Properties

Label 2188.2.a.d.1.3
Level $2188$
Weight $2$
Character 2188.1
Self dual yes
Analytic conductor $17.471$
Analytic rank $1$
Dimension $19$
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] = [2188,2,Mod(1,2188)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2188, 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("2188.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2188 = 2^{2} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2188.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: \(17.4712679623\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 8 x^{18} - 6 x^{17} + 187 x^{16} - 201 x^{15} - 1757 x^{14} + 3000 x^{13} + 8703 x^{12} + \cdots - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.70555\) of defining polynomial
Character \(\chi\) \(=\) 2188.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.70555 q^{3} +2.14706 q^{5} +4.69407 q^{7} +4.32000 q^{9} +O(q^{10})\) \(q-2.70555 q^{3} +2.14706 q^{5} +4.69407 q^{7} +4.32000 q^{9} +0.590837 q^{11} -5.74905 q^{13} -5.80897 q^{15} -4.15474 q^{17} -1.46951 q^{19} -12.7000 q^{21} -9.20632 q^{23} -0.390142 q^{25} -3.57132 q^{27} +7.19187 q^{29} -9.61556 q^{31} -1.59854 q^{33} +10.0784 q^{35} +1.37912 q^{37} +15.5544 q^{39} -1.27885 q^{41} -9.14522 q^{43} +9.27529 q^{45} -8.01716 q^{47} +15.0343 q^{49} +11.2409 q^{51} +8.13076 q^{53} +1.26856 q^{55} +3.97582 q^{57} -4.91482 q^{59} -5.28465 q^{61} +20.2784 q^{63} -12.3436 q^{65} +4.67199 q^{67} +24.9082 q^{69} -1.33994 q^{71} -3.84901 q^{73} +1.05555 q^{75} +2.77343 q^{77} +15.7596 q^{79} -3.29761 q^{81} -14.8295 q^{83} -8.92047 q^{85} -19.4580 q^{87} -12.7019 q^{89} -26.9864 q^{91} +26.0154 q^{93} -3.15511 q^{95} +7.39929 q^{97} +2.55242 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 8 q^{3} - q^{5} - 9 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 8 q^{3} - q^{5} - 9 q^{7} + 19 q^{9} - 10 q^{11} - 4 q^{13} - 11 q^{15} - 32 q^{17} - 14 q^{19} - 8 q^{21} - 48 q^{23} + 14 q^{25} - 47 q^{27} - 10 q^{29} + 7 q^{31} - 16 q^{33} - 25 q^{35} + 8 q^{37} - 11 q^{39} - 9 q^{41} - 4 q^{43} + 8 q^{45} - 53 q^{47} + 6 q^{49} - q^{51} + 4 q^{53} - 31 q^{55} - 22 q^{57} - 14 q^{59} + 14 q^{61} - 45 q^{63} - 36 q^{65} - 29 q^{67} + 18 q^{69} - 41 q^{71} - 34 q^{73} - 61 q^{75} + 13 q^{77} - 12 q^{79} + 11 q^{81} - 87 q^{83} - 25 q^{85} - 39 q^{87} - 46 q^{89} - 35 q^{91} - 24 q^{93} - 44 q^{95} - 46 q^{97} - 51 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.70555 −1.56205 −0.781025 0.624500i \(-0.785303\pi\)
−0.781025 + 0.624500i \(0.785303\pi\)
\(4\) 0 0
\(5\) 2.14706 0.960194 0.480097 0.877216i \(-0.340602\pi\)
0.480097 + 0.877216i \(0.340602\pi\)
\(6\) 0 0
\(7\) 4.69407 1.77419 0.887095 0.461586i \(-0.152720\pi\)
0.887095 + 0.461586i \(0.152720\pi\)
\(8\) 0 0
\(9\) 4.32000 1.44000
\(10\) 0 0
\(11\) 0.590837 0.178144 0.0890721 0.996025i \(-0.471610\pi\)
0.0890721 + 0.996025i \(0.471610\pi\)
\(12\) 0 0
\(13\) −5.74905 −1.59450 −0.797250 0.603649i \(-0.793713\pi\)
−0.797250 + 0.603649i \(0.793713\pi\)
\(14\) 0 0
\(15\) −5.80897 −1.49987
\(16\) 0 0
\(17\) −4.15474 −1.00767 −0.503836 0.863799i \(-0.668078\pi\)
−0.503836 + 0.863799i \(0.668078\pi\)
\(18\) 0 0
\(19\) −1.46951 −0.337128 −0.168564 0.985691i \(-0.553913\pi\)
−0.168564 + 0.985691i \(0.553913\pi\)
\(20\) 0 0
\(21\) −12.7000 −2.77137
\(22\) 0 0
\(23\) −9.20632 −1.91965 −0.959825 0.280598i \(-0.909467\pi\)
−0.959825 + 0.280598i \(0.909467\pi\)
\(24\) 0 0
\(25\) −0.390142 −0.0780284
\(26\) 0 0
\(27\) −3.57132 −0.687301
\(28\) 0 0
\(29\) 7.19187 1.33550 0.667749 0.744387i \(-0.267258\pi\)
0.667749 + 0.744387i \(0.267258\pi\)
\(30\) 0 0
\(31\) −9.61556 −1.72701 −0.863503 0.504344i \(-0.831734\pi\)
−0.863503 + 0.504344i \(0.831734\pi\)
\(32\) 0 0
\(33\) −1.59854 −0.278270
\(34\) 0 0
\(35\) 10.0784 1.70357
\(36\) 0 0
\(37\) 1.37912 0.226727 0.113363 0.993554i \(-0.463838\pi\)
0.113363 + 0.993554i \(0.463838\pi\)
\(38\) 0 0
\(39\) 15.5544 2.49069
