 0
\(358\) −22.5021 −1.18927
\(359\) −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
\(362\) −2.05861 −0.108198
\(363\) 0 0
\(364\) −5.70931 −0.299249
\(365\) 6.98880 0.365810
\(366\) 0 0
\(367\) 5.02023 0.262054 0.131027 0.991379i \(-0.458173\pi\)
0.131027 + 0.991379i \(0.458173\pi\)
\(368\) 39.4064 2.05420
\(369\) 0 0
\(370\) −7.52201 −0.391051
\(371\) −17.7733 −0.922742
\(372\) 0 0
\(373\) −3.12699 −0.161910 −0.0809548 0.996718i \(-0.525797\pi\)
−0.0809548 + 0.996718i \(0.525797\pi\)
\(374\) 3.91449 0.202413
\(375\) 0 0
\(376\) −0.640790 −0.0330462
\(377\) 9.86776 0.508215
\(378\) 0 0
\(379\) 16.8372 0.864867 0.432433 0.901666i \(-0.357655\pi\)
0.432433 + 0.901666i \(0.357655\pi\)
\(380\) 0.992025 0.0508898
\(381\) 0 0
\(382\) 2.71131 0.138723
\(383\) 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
\(386\) −2.65297 −0.135032
\(387\) 0 0
\(388\) −20.1936 −1.02517
\(389\) 13.8035 0.699867 0.349933 0.936775i \(-0.386204\pi\)
0.349933 + 0.936775i \(0.386204\pi\)
\(390\) 0 0
\(391\) 3.27538 0.165643
\(392\) −3.64250 −0.183974
\(393\) 0 0
\(394\) −9.65096 −0.486208
\(395\) −10.1882 −0.512625
\(396\) 0 0
\(397\) 13.6460 0.684875 0.342438 0.939541i \(-0.388747\pi\)
0.342438 + 0.939541i \(0.388747\pi\)
\(398\) 31.4360 1.57574
\(399\) 0 0
\(400\) 15.5710 0.778548
\(401\) 4.21927 0.210700 0.105350 0.994435i \(-0.466404\pi\)
0.105350 + 0.994435i \(0.466404\pi\)
\(402\) 0 0
\(403\) −6.48558 −0.323070
\(404\) 7.71484 0.383828
\(405\) 0 0
\(406\) 14.5062 0.719929
\(407\) 16.2883 0.807381
\(408\) 0 0
\(409\) 3.32926 0.164621 0.0823105 0.996607i \(-0.473770\pi\)
0.0823105 + 0.996607i \(0.473770\pi\)
\(410\) −24.7312 −1.22139
\(411\) 0 0
\(412\) −1.92999 −0.0950836
\(413\) −11.2833 −0.555217
\(414\) 0 0
\(415\) 23.5084 1.15398
\(416\) 15.4895 0.759436
\(417\) 0 0
\(418\) −4.97056 −0.243118
\(419\) 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 \(0.454022\pi\)
\(422\) 22.0472 1.07324
\(423\) 0 0
\(424\) 9.29818 0.451559
\(425\) 1.29423 0.0627792
\(426\) 0 0
\(427\) 1.52401 0.0737521
\(428\) −17.5400 −0.847828
\(429\) 0 0
\(430\) −24.3496 −1.17424
\(431\) 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 \(0.543356\pi\)
\(434\) −9.53418 −0.457655
\(435\) 0 0
\(436\) 2.81157 0.134650
\(437\) −4.15903 −0.198953
\(438\) 0 0
\(439\) −26.7713 −1.27772 −0.638861 0.769322i \(-0.720594\pi\)
−0.638861 + 0.769322i \(0.720594\pi\)
\(440\) 6.21755 0.296410
\(441\) 0 0
\(442\) 1.61364 0.0767532
\(443\) −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
\(446\) −50.6646 −2.39904
\(447\) 0 0
\(448\) 6.56497 0.310166
\(449\) 7.11797 0.335918 0.167959 0.985794i \(-0.446282\pi\)
0.167959 + 0.985794i \(0.446282\pi\)
\(450\) 0 0
\(451\) 53.5533 2.52173