\(40\) 0 0
\(41\) −1.27885 −0.199722 −0.0998611 0.995001i \(-0.531840\pi\)
−0.0998611 + 0.995001i \(0.531840\pi\)
\(42\) 0 0
\(43\) −9.14522 −1.39463 −0.697317 0.716763i \(-0.745623\pi\)
−0.697317 + 0.716763i \(0.745623\pi\)
\(44\) 0 0
\(45\) 9.27529 1.38268
\(46\) 0 0
\(47\) −8.01716 −1.16942 −0.584711 0.811241i \(-0.698792\pi\)
−0.584711 + 0.811241i \(0.698792\pi\)
\(48\) 0 0
\(49\) 15.0343 2.14775
\(50\) 0 0
\(51\) 11.2409 1.57403
\(52\) 0 0
\(53\) 8.13076 1.11685 0.558423 0.829557i \(-0.311407\pi\)
0.558423 + 0.829557i \(0.311407\pi\)
\(54\) 0 0
\(55\) 1.26856 0.171053
\(56\) 0 0
\(57\) 3.97582 0.526610
\(58\) 0 0
\(59\) −4.91482 −0.639855 −0.319928 0.947442i \(-0.603659\pi\)
−0.319928 + 0.947442i \(0.603659\pi\)
\(60\) 0 0
\(61\) −5.28465 −0.676630 −0.338315 0.941033i \(-0.609857\pi\)
−0.338315 + 0.941033i \(0.609857\pi\)
\(62\) 0 0
\(63\) 20.2784 2.55483
\(64\) 0 0
\(65\) −12.3436 −1.53103
\(66\) 0 0
\(67\) 4.67199 0.570774 0.285387 0.958412i \(-0.407878\pi\)
0.285387 + 0.958412i \(0.407878\pi\)
\(68\) 0 0
\(69\) 24.9082 2.99859
\(70\) 0 0
\(71\) −1.33994 −0.159021 −0.0795107 0.996834i \(-0.525336\pi\)
−0.0795107 + 0.996834i \(0.525336\pi\)
\(72\) 0 0
\(73\) −3.84901 −0.450493 −0.225246 0.974302i \(-0.572319\pi\)
−0.225246 + 0.974302i \(0.572319\pi\)
\(74\) 0 0
\(75\) 1.05555 0.121884
\(76\) 0 0
\(77\) 2.77343 0.316062
\(78\) 0 0
\(79\) 15.7596 1.77310 0.886549 0.462634i \(-0.153096\pi\)
0.886549 + 0.462634i \(0.153096\pi\)
\(80\) 0 0
\(81\) −3.29761 −0.366401
\(82\) 0 0
\(83\) −14.8295 −1.62775 −0.813875 0.581040i \(-0.802646\pi\)
−0.813875 + 0.581040i \(0.802646\pi\)
\(84\) 0 0
\(85\) −8.92047 −0.967560
\(86\) 0 0
\(87\) −19.4580 −2.08611
\(88\) 0 0
\(89\) −12.7019 −1.34639 −0.673197 0.739463i \(-0.735080\pi\)
−0.673197 + 0.739463i \(0.735080\pi\)
\(90\) 0 0
\(91\) −26.9864 −2.82895
\(92\) 0 0
\(93\) 26.0154 2.69767
\(94\) 0 0
\(95\) −3.15511 −0.323708
\(96\) 0 0
\(97\) 7.39929 0.751284 0.375642 0.926765i \(-0.377422\pi\)
0.375642 + 0.926765i \(0.377422\pi\)
\(98\) 0 0
\(99\) 2.55242 0.256528
\(100\) 0 0
\(101\) 15.9619 1.58827 0.794135 0.607741i \(-0.207924\pi\)
0.794135 + 0.607741i \(0.207924\pi\)
\(102\) 0 0
\(103\) −13.8375 −1.36345 −0.681725 0.731609i \(-0.738770\pi\)
−0.681725 + 0.731609i \(0.738770\pi\)
\(104\) 0 0
\(105\) −27.2677 −2.66106
\(106\) 0 0
\(107\) 5.31245 0.513574 0.256787 0.966468i \(-0.417336\pi\)
0.256787 + 0.966468i \(0.417336\pi\)
\(108\) 0 0
\(109\) 9.15861 0.877236 0.438618 0.898674i \(-0.355468\pi\)
0.438618 + 0.898674i \(0.355468\pi\)
\(110\) 0 0
\(111\) −3.73129 −0.354158
\(112\) 0 0
\(113\) −3.84691 −0.361886 −0.180943 0.983494i \(-0.557915\pi\)
−0.180943 + 0.983494i \(0.557915\pi\)
\(114\) 0 0
\(115\) −19.7665 −1.84324
\(116\) 0 0
\(117\) −24.8359 −2.29608
\(118\) 0 0
\(119\) −19.5026 −1.78780
\(120\) 0 0
\(121\) −10.6509 −0.968265
\(122\) 0 0
\(123\) 3.45998 0.311976
\(124\) 0 0
\(125\) −11.5729 −1.03512
\(126\) 0 0
\(127\) −8.37675 −0.743317 −0.371658 0.928370i \(-0.621211\pi\)
−0.371658 + 0.928370i \(0.621211\pi\)
\(128\) 0 0
\(129\) 24.7428 2.17849
\(130\) 0 0
\(131\) 9.70641 0.848053 0.424027 0.905650i \(-0.360616\pi\)
0.424027 + 0.905650i \(0.360616\pi\)
\(132\) 0 0
\(133\) −6.89796 −0.598129
\(134\) 0 0
\(135\) −7.66783 −0.659942
\(136\) 0 0
\(137\) 1.99461 0.170411 0.0852055 0.996363i \(-0.472845\pi\)
0.0852055 + 0.996363i \(0.472845\pi\)
\(138\) 0 0
\(139\) 10.6719 0.905181 0.452590 0.891718i \(-0.350500\pi\)
0.452590 + 0.891718i \(0.350500\pi\)
\(140\) 0 0
\(141\) 21.6908 1.82670
\(142\) 0 0
\(143\) −3.39676 −0.284051
\(144\) 0 0
\(145\) 15.4414 1.28234
\(146\) 0 0
\(147\) −40.6759 −3.35489
\(148\) 0 0
\(149\) −8.88883 −0.728201 −0.364101 0.931360i \(-0.618624\pi\)
−0.364101 + 0.931360i \(0.618624\pi\)
\(150\) 0 0
\(151\) −13.8575 −1.12771 −0.563855 0.825874i \(-0.690682\pi\)
−0.563855 + 0.825874i \(0.690682\pi\)
\(152\) 0 0
\(153\) −17.9485 −1.45105
\(154\) 0 0
\(155\) −20.6452 −1.65826
\(156\) 0 0
\(157\) 19.6167 1.56558 0.782791 0.622285i \(-0.213796\pi\)
0.782791 + 0.622285i \(0.213796\pi\)
\(158\) 0 0
\(159\) −21.9982 −1.74457
\(160\) 0 0
\(161\) −43.2151 −3.40583
\(162\) 0 0
\(163\) 6.86906 0.538027 0.269013 0.963136i \(-0.413302\pi\)
0.269013 + 0.963136i \(0.413302\pi\)
\(164\) 0 0
\(165\) −3.43216 −0.267193
\(166\) 0 0
\(167\) −4.02765 −0.311669 −0.155834 0.987783i \(-0.549807\pi\)
−0.155834 + 0.987783i \(0.549807\pi\)
\(168\) 0 0
\(169\) 20.0516 1.54243
\(170\) 0 0
\(171\) −6.34826 −0.485464
\(172\) 0 0
\(173\) −16.9875 −1.29154 −0.645769 0.763533i \(-0.723463\pi\)
−0.645769 + 0.763533i \(0.723463\pi\)
\(174\) 0 0
\(175\) −1.83135 −0.138437
\(176\) 0 0
\(177\) 13.2973 0.999485
\(178\) 0 0
\(179\) −13.1553 −0.983276 −0.491638 0.870800i \(-0.663602\pi\)
−0.491638 + 0.870800i \(0.663602\pi\)
\(180\) 0 0
\(181\) 8.47748 0.630126 0.315063 0.949071i \(-0.397974\pi\)
0.315063 + 0.949071i \(0.397974\pi\)
\(182\) 0 0
\(183\) 14.2979 1.05693
\(184\) 0 0
\(185\) 2.96106 0.217701
\(186\) 0 0
\(187\) −2.45478 −0.179511
\(188\) 0 0
\(189\) −16.7640 −1.21940
\(190\) 0 0
\(191\) −4.26070 −0.308293 −0.154147 0.988048i \(-0.549263\pi\)
−0.154147 + 0.988048i \(0.549263\pi\)
\(192\) 0 0
\(193\) 10.0626 0.724319 0.362160 0.932116i \(-0.382040\pi\)
0.362160 + 0.932116i \(0.382040\pi\)
\(194\) 0 0
\(195\) 33.3961 2.39154
\(196\) 0 0
\(197\) 6.79741 0.484296 0.242148 0.970239i \(-0.422148\pi\)
0.242148 + 0.970239i \(0.422148\pi\)
\(198\) 0 0
\(199\) −2.42106 −0.171624 −0.0858122 0.996311i \(-0.527348\pi\)
−0.0858122 + 0.996311i \(0.527348\pi\)
\(200\) 0 0
\(201\) −12.6403 −0.891578
\(202\) 0 0
\(203\) 33.7591 2.36943
\(204\) 0 0
\(205\) −2.74576 −0.191772
\(206\) 0 0
\(207\) −39.7713 −2.76430
\(208\) 0 0
\(209\) −0.868239 −0.0600573
\(210\) 0 0
\(211\) −3.79643 −0.261357 −0.130679 0.991425i \(-0.541716\pi\)
−0.130679 + 0.991425i \(0.541716\pi\)
\(212\) 0 0
\(213\) 3.62527 0.248399
\(214\) 0 0
\(215\) −19.6353 −1.33912
\(216\) 0 0
\(217\) −45.1361 −3.06404
\(218\) 0 0
\(219\) 10.4137 0.703692
\(220\) 0 0
\(221\) 23.8858 1.60673
\(222\) 0 0
\(223\) 12.2596 0.820966 0.410483 0.911868i \(-0.365360\pi\)
0.410483 + 0.911868i \(0.365360\pi\)
\(224\) 0 0
\(225\) −1.68541 −0.112361
\(226\) 0 0
\(227\) −18.5387 −1.23045 −0.615227 0.788350i \(-0.710936\pi\)
−0.615227 + 0.788350i \(0.710936\pi\)
\(228\) 0 0
\(229\) 2.50004 0.165207 0.0826036 0.996582i \(-0.473676\pi\)
0.0826036 + 0.996582i \(0.473676\pi\)
\(230\) 0 0
\(231\) −7.50365 −0.493704
\(232\) 0 0
\(233\) −3.32097 −0.217564 −0.108782 0.994066i \(-0.534695\pi\)
−0.108782 + 0.994066i \(0.534695\pi\)
\(234\) 0 0
\(235\) −17.2133 −1.12287
\(236\) 0 0
\(237\) −42.6385 −2.76967
\(238\) 0 0
\(239\) 1.26627 0.0819080 0.0409540 0.999161i \(-0.486960\pi\)
0.0409540 + 0.999161i \(0.486960\pi\)
\(240\) 0 0
\(241\) 17.3866 1.11997 0.559986 0.828502i \(-0.310806\pi\)
0.559986 + 0.828502i \(0.310806\pi\)
\(242\) 0 0
\(243\) 19.6358 1.25964
\(244\) 0 0
\(245\) 32.2794 2.06226
\(246\) 0 0
\(247\) 8.44827 0.537550
\(248\) 0 0
\(249\) 40.1220 2.54263
\(250\) 0 0
\(251\) 24.6180 1.55388 0.776938 0.629577i \(-0.216772\pi\)
0.776938 + 0.629577i \(0.216772\pi\)
\(252\) 0 0
\(253\) −5.43944 −0.341975
\(254\) 0 0
\(255\) 24.1348 1.51138
\(256\) 0 0
\(257\) 15.1970 0.947961 0.473981 0.880535i \(-0.342817\pi\)
0.473981 + 0.880535i \(0.342817\pi\)
\(258\) 0 0
\(259\) 6.47370 0.402256
\(260\) 0 0
\(261\) 31.0689 1.92312
\(262\) 0 0
\(263\) −11.4014 −0.703038 −0.351519 0.936181i \(-0.614335\pi\)
−0.351519 + 0.936181i \(0.614335\pi\)
\(264\) 0 0
\(265\) 17.4572 1.07239
\(266\) 0 0
\(267\) 34.3655 2.10314
\(268\) 0 0
\(269\) −18.2805 −1.11458 −0.557291 0.830317i \(-0.688159\pi\)
−0.557291 + 0.830317i \(0.688159\pi\)
\(270\) 0 0
\(271\) −24.6033 −1.49454 −0.747272 0.664518i \(-0.768637\pi\)
−0.747272 + 0.664518i \(0.768637\pi\)
\(272\) 0 0
\(273\) 73.0132 4.41896
\(274\) 0 0
\(275\) −0.230511 −0.0139003