\(452\) 16.4682 0.774597
\(453\) 0 0
\(454\) −11.5660 −0.542821
\(455\) −4.89916 −0.229676
\(456\) 0 0
\(457\) 25.8336 1.20845 0.604223 0.796815i \(-0.293484\pi\)
0.604223 + 0.796815i \(0.293484\pi\)
\(458\) −9.46265 −0.442161
\(459\) 0 0
\(460\) −16.5744 −0.772783
\(461\) 38.8134 1.80772 0.903860 0.427829i \(-0.140721\pi\)
0.903860 + 0.427829i \(0.140721\pi\)
\(462\) 0 0
\(463\) 22.8908 1.06383 0.531913 0.846799i \(-0.321473\pi\)
0.531913 + 0.846799i \(0.321473\pi\)
\(464\) −21.3177 −0.989648
\(465\) 0 0
\(466\) −15.4083 −0.713775
\(467\) 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
\(470\) 1.75180 0.0808044
\(471\) 0 0
\(472\) 5.90294 0.271705
\(473\) 52.7270 2.42439
\(474\) 0 0
\(475\) −1.64339 −0.0754039
\(476\) 1.02518 0.0469891
\(477\) 0 0
\(478\) 47.7125 2.18232
\(479\) 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
\(482\) −50.2479 −2.28873
\(483\) 0 0
\(484\) 26.1493 1.18860
\(485\) −17.3281 −0.786828
\(486\) 0 0
\(487\) 7.59211 0.344031 0.172016 0.985094i \(-0.444972\pi\)
0.172016 + 0.985094i \(0.444972\pi\)
\(488\) −0.797295 −0.0360918
\(489\) 0 0
\(490\) 9.95790 0.449852
\(491\) −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
\(494\) −2.04898 −0.0921881
\(495\) 0 0
\(496\) 14.0110 0.629114
\(497\) 19.5810 0.878328
\(498\) 0 0
\(499\) 5.76535 0.258093 0.129046 0.991639i \(-0.458808\pi\)
0.129046 + 0.991639i \(0.458808\pi\)
\(500\) −16.4907 −0.737486
\(501\) 0 0
\(502\) 44.5864 1.98999
\(503\) 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
\(506\) 83.0462 3.69185
\(507\) 0 0
\(508\) 25.8408 1.14650
\(509\) −18.1907 −0.806289 −0.403145 0.915136i \(-0.632083\pi\)
−0.403145 + 0.915136i \(0.632083\pi\)
\(510\) 0 0
\(511\) −9.17088 −0.405696
\(512\) −24.9846 −1.10417
\(513\) 0 0
\(514\) −12.9906 −0.572991
\(515\) −1.65612 −0.0729773
\(516\) 0 0
\(517\) −3.79337 −0.166832
\(518\) 9.87058 0.433688
\(519\) 0 0
\(520\) 2.56302 0.112396
\(521\) −21.4553 −0.939976 −0.469988 0.882673i \(-0.655742\pi\)
−0.469988 + 0.882673i \(0.655742\pi\)
\(522\) 0 0
\(523\) 11.6184 0.508038 0.254019 0.967199i \(-0.418247\pi\)
0.254019 + 0.967199i \(0.418247\pi\)
\(524\) −0.416899 −0.0182123
\(525\) 0 0
\(526\) −16.9309 −0.738221
\(527\) 1.16457 0.0507294
\(528\) 0 0
\(529\) 46.4874 2.02119
\(530\) −25.4195 −1.10415
\(531\) 0 0
\(532\) −1.30176 −0.0564385
\(533\) 22.0759 0.956214
\(534\) 0 0
\(535\) −15.0511 −0.650714
\(536\) 4.19655 0.181264
\(537\) 0 0
\(538\) 13.9479 0.601335
\(539\) −21.5630 −0.928785
\(540\) 0 0
\(541\) 15.8443 0.681198 0.340599 0.940209i \(-0.389370\pi\)
0.340599 + 0.940209i \(0.389370\pi\)
\(542\) −45.7467 −1.96499
\(543\) 0 0
\(544\) −2.78134 −0.119249
\(545\) 2.41260 0.103345
\(546\) 0 0
\(547\) −1.00000 −0.0427569