\(276\) 0 0
\(277\) −13.8532 −0.832360 −0.416180 0.909282i \(-0.636631\pi\)
−0.416180 + 0.909282i \(0.636631\pi\)
\(278\) 0 0
\(279\) −41.5392 −2.48689
\(280\) 0 0
\(281\) 1.38849 0.0828304 0.0414152 0.999142i \(-0.486813\pi\)
0.0414152 + 0.999142i \(0.486813\pi\)
\(282\) 0 0
\(283\) −5.99481 −0.356355 −0.178177 0.983998i \(-0.557020\pi\)
−0.178177 + 0.983998i \(0.557020\pi\)
\(284\) 0 0
\(285\) 8.53632 0.505648
\(286\) 0 0
\(287\) −6.00299 −0.354345
\(288\) 0 0
\(289\) 0.261861 0.0154036
\(290\) 0 0
\(291\) −20.0192 −1.17354
\(292\) 0 0
\(293\) 10.9440 0.639356 0.319678 0.947526i \(-0.396425\pi\)
0.319678 + 0.947526i \(0.396425\pi\)
\(294\) 0 0
\(295\) −10.5524 −0.614385
\(296\) 0 0
\(297\) −2.11007 −0.122439
\(298\) 0 0
\(299\) 52.9276 3.06088
\(300\) 0 0
\(301\) −42.9283 −2.47434
\(302\) 0 0
\(303\) −43.1858 −2.48096
\(304\) 0 0
\(305\) −11.3465 −0.649696
\(306\) 0 0
\(307\) 10.3830 0.592588 0.296294 0.955097i \(-0.404249\pi\)
0.296294 + 0.955097i \(0.404249\pi\)
\(308\) 0 0
\(309\) 37.4381 2.12978
\(310\) 0 0
\(311\) 10.4132 0.590477 0.295239 0.955424i \(-0.404601\pi\)
0.295239 + 0.955424i \(0.404601\pi\)
\(312\) 0 0
\(313\) −23.1140 −1.30648 −0.653240 0.757151i \(-0.726591\pi\)
−0.653240 + 0.757151i \(0.726591\pi\)
\(314\) 0 0
\(315\) 43.5388 2.45313
\(316\) 0 0
\(317\) 17.9414 1.00769 0.503846 0.863794i \(-0.331918\pi\)
0.503846 + 0.863794i \(0.331918\pi\)
\(318\) 0 0
\(319\) 4.24923 0.237911
\(320\) 0 0
\(321\) −14.3731 −0.802228
\(322\) 0 0
\(323\) 6.10541 0.339714
\(324\) 0 0
\(325\) 2.24295 0.124416
\(326\) 0 0
\(327\) −24.7791 −1.37029
\(328\) 0 0
\(329\) −37.6331 −2.07478
\(330\) 0 0
\(331\) 12.6507 0.695347 0.347674 0.937616i \(-0.386972\pi\)
0.347674 + 0.937616i \(0.386972\pi\)
\(332\) 0 0
\(333\) 5.95781 0.326486
\(334\) 0 0
\(335\) 10.0310 0.548054
\(336\) 0 0
\(337\) 34.9183 1.90212 0.951061 0.309002i \(-0.0999951\pi\)
0.951061 + 0.309002i \(0.0999951\pi\)
\(338\) 0 0
\(339\) 10.4080 0.565285
\(340\) 0 0
\(341\) −5.68123 −0.307656
\(342\) 0 0
\(343\) 37.7134 2.03633
\(344\) 0 0
\(345\) 53.4793 2.87923
\(346\) 0 0
\(347\) −30.8185 −1.65442 −0.827212 0.561890i \(-0.810074\pi\)
−0.827212 + 0.561890i \(0.810074\pi\)
\(348\) 0 0
\(349\) −8.68565 −0.464932 −0.232466 0.972605i \(-0.574679\pi\)
−0.232466 + 0.972605i \(0.574679\pi\)
\(350\) 0 0
\(351\) 20.5317 1.09590
\(352\) 0 0
\(353\) −29.0976 −1.54871 −0.774356 0.632751i \(-0.781926\pi\)
−0.774356 + 0.632751i \(0.781926\pi\)
\(354\) 0 0
\(355\) −2.87692 −0.152691
\(356\) 0 0
\(357\) 52.7653 2.79264
\(358\) 0 0
\(359\) −22.5265 −1.18890 −0.594452 0.804131i \(-0.702631\pi\)
−0.594452 + 0.804131i \(0.702631\pi\)
\(360\) 0 0
\(361\) −16.8406 −0.886345
\(362\) 0 0
\(363\) 28.8166 1.51248
\(364\) 0 0
\(365\) −8.26405 −0.432560
\(366\) 0 0
\(367\) 13.4363 0.701370 0.350685 0.936493i \(-0.385949\pi\)
0.350685 + 0.936493i \(0.385949\pi\)
\(368\) 0 0
\(369\) −5.52461 −0.287600
\(370\) 0 0
\(371\) 38.1663 1.98150
\(372\) 0 0
\(373\) −12.0426 −0.623541 −0.311771 0.950157i \(-0.600922\pi\)
−0.311771 + 0.950157i \(0.600922\pi\)
\(374\) 0 0
\(375\) 31.3112 1.61690
\(376\) 0 0
\(377\) −41.3465 −2.12945
\(378\) 0 0
\(379\) −28.4063 −1.45913 −0.729567 0.683909i \(-0.760279\pi\)
−0.729567 + 0.683909i \(0.760279\pi\)
\(380\) 0 0
\(381\) 22.6637 1.16110
\(382\) 0 0
\(383\) 0.299679 0.0153129 0.00765645 0.999971i \(-0.497563\pi\)
0.00765645 + 0.999971i \(0.497563\pi\)
\(384\) 0 0
\(385\) 5.95472 0.303480
\(386\) 0 0
\(387\) −39.5073 −2.00827
\(388\) 0 0
\(389\) −21.8178 −1.10621 −0.553103 0.833113i \(-0.686556\pi\)
−0.553103 + 0.833113i \(0.686556\pi\)
\(390\) 0 0
\(391\) 38.2499 1.93438
\(392\) 0 0
\(393\) −26.2612 −1.32470
\(394\) 0 0
\(395\) 33.8369 1.70252
\(396\) 0 0
\(397\) 7.00401 0.351521 0.175760 0.984433i \(-0.443762\pi\)
0.175760 + 0.984433i \(0.443762\pi\)
\(398\) 0 0
\(399\) 18.6628 0.934307
\(400\) 0 0
\(401\) −8.06453 −0.402723 −0.201362 0.979517i \(-0.564537\pi\)
−0.201362 + 0.979517i \(0.564537\pi\)
\(402\) 0 0
\(403\) 55.2804 2.75371
\(404\) 0 0
\(405\) −7.08016 −0.351816
\(406\) 0 0
\(407\) 0.814838 0.0403900
\(408\) 0 0
\(409\) 19.3853 0.958544 0.479272 0.877666i \(-0.340901\pi\)
0.479272 + 0.877666i \(0.340901\pi\)
\(410\) 0 0
\(411\) −5.39651 −0.266190
\(412\) 0 0
\(413\) −23.0705 −1.13522
\(414\) 0 0
\(415\) −31.8398 −1.56295
\(416\) 0 0
\(417\) −28.8734 −1.41394
\(418\) 0 0
\(419\) −21.0266 −1.02722 −0.513609 0.858024i \(-0.671692\pi\)
−0.513609 + 0.858024i \(0.671692\pi\)
\(420\) 0 0
\(421\) 36.2193 1.76522 0.882611 0.470105i \(-0.155784\pi\)
0.882611 + 0.470105i \(0.155784\pi\)
\(422\) 0 0
\(423\) −34.6341 −1.68397
\(424\) 0 0
\(425\) 1.62094 0.0786271
\(426\) 0 0
\(427\) −24.8065 −1.20047
\(428\) 0 0
\(429\) 9.19009 0.443702
\(430\) 0 0
\(431\) 23.9972 1.15591 0.577953 0.816070i \(-0.303852\pi\)
0.577953 + 0.816070i \(0.303852\pi\)
\(432\) 0 0
\(433\) 22.2587 1.06969 0.534843 0.844952i \(-0.320371\pi\)
0.534843 + 0.844952i \(0.320371\pi\)
\(434\) 0 0
\(435\) −41.7774 −2.00307
\(436\) 0 0
\(437\) 13.5287 0.647167
\(438\) 0 0
\(439\) −35.9300 −1.71485 −0.857423 0.514612i \(-0.827936\pi\)
−0.857423 + 0.514612i \(0.827936\pi\)
\(440\) 0 0
\(441\) 64.9480 3.09276
\(442\) 0 0
\(443\) 18.0219 0.856245 0.428122 0.903721i \(-0.359175\pi\)
0.428122 + 0.903721i \(0.359175\pi\)
\(444\) 0 0
\(445\) −27.2716 −1.29280
\(446\) 0 0
\(447\) 24.0492 1.13749
\(448\) 0 0
\(449\) 15.4794 0.730517 0.365258 0.930906i \(-0.380981\pi\)
0.365258 + 0.930906i \(0.380981\pi\)
\(450\) 0 0
\(451\) −0.755590 −0.0355794
\(452\) 0 0
\(453\) 37.4922 1.76154
\(454\) 0 0
\(455\) −57.9415 −2.71634
\(456\) 0 0
\(457\) 8.50970 0.398067 0.199033 0.979993i \(-0.436220\pi\)
0.199033 + 0.979993i \(0.436220\pi\)
\(458\) 0 0
\(459\) 14.8379 0.692574
\(460\) 0 0
\(461\) −28.9586 −1.34873 −0.674367 0.738396i \(-0.735584\pi\)
−0.674367 + 0.738396i \(0.735584\pi\)
\(462\) 0 0
\(463\) 1.09061 0.0506850 0.0253425 0.999679i \(-0.491932\pi\)
0.0253425 + 0.999679i \(0.491932\pi\)
\(464\) 0 0
\(465\) 55.8565 2.59028
\(466\) 0 0
\(467\) 13.3623 0.618333 0.309167 0.951008i \(-0.399950\pi\)
0.309167 + 0.951008i \(0.399950\pi\)
\(468\) 0 0
\(469\) 21.9306 1.01266
\(470\) 0 0
\(471\) −53.0739 −2.44552
\(472\) 0 0
\(473\) −5.40334 −0.248446
\(474\) 0 0
\(475\) 0.573316 0.0263055
\(476\) 0 0
\(477\) 35.1249 1.60826
\(478\) 0 0
\(479\) 5.84759 0.267183 0.133592 0.991036i \(-0.457349\pi\)
0.133592 + 0.991036i \(0.457349\pi\)
\(480\) 0 0
\(481\) −7.92866 −0.361516
\(482\) 0 0
\(483\) 116.921 5.32007
\(484\) 0 0
\(485\) 15.8867 0.721378
\(486\) 0 0
\(487\) −9.40839 −0.426335 −0.213168 0.977016i \(-0.568378\pi\)
−0.213168 + 0.977016i \(0.568378\pi\)
\(488\) 0 0
\(489\) −18.5846 −0.840424
\(490\) 0 0
\(491\) 23.8787 1.07763 0.538814 0.842425i \(-0.318872\pi\)
0.538814 + 0.842425i \(0.318872\pi\)
\(492\) 0 0
\(493\) −29.8804 −1.34574
\(494\) 0 0
\(495\) 5.48019 0.246316
\(496\) 0 0
\(497\) −6.28976 −0.282134
\(498\) 0 0
\(499\) 29.2196 1.30805 0.654025 0.756473i \(-0.273079\pi\)
0.654025 + 0.756473i \(0.273079\pi\)
\(500\) 0 0
\(501\) 10.8970 0.486842
\(502\) 0 0
\(503\) 38.9542 1.73688 0.868440 0.495794i \(-0.165123\pi\)
0.868440 + 0.495794i \(0.165123\pi\)
\(504\) 0 0
\(505\) 34.2712 1.52505
\(506\) 0 0
\(507\) −54.2507 −2.40936
\(508\) 0 0
\(509\) −0.811469 −0.0359677 −0.0179839 0.999838i \(-0.505725\pi\)
−0.0179839 + 0.999838i \(0.505725\pi\)
\(510\) 0 0
\(511\) −18.0675 −0.799260
\(512\) 0 0
\(513\) 5.24807 0.231708
\(514\) 0 0
\(515\) −29.7099 −1.30918
\(516\) 0 0
\(517\) −4.73684 −0.208326
\(518\) 0 0
\(519\) 45.9606 2.01745
\(520\) 0 0
\(521\) −17.3710 −0.761038 −0.380519 0.924773i \(-0.624255\pi\)
−0.380519 + 0.924773i \(0.624255\pi\)
\(522\) 0 0
\(523\) −37.4711 −1.63850 −0.819249 0.573438i \(-0.805609\pi\)
−0.819249 + 0.573438i \(0.805609\pi\)
\(524\) 0 0
\(525\) 4.95482 0.216246
\(526\) 0 0
\(527\) 39.9501 1.74026
\(528\) 0 0
\(529\) 61.7563 2.68506
\(530\) 0 0