\(548\) 8.57325 0.366231
\(549\) 0 0
\(550\) 32.8147 1.39922
\(551\) 2.24991 0.0958495
\(552\) 0 0
\(553\) 13.3692 0.568517
\(554\) −29.7377 −1.26343
\(555\) 0 0
\(556\) −34.5566 −1.46553
\(557\) −11.0799 −0.469471 −0.234735 0.972059i \(-0.575422\pi\)
−0.234735 + 0.972059i \(0.575422\pi\)
\(558\) 0 0
\(559\) 21.7353 0.919305
\(560\) 10.5838 0.447248
\(561\) 0 0
\(562\) 10.4221 0.439632
\(563\) −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
\(566\) 40.4106 1.69858
\(567\) 0 0
\(568\) −10.2439 −0.429825
\(569\) 9.52628 0.399363 0.199681 0.979861i \(-0.436009\pi\)
0.199681 + 0.979861i \(0.436009\pi\)
\(570\) 0 0
\(571\) 4.76393 0.199364 0.0996822 0.995019i \(-0.468217\pi\)
0.0996822 + 0.995019i \(0.468217\pi\)
\(572\) 17.6817 0.739310
\(573\) 0 0
\(574\) 32.4529 1.35456
\(575\) 27.4571 1.14504
\(576\) 0 0
\(577\) 33.8345 1.40855 0.704274 0.709929i \(-0.251273\pi\)
0.704274 + 0.709929i \(0.251273\pi\)
\(578\) 31.6151 1.31501
\(579\) 0 0
\(580\) 8.96623 0.372303
\(581\) −30.8483 −1.27980
\(582\) 0 0
\(583\) 55.0437 2.27968
\(584\) 4.79779 0.198534
\(585\) 0 0
\(586\) −44.9374 −1.85635
\(587\) 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
\(590\) −16.1375 −0.664371
\(591\) 0 0
\(592\) −14.5054 −0.596168
\(593\) −32.7830 −1.34624 −0.673118 0.739535i \(-0.735046\pi\)
−0.673118 + 0.739535i \(0.735046\pi\)
\(594\) 0 0
\(595\) 0.879707 0.0360645
\(596\) 4.92984 0.201934
\(597\) 0 0
\(598\) 34.2336 1.39992
\(599\) 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 \(0.592857\pi\)
\(602\) 31.9521 1.30227
\(603\) 0 0
\(604\) 27.9000 1.13523
\(605\) 22.4387 0.912262
\(606\) 0 0
\(607\) 16.3651 0.664239 0.332120 0.943237i \(-0.392236\pi\)
0.332120 + 0.943237i \(0.392236\pi\)
\(608\) 3.53171 0.143230
\(609\) 0 0
\(610\) 2.17965 0.0882516
\(611\) −1.56372 −0.0632612
\(612\) 0 0
\(613\) −13.3441 −0.538961 −0.269481 0.963006i \(-0.586852\pi\)
−0.269481 + 0.963006i \(0.586852\pi\)
\(614\) −2.47282 −0.0997948
\(615\) 0 0
\(616\) −8.15882 −0.328728
\(617\) −34.5802 −1.39215 −0.696074 0.717971i \(-0.745071\pi\)
−0.696074 + 0.717971i \(0.745071\pi\)
\(618\) 0 0
\(619\) −20.0240 −0.804832 −0.402416 0.915457i \(-0.631829\pi\)
−0.402416 + 0.915457i \(0.631829\pi\)
\(620\) −5.89306 −0.236671
\(621\) 0 0
\(622\) −41.2837 −1.65533
\(623\) 22.9137 0.918019
\(624\) 0 0
\(625\) 2.31852 0.0927408
\(626\) 1.67373 0.0668957
\(627\) 0 0
\(628\) −3.19868 −0.127641
\(629\) −1.20566 −0.0480728
\(630\) 0 0
\(631\) −1.64737 −0.0655808 −0.0327904 0.999462i \(-0.510439\pi\)
−0.0327904 + 0.999462i \(0.510439\pi\)
\(632\) −6.99418 −0.278213
\(633\) 0 0
\(634\) −45.6579 −1.81331
\(635\) 22.1740 0.879948
\(636\) 0 0
\(637\) −8.88878 −0.352186
\(638\) −44.9255 −1.77862
\(639\) 0 0