\(531\) −21.2320 −0.921391
\(532\) 0 0
\(533\) 7.35216 0.318457
\(534\) 0 0
\(535\) 11.4061 0.493130
\(536\) 0 0
\(537\) 35.5924 1.53593
\(538\) 0 0
\(539\) 8.88280 0.382609
\(540\) 0 0
\(541\) −15.9974 −0.687780 −0.343890 0.939010i \(-0.611745\pi\)
−0.343890 + 0.939010i \(0.611745\pi\)
\(542\) 0 0
\(543\) −22.9362 −0.984289
\(544\) 0 0
\(545\) 19.6641 0.842316
\(546\) 0 0
\(547\) 1.00000 0.0427569
\(548\) 0 0
\(549\) −22.8297 −0.974347
\(550\) 0 0
\(551\) −10.5685 −0.450233
\(552\) 0 0
\(553\) 73.9768 3.14581
\(554\) 0 0
\(555\) −8.01129 −0.340060
\(556\) 0 0
\(557\) −39.9182 −1.69139 −0.845695 0.533667i \(-0.820814\pi\)
−0.845695 + 0.533667i \(0.820814\pi\)
\(558\) 0 0
\(559\) 52.5764 2.22374
\(560\) 0 0
\(561\) 6.64152 0.280405
\(562\) 0 0
\(563\) −31.8813 −1.34363 −0.671817 0.740717i \(-0.734486\pi\)
−0.671817 + 0.740717i \(0.734486\pi\)
\(564\) 0 0
\(565\) −8.25953 −0.347481
\(566\) 0 0
\(567\) −15.4792 −0.650065
\(568\) 0 0
\(569\) −4.19999 −0.176073 −0.0880363 0.996117i \(-0.528059\pi\)
−0.0880363 + 0.996117i \(0.528059\pi\)
\(570\) 0 0
\(571\) 15.3317 0.641611 0.320805 0.947145i \(-0.396046\pi\)
0.320805 + 0.947145i \(0.396046\pi\)
\(572\) 0 0
\(573\) 11.5275 0.481570
\(574\) 0 0
\(575\) 3.59177 0.149787
\(576\) 0 0
\(577\) −44.1755 −1.83905 −0.919526 0.393030i \(-0.871427\pi\)
−0.919526 + 0.393030i \(0.871427\pi\)
\(578\) 0 0
\(579\) −27.2248 −1.13142
\(580\) 0 0
\(581\) −69.6107 −2.88794
\(582\) 0 0
\(583\) 4.80396 0.198960
\(584\) 0 0
\(585\) −53.3241 −2.20468
\(586\) 0 0
\(587\) −7.59967 −0.313672 −0.156836 0.987625i \(-0.550129\pi\)
−0.156836 + 0.987625i \(0.550129\pi\)
\(588\) 0 0
\(589\) 14.1301 0.582221
\(590\) 0 0
\(591\) −18.3907 −0.756494
\(592\) 0 0
\(593\) −37.1183 −1.52426 −0.762132 0.647422i \(-0.775847\pi\)
−0.762132 + 0.647422i \(0.775847\pi\)
\(594\) 0 0
\(595\) −41.8733 −1.71664
\(596\) 0 0
\(597\) 6.55029 0.268086
\(598\) 0 0
\(599\) −5.78766 −0.236477 −0.118239 0.992985i \(-0.537725\pi\)
−0.118239 + 0.992985i \(0.537725\pi\)
\(600\) 0 0
\(601\) 40.0019 1.63171 0.815856 0.578255i \(-0.196266\pi\)
0.815856 + 0.578255i \(0.196266\pi\)
\(602\) 0 0
\(603\) 20.1830 0.821915
\(604\) 0 0
\(605\) −22.8681 −0.929721
\(606\) 0 0
\(607\) −9.15068 −0.371415 −0.185707 0.982605i \(-0.559458\pi\)
−0.185707 + 0.982605i \(0.559458\pi\)
\(608\) 0 0
\(609\) −91.3370 −3.70116
\(610\) 0 0
\(611\) 46.0911 1.86465
\(612\) 0 0
\(613\) −38.0698 −1.53763 −0.768813 0.639474i \(-0.779152\pi\)
−0.768813 + 0.639474i \(0.779152\pi\)
\(614\) 0 0
\(615\) 7.42878 0.299557
\(616\) 0 0
\(617\) −24.2439 −0.976023 −0.488011 0.872837i \(-0.662278\pi\)
−0.488011 + 0.872837i \(0.662278\pi\)
\(618\) 0 0
\(619\) 0.304762 0.0122494 0.00612471 0.999981i \(-0.498050\pi\)
0.00612471 + 0.999981i \(0.498050\pi\)
\(620\) 0 0
\(621\) 32.8787 1.31938
\(622\) 0 0
\(623\) −59.6234 −2.38876
\(624\) 0 0
\(625\) −22.8971 −0.915883
\(626\) 0 0
\(627\) 2.34906 0.0938125
\(628\) 0 0
\(629\) −5.72990 −0.228466
\(630\) 0 0
\(631\) 11.9553 0.475935 0.237967 0.971273i \(-0.423519\pi\)
0.237967 + 0.971273i \(0.423519\pi\)
\(632\) 0 0
\(633\) 10.2714 0.408253
\(634\) 0 0
\(635\) −17.9854 −0.713728
\(636\) 0 0
\(637\) −86.4328 −3.42459
\(638\) 0 0
\(639\) −5.78853 −0.228991
\(640\) 0 0
\(641\) 12.7547 0.503782 0.251891 0.967756i \(-0.418948\pi\)
0.251891 + 0.967756i \(0.418948\pi\)
\(642\) 0 0
\(643\) 22.1093 0.871907 0.435954 0.899969i \(-0.356411\pi\)
0.435954 + 0.899969i \(0.356411\pi\)
\(644\) 0 0
\(645\) 53.1243 2.09177
\(646\) 0 0
\(647\) −42.8684 −1.68533 −0.842666 0.538436i \(-0.819015\pi\)
−0.842666 + 0.538436i \(0.819015\pi\)
\(648\) 0 0
\(649\) −2.90386 −0.113986
\(650\) 0 0
\(651\) 122.118 4.78618
\(652\) 0 0
\(653\) −23.0620 −0.902484 −0.451242 0.892402i \(-0.649019\pi\)
−0.451242 + 0.892402i \(0.649019\pi\)
\(654\) 0 0
\(655\) 20.8402 0.814295
\(656\) 0 0
\(657\) −16.6277 −0.648709
\(658\) 0 0
\(659\) 24.7761 0.965138 0.482569 0.875858i \(-0.339704\pi\)