\(640\) −9.10283 −0.359821
\(641\) −30.9733 −1.22337 −0.611686 0.791100i \(-0.709509\pi\)
−0.611686 + 0.791100i \(0.709509\pi\)
\(642\) 0 0
\(643\) 24.8226 0.978907 0.489454 0.872029i \(-0.337196\pi\)
0.489454 + 0.872029i \(0.337196\pi\)
\(644\) 21.7493 0.857042
\(645\) 0 0
\(646\) 0.367921 0.0144757
\(647\) −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
\(650\) 13.5270 0.530572
\(651\) 0 0
\(652\) −35.1182 −1.37533
\(653\) 26.5164 1.03767 0.518834 0.854875i \(-0.326366\pi\)
0.518834 + 0.854875i \(0.326366\pi\)
\(654\) 0 0
\(655\) −0.357740 −0.0139781
\(656\) −47.6914 −1.86204
\(657\) 0 0
\(658\) −2.29875 −0.0896147
\(659\) 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 \(0.803764\pi\)
\(662\) −52.0871 −2.02442
\(663\) 0 0
\(664\) 16.1384 0.626292
\(665\) −1.11704 −0.0433169
\(666\) 0 0
\(667\) −37.5906 −1.45551
\(668\) 17.2680 0.668120
\(669\) 0 0
\(670\) −11.4726 −0.443225
\(671\) −4.71986 −0.182208
\(672\) 0 0
\(673\) −3.20648 −0.123601 −0.0618003 0.998089i \(-0.519684\pi\)
−0.0618003 + 0.998089i \(0.519684\pi\)
\(674\) −11.4621 −0.441503
\(675\) 0 0
\(676\) −12.4998 −0.480763
\(677\) −4.84844 −0.186341 −0.0931704 0.995650i \(-0.529700\pi\)
−0.0931704 + 0.995650i \(0.529700\pi\)
\(678\) 0 0
\(679\) 22.7384 0.872618
\(680\) −0.460223 −0.0176487
\(681\) 0 0
\(682\) 29.5273 1.13066
\(683\) 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
\(686\) −35.5847 −1.35863
\(687\) 0 0
\(688\) −46.9555 −1.79016
\(689\) 22.6903 0.864432
\(690\) 0 0
\(691\) −9.59431 −0.364985 −0.182492 0.983207i \(-0.558416\pi\)
−0.182492 + 0.983207i \(0.558416\pi\)
\(692\) 26.1985 0.995916
\(693\) 0 0
\(694\) 52.0689 1.97651
\(695\) −29.6530 −1.12480
\(696\) 0 0
\(697\) −3.96401 −0.150148
\(698\) 26.9482 1.02000
\(699\) 0 0
\(700\) 8.59397 0.324821
\(701\) 48.0675 1.81549 0.907743 0.419527i \(-0.137804\pi\)
0.907743 + 0.419527i \(0.137804\pi\)
\(702\) 0 0
\(703\) 1.53093 0.0577401
\(704\) −20.3317 −0.766279
\(705\) 0 0
\(706\) 3.55568 0.133820
\(707\) −8.68706 −0.326710
\(708\) 0 0
\(709\) 12.6771 0.476099 0.238049 0.971253i \(-0.423492\pi\)
0.238049 + 0.971253i \(0.423492\pi\)
\(710\) 28.0049 1.05100
\(711\) 0 0
\(712\) −11.9874 −0.449248
\(713\) 24.7064 0.925263
\(714\) 0 0
\(715\) 15.1727 0.567426
\(716\) 18.2511 0.682076
\(717\) 0 0
\(718\) 18.8372 0.702999
\(719\) −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
\(722\) 35.1911 1.30968
\(723\) 0 0
\(724\) 1.66971 0.0620541
\(725\) −14.8535 −0.551644
\(726\) 0 0
\(727\) −16.9039 −0.626932 −0.313466 0.949599i \(-0.601490\pi\)
−0.313466 + 0.949599i \(0.601490\pi\)
\(728\) −3.36326 −0.124651
\(729\) 0 0
\(730\) −13.1163 −0.485454
\(731\) −3.90285 −0.144352
\(732\) 0 0
\(733\) 17.0731 0.630609 0.315305 0.948991i \(-0.397893\pi\)
0.315305 + 0.948991i \(0.397893\pi\)
\(734\) −9.42174 −0.347763
\(735\) 0 0
\(736\) −59.0063 −2.17500
\(737\) 24.8429 0.915101
\(738\) 0 0
\(739\) −0.313121 −0.0115183 −0.00575917 0.999983i \(-0.501833\pi\)
−0.00575917 + 0.999983i \(0.501833\pi\)
\(740\) 6.10098 0.224277
\(741\) 0 0
\(742\) 33.3561 1.22454
\(743\) 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
\(746\) 5.86860 0.214865
\(747\) 0 0
\(748\) −3.17498 −0.116089
\(749\) 19.7504 0.721663
\(750\) 0 0
\(751\) −14.1901 −0.517803 −0.258901 0.965904i \(-0.583360\pi\)
−0.258901 + 0.965904i \(0.583360\pi\)
\(752\) 3.37816 0.123189
\(753\) 0 0
\(754\) −18.5194 −0.674435
\(755\) 23.9410 0.871301
\(756\) 0 0
\(757\) −32.0979 −1.16662 −0.583309 0.812250i \(-0.698242\pi\)
−0.583309 + 0.812250i \(0.698242\pi\)
\(758\) −31.5992 −1.14773
\(759\) 0 0
\(760\) 0.584385 0.0211979
\(761\) −27.3107 −0.990013 −0.495006 0.868889i \(-0.664834\pi\)
−0.495006 + 0.868889i \(0.664834\pi\)
\(762\) 0 0
\(763\) −3.16588 −0.114613
\(764\) −2.19910 −0.0795607
\(765\) 0 0
\(766\) −6.22997 −0.225098
\(767\) 14.4049 0.520132
\(768\) 0 0
\(769\) 15.5549 0.560923 0.280461 0.959865i \(-0.409513\pi\)
0.280461 + 0.959865i \(0.409513\pi\)
\(770\) 22.3047 0.803805
\(771\) 0 0
\(772\) 2.15178 0.0774442
\(773\) 49.6932 1.78734 0.893670 0.448724i \(-0.148122\pi\)
0.893670 + 0.448724i \(0.148122\pi\)
\(774\) 0 0
\(775\) 9.76245 0.350678
\(776\) −11.8957 −0.427030
\(777\) 0 0
\(778\) −25.9058 −0.928769
\(779\) 5.03345 0.180342
\(780\) 0 0
\(781\) −60.6423 −2.16995
\(782\) −6.14708 −0.219819
\(783\) 0 0
\(784\) 19.2028 0.685813
\(785\) −2.74478 −0.0979654
\(786\) 0 0
\(787\) −14.9449 −0.532729 −0.266365 0.963872i \(-0.585823\pi\)
−0.266365 + 0.963872i \(0.585823\pi\)
\(788\) 7.82774 0.278852
\(789\) 0 0
\(790\) 19.1208 0.680286
\(791\) −18.5435 −0.659330
\(792\) 0 0
\(793\) −1.94564 −0.0690916
\(794\) −25.6103 −0.908874
\(795\) 0 0
\(796\) −25.4972 −0.903726
\(797\) 13.1154 0.464573 0.232286 0.972647i \(-0.425379\pi\)
0.232286 + 0.972647i \(0.425379\pi\)
\(798\) 0 0
\(799\) 0.280785 0.00993347
\(800\) −23.3157 −0.824333
\(801\) 0 0
\(802\) −7.91853 −0.279613
\(803\) 28.4022 1.00229
\(804\) 0 0
\(805\) 18.6630 0.657786
\(806\) 12.1718 0.428735
\(807\) 0 0
\(808\) 4.54468 0.159881
\(809\) 19.4909 0.685262 0.342631 0.939470i \(-0.388682\pi\)
0.342631 + 0.939470i \(0.388682\pi\)
\(810\) 0 0
\(811\) 18.5367 0.650912 0.325456 0.945557i \(-0.394482\pi\)
0.325456 + 0.945557i \(0.394482\pi\)
\(812\) −11.7657 −0.412896
\(813\) 0 0
\(814\) −30.5691 −1.07145
\(815\) −30.1349 −1.05558
\(816\) 0 0
\(817\) 4.95579 0.173381
\(818\) −6.24819 −0.218463
\(819\) 0 0
\(820\) 20.0590 0.700492
\(821\) 12.1526 0.424130 0.212065 0.977256i \(-0.431981\pi\)