0.482569 + 0.875858i \(0.339704\pi\)
\(660\) 0 0
\(661\) 29.6199 1.15208 0.576039 0.817422i \(-0.304597\pi\)
0.576039 + 0.817422i \(0.304597\pi\)
\(662\) 0 0
\(663\) −64.6243 −2.50980
\(664\) 0 0
\(665\) −14.8103 −0.574319
\(666\) 0 0
\(667\) −66.2107 −2.56369
\(668\) 0 0
\(669\) −33.1690 −1.28239
\(670\) 0 0
\(671\) −3.12237 −0.120538
\(672\) 0 0
\(673\) −27.5103 −1.06045 −0.530223 0.847858i \(-0.677892\pi\)
−0.530223 + 0.847858i \(0.677892\pi\)
\(674\) 0 0
\(675\) 1.39332 0.0536290
\(676\) 0 0
\(677\) 43.7448 1.68125 0.840624 0.541619i \(-0.182188\pi\)
0.840624 + 0.541619i \(0.182188\pi\)
\(678\) 0 0
\(679\) 34.7328 1.33292
\(680\) 0 0
\(681\) 50.1573 1.92203
\(682\) 0 0
\(683\) 34.9503 1.33734 0.668668 0.743561i \(-0.266865\pi\)
0.668668 + 0.743561i \(0.266865\pi\)
\(684\) 0 0
\(685\) 4.28254 0.163627
\(686\) 0 0
\(687\) −6.76398 −0.258062
\(688\) 0 0
\(689\) −46.7442 −1.78081
\(690\) 0 0
\(691\) 35.4820 1.34980 0.674899 0.737910i \(-0.264187\pi\)
0.674899 + 0.737910i \(0.264187\pi\)
\(692\) 0 0
\(693\) 11.9812 0.455129
\(694\) 0 0
\(695\) 22.9132 0.869149
\(696\) 0 0
\(697\) 5.31327 0.201255
\(698\) 0 0
\(699\) 8.98505 0.339846
\(700\) 0 0
\(701\) −45.8462 −1.73159 −0.865794 0.500400i \(-0.833186\pi\)
−0.865794 + 0.500400i \(0.833186\pi\)
\(702\) 0 0
\(703\) −2.02663 −0.0764358
\(704\) 0 0
\(705\) 46.5714 1.75398
\(706\) 0 0
\(707\) 74.9263 2.81789
\(708\) 0 0
\(709\) 19.1166 0.717939 0.358969 0.933349i \(-0.383128\pi\)
0.358969 + 0.933349i \(0.383128\pi\)
\(710\) 0 0
\(711\) 68.0816 2.55326
\(712\) 0 0
\(713\) 88.5239 3.31525
\(714\) 0 0
\(715\) −7.29303 −0.272744
\(716\) 0 0
\(717\) −3.42595 −0.127944
\(718\) 0 0
\(719\) 10.4573 0.389993 0.194997 0.980804i \(-0.437530\pi\)
0.194997 + 0.980804i \(0.437530\pi\)
\(720\) 0 0
\(721\) −64.9542 −2.41902
\(722\) 0 0
\(723\) −47.0404 −1.74945
\(724\) 0 0
\(725\) −2.80585 −0.104207
\(726\) 0 0
\(727\) −13.4503 −0.498844 −0.249422 0.968395i \(-0.580241\pi\)
−0.249422 + 0.968395i \(0.580241\pi\)
\(728\) 0 0
\(729\) −43.2328 −1.60122
\(730\) 0 0
\(731\) 37.9960 1.40533
\(732\) 0 0
\(733\) −40.8592 −1.50917 −0.754584 0.656203i \(-0.772161\pi\)
−0.754584 + 0.656203i \(0.772161\pi\)
\(734\) 0 0
\(735\) −87.3336 −3.22135
\(736\) 0 0
\(737\) 2.76039 0.101680
\(738\) 0 0
\(739\) 32.2376 1.18588 0.592939 0.805247i \(-0.297967\pi\)
0.592939 + 0.805247i \(0.297967\pi\)
\(740\) 0 0
\(741\) −22.8572 −0.839680
\(742\) 0 0
\(743\) 15.5927 0.572040 0.286020 0.958224i \(-0.407668\pi\)
0.286020 + 0.958224i \(0.407668\pi\)
\(744\) 0 0
\(745\) −19.0848 −0.699214
\(746\) 0 0
\(747\) −64.0634 −2.34396
\(748\) 0 0
\(749\) 24.9370 0.911178
\(750\) 0 0
\(751\) 43.8336 1.59951 0.799755 0.600327i \(-0.204963\pi\)
0.799755 + 0.600327i \(0.204963\pi\)
\(752\) 0 0
\(753\) −66.6053 −2.42723
\(754\) 0 0
\(755\) −29.7529 −1.08282
\(756\) 0 0
\(757\) −9.28134 −0.337336 −0.168668 0.985673i \(-0.553947\pi\)
−0.168668 + 0.985673i \(0.553947\pi\)
\(758\) 0 0
\(759\) 14.7167 0.534181
\(760\) 0 0
\(761\) −42.5353 −1.54190 −0.770952 0.636893i \(-0.780219\pi\)
−0.770952 + 0.636893i \(0.780219\pi\)
\(762\) 0 0
\(763\) 42.9911 1.55638
\(764\) 0 0
\(765\) −38.5364 −1.39329
\(766\) 0 0
\(767\) 28.2556 1.02025
\(768\) 0 0
\(769\) 40.0893 1.44566 0.722829 0.691027i \(-0.242842\pi\)
0.722829 + 0.691027i \(0.242842\pi\)
\(770\) 0 0
\(771\) −41.1162 −1.48076
\(772\) 0 0
\(773\) −19.6626 −0.707216 −0.353608 0.935394i \(-0.615045\pi\)
−0.353608 + 0.935394i \(0.615045\pi\)
\(774\) 0 0
\(775\) 3.75143 0.134756
\(776\) 0 0
\(777\) −17.5149 −0.628344
\(778\) 0 0
\(779\) 1.87927 0.0673319
\(780\) 0 0
\(781\) −0.791686 −0.0283287
\(782\) 0 0
\(783\) −25.6845 −0.917889
\(784\) 0 0
\(785\) 42.1181 1.50326
\(786\) 0 0
\(787\) 23.6239 0.842103 0.421051 0.907037i \(-0.361661\pi\)
0.421051 + 0.907037i \(0.361661\pi\)
\(788\) 0 0
\(789\) 30.8470 1.09818
\(790\) 0 0
\(791\) −18.0576 −0.642055
\(792\) 0 0
\(793\) 30.3818 1.07889
\(794\) 0 0
\(795\) −47.2313 −1.67512