0.212065 + 0.977256i \(0.431981\pi\)
\(822\) 0 0
\(823\) 15.5837 0.543214 0.271607 0.962408i \(-0.412445\pi\)
0.271607 + 0.962408i \(0.412445\pi\)
\(824\) −1.13692 −0.0396065
\(825\) 0 0
\(826\) 21.1761 0.736809
\(827\) −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 \(0.129403\pi\)
\(830\) −44.1194 −1.53141
\(831\) 0 0
\(832\) −8.38120 −0.290566
\(833\) 1.59609 0.0553014
\(834\) 0 0
\(835\) 14.8177 0.512787
\(836\) 4.03154 0.139434
\(837\) 0 0
\(838\) −32.2972 −1.11569
\(839\) 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
\(842\) −11.0858 −0.382041
\(843\) 0 0
\(844\) −17.8821 −0.615529
\(845\) −10.7261 −0.368989
\(846\) 0 0
\(847\) −29.4446 −1.01173
\(848\) −49.0187 −1.68331
\(849\) 0 0
\(850\) −2.42894 −0.0833121
\(851\) −25.5782 −0.876808
\(852\) 0 0
\(853\) 23.7883 0.814498 0.407249 0.913317i \(-0.366488\pi\)
0.407249 + 0.913317i \(0.366488\pi\)
\(854\) −2.86020 −0.0978739
\(855\) 0 0
\(856\) −10.3325 −0.353158
\(857\) 1.27557 0.0435727 0.0217863 0.999763i \(-0.493065\pi\)
0.0217863 + 0.999763i \(0.493065\pi\)
\(858\) 0 0
\(859\) −38.6456 −1.31857 −0.659285 0.751893i \(-0.729141\pi\)
−0.659285 + 0.751893i \(0.729141\pi\)
\(860\) 19.7495 0.673454
\(861\) 0 0
\(862\) −50.7937 −1.73004
\(863\) −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
\(866\) 10.6056 0.360393
\(867\) 0 0
\(868\) 7.73302 0.262476
\(869\) −41.4044 −1.40455
\(870\) 0 0
\(871\) 10.2408 0.346998
\(872\) 1.65625 0.0560876
\(873\) 0 0
\(874\) 7.80547 0.264024
\(875\) 18.5688 0.627741
\(876\) 0 0
\(877\) −5.39699 −0.182244 −0.0911218 0.995840i \(-0.529045\pi\)
−0.0911218 + 0.995840i \(0.529045\pi\)
\(878\) 50.2431 1.69562
\(879\) 0 0
\(880\) −32.7781 −1.10495
\(881\) 5.04786 0.170067 0.0850333 0.996378i \(-0.472900\pi\)
0.0850333 + 0.996378i \(0.472900\pi\)
\(882\) 0 0
\(883\) −36.4145 −1.22545 −0.612723 0.790298i \(-0.709926\pi\)
−0.612723 + 0.790298i \(0.709926\pi\)
\(884\) −1.30880 −0.0440197
\(885\) 0 0
\(886\) 64.1841 2.15631
\(887\) 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
\(890\) 32.7714 1.09850
\(891\) 0 0
\(892\) 41.0932 1.37590
\(893\) −0.356538 −0.0119311
\(894\) 0 0
\(895\) 15.6613 0.523498
\(896\) 11.9450 0.399053
\(897\) 0 0
\(898\) −13.3587 −0.445785
\(899\) −13.3654 −0.445763
\(900\) 0 0
\(901\) −4.07434 −0.135736
\(902\) −100.506 −3.34649
\(903\) 0 0
\(904\) 9.70111 0.322654
\(905\) 1.43277 0.0476270
\(906\) 0 0
\(907\) −27.1951 −0.903000 −0.451500 0.892271i \(-0.649111\pi\)
−0.451500 + 0.892271i \(0.649111\pi\)
\(908\) 9.38102 0.311320
\(909\) 0 0
\(910\) 9.19451 0.304795
\(911\) 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
\(914\) −48.4834 −1.60369
\(915\) 0 0
\(916\) 7.67500 0.253589
\(917\) 0.469436 0.0155021
\(918\) 0 0
\(919\) −20.3674 −0.671857 −0.335928 0.941887i \(-0.609050\pi\)
−0.335928 + 0.941887i \(0.609050\pi\)
\(920\) −9.76366 −0.321898
\(921\) 0 0
\(922\) −72.8432 −2.39896
\(923\) −24.9982 −0.822825
\(924\) 0 0
\(925\) −10.1069 −0.332313
\(926\) −42.9604 −1.41177
\(927\) 0 0
\(928\) 31.9207 1.04785
\(929\) −3.43001 −0.112535 −0.0562675 0.998416i \(-0.517920\pi\)
−0.0562675 + 0.998416i \(0.517920\pi\)
\(930\) 0 0
\(931\) −2.02670 −0.0664224
\(932\) 12.4974 0.409367
\(933\) 0 0
\(934\) −22.8778 −0.748583
\(935\) −2.72445 −0.0890989
\(936\) 0 0
\(937\) −55.8441 −1.82435 −0.912173 0.409805i \(-0.865597\pi\)
−0.912173 + 0.409805i \(0.865597\pi\)
\(938\) 15.0546 0.491551
\(939\) 0 0
\(940\) −1.42086 −0.0463432
\(941\) 35.2757 1.14996 0.574978 0.818169i \(-0.305011\pi\)
0.574978 + 0.818169i \(0.305011\pi\)
\(942\) 0 0
\(943\) −84.0969 −2.73857
\(944\) −31.1195 −1.01285
\(945\) 0 0
\(946\) −98.9555 −3.21732
\(947\) 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
\(950\) 3.08424 0.100066
\(951\) 0 0
\(952\) 0.603916 0.0195730
\(953\) −39.0690 −1.26557 −0.632784 0.774328i \(-0.718088\pi\)
−0.632784 + 0.774328i \(0.718088\pi\)
\(954\) 0 0
\(955\) −1.88705 −0.0610634
\(956\) −38.6988 −1.25161
\(957\) 0 0
\(958\) −65.9168 −2.12968
\(959\) −9.65365 −0.311733
\(960\) 0 0
\(961\) −22.2156 −0.716631
\(962\) −12.6013 −0.406283
\(963\) 0 0
\(964\) 40.7553 1.31264
\(965\) 1.84644 0.0594390
\(966\) 0 0
\(967\) 11.8771 0.381942 0.190971 0.981596i \(-0.438836\pi\)
0.190971 + 0.981596i \(0.438836\pi\)
\(968\) 15.4041 0.495106
\(969\) 0 0
\(970\) 32.5206 1.04417
\(971\) 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
\(974\) −14.2485 −0.456552
\(975\) 0 0
\(976\) 4.20323 0.134542
\(977\) −30.9084 −0.988846 −0.494423 0.869221i \(-0.664621\pi\)
−0.494423 + 0.869221i \(0.664621\pi\)
\(978\) 0 0
\(979\) −70.9637 −2.26801
\(980\) −8.07669 −0.258001
\(981\) 0 0
\(982\) 34.3308 1.09554
\(983\) −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
\(986\) 3.32539 0.105902
\(987\) 0 0
\(988\) 1.66190 0.0528720
\(989\) −82.7992 −2.63286
\(990\) 0 0
\(991\) −22.7329 −0.722135 −0.361067 0.932540i \(-0.617588\pi\)
−0.361067 + 0.932540i \(0.617588\pi\)
\(992\) −20.9799 −0.666112
\(993\) 0 0
\(994\) −36.7487 −1.16560
\(995\) −21.8792 −0.693616
\(996\) 0 0
\(997\) −10.6754 −0.338095 −0.169047 0.985608i \(-0.554069\pi\)
−0.169047 + 0.985608i \(0.554069\pi\)
\(998\) −10.8201 −0.342506
\(999\) 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 4923.2.a.l.1.3 18
3.2 odd 2 547.2.a.b.1.16 18
12.11 even 2 8752.2.a.s.1.11 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.16 18 3.2 odd 2
4923.2.a.l.1.3 18 1.1 even 1 trivial
8752.2.a.s.1.11 18 12.11 even 2