\(796\) 0 0
\(797\) 9.45081 0.334765 0.167382 0.985892i \(-0.446469\pi\)
0.167382 + 0.985892i \(0.446469\pi\)
\(798\) 0 0
\(799\) 33.3092 1.17839
\(800\) 0 0
\(801\) −54.8720 −1.93881
\(802\) 0 0
\(803\) −2.27414 −0.0802527
\(804\) 0 0
\(805\) −92.7853 −3.27025
\(806\) 0 0
\(807\) 49.4588 1.74103
\(808\) 0 0
\(809\) 31.8431 1.11955 0.559773 0.828646i \(-0.310888\pi\)
0.559773 + 0.828646i \(0.310888\pi\)
\(810\) 0 0
\(811\) 28.3073 0.994003 0.497001 0.867750i \(-0.334434\pi\)
0.497001 + 0.867750i \(0.334434\pi\)
\(812\) 0 0
\(813\) 66.5655 2.33455
\(814\) 0 0
\(815\) 14.7483 0.516610
\(816\) 0 0
\(817\) 13.4390 0.470169
\(818\) 0 0
\(819\) −116.581 −4.07368
\(820\) 0 0
\(821\) −42.9066 −1.49745 −0.748726 0.662880i \(-0.769334\pi\)
−0.748726 + 0.662880i \(0.769334\pi\)
\(822\) 0 0
\(823\) 18.5656 0.647157 0.323579 0.946201i \(-0.395114\pi\)
0.323579 + 0.946201i \(0.395114\pi\)
\(824\) 0 0
\(825\) 0.623658 0.0217130
\(826\) 0 0
\(827\) 6.00267 0.208733 0.104367 0.994539i \(-0.466718\pi\)
0.104367 + 0.994539i \(0.466718\pi\)
\(828\) 0 0
\(829\) 1.15215 0.0400159 0.0200080 0.999800i \(-0.493631\pi\)
0.0200080 + 0.999800i \(0.493631\pi\)
\(830\) 0 0
\(831\) 37.4806 1.30019
\(832\) 0 0
\(833\) −62.4634 −2.16423
\(834\) 0 0
\(835\) −8.64759 −0.299262
\(836\) 0 0
\(837\) 34.3402 1.18697
\(838\) 0 0
\(839\) 13.7860 0.475947 0.237973 0.971272i \(-0.423517\pi\)
0.237973 + 0.971272i \(0.423517\pi\)
\(840\) 0 0
\(841\) 22.7231 0.783554
\(842\) 0 0
\(843\) −3.75663 −0.129385
\(844\) 0 0
\(845\) 43.0520 1.48103
\(846\) 0 0
\(847\) −49.9961 −1.71789
\(848\) 0 0
\(849\) 16.2193 0.556644
\(850\) 0 0
\(851\) −12.6967 −0.435236
\(852\) 0 0
\(853\) 42.9006 1.46889 0.734445 0.678668i \(-0.237443\pi\)
0.734445 + 0.678668i \(0.237443\pi\)
\(854\) 0 0
\(855\) −13.6301 −0.466139
\(856\) 0 0
\(857\) 14.4266 0.492802 0.246401 0.969168i \(-0.420752\pi\)
0.246401 + 0.969168i \(0.420752\pi\)
\(858\) 0 0
\(859\) −29.6901 −1.01301 −0.506506 0.862236i \(-0.669063\pi\)
−0.506506 + 0.862236i \(0.669063\pi\)
\(860\) 0 0
\(861\) 16.2414 0.553505
\(862\) 0 0
\(863\) −17.9177 −0.609925 −0.304963 0.952364i \(-0.598644\pi\)
−0.304963 + 0.952364i \(0.598644\pi\)
\(864\) 0 0
\(865\) −36.4732 −1.24013
\(866\) 0 0
\(867\) −0.708479 −0.0240612
\(868\) 0 0
\(869\) 9.31139 0.315867
\(870\) 0 0
\(871\) −26.8595 −0.910100
\(872\) 0 0
\(873\) 31.9649 1.08185
\(874\) 0 0
\(875\) −54.3242 −1.83649
\(876\) 0 0
\(877\) 40.1479 1.35570 0.677849 0.735201i \(-0.262912\pi\)
0.677849 + 0.735201i \(0.262912\pi\)
\(878\) 0 0
\(879\) −29.6096 −0.998706
\(880\) 0 0
\(881\) 32.1429 1.08292 0.541461 0.840726i \(-0.317871\pi\)
0.541461 + 0.840726i \(0.317871\pi\)
\(882\) 0 0
\(883\) −43.7505 −1.47232 −0.736161 0.676807i \(-0.763363\pi\)
−0.736161 + 0.676807i \(0.763363\pi\)
\(884\) 0 0
\(885\) 28.5500 0.959699
\(886\) 0 0
\(887\) 26.7741 0.898986 0.449493 0.893284i \(-0.351605\pi\)
0.449493 + 0.893284i \(0.351605\pi\)
\(888\) 0 0
\(889\) −39.3210 −1.31879
\(890\) 0 0
\(891\) −1.94835 −0.0652722
\(892\) 0 0
\(893\) 11.7813 0.394245
\(894\) 0 0
\(895\) −28.2453 −0.944136
\(896\) 0 0
\(897\) −143.198 −4.78125
\(898\) 0 0
\(899\) −69.1539 −2.30641
\(900\) 0 0
\(901\) −33.7812 −1.12541
\(902\) 0 0
\(903\) 116.145 3.86505
\(904\) 0 0
\(905\) 18.2016 0.605043
\(906\) 0 0
\(907\) −17.6942 −0.587525 −0.293762 0.955878i \(-0.594907\pi\)
−0.293762 + 0.955878i \(0.594907\pi\)
\(908\) 0 0
\(909\) 68.9555 2.28711
\(910\) 0 0
\(911\) −41.8408 −1.38625 −0.693124 0.720818i \(-0.743766\pi\)
−0.693124 + 0.720818i \(0.743766\pi\)
\(912\) 0 0
\(913\) −8.76183 −0.289974
\(914\) 0 0
\(915\) 30.6984 1.01486
\(916\) 0 0
\(917\) 45.5626 1.50461
\(918\) 0 0
\(919\) 1.91865 0.0632905 0.0316452 0.999499i \(-0.489925\pi\)
0.0316452 + 0.999499i \(0.489925\pi\)
\(920\) 0 0
\(921\) −28.0917 −0.925652
\(922\) 0 0
\(923\) 7.70338 0.253560
\(924\) 0 0
\(925\) −0.538055 −0.0176911
\(926\) 0 0
\(927\) −59.7780 −1.96337
\(928\) 0 0
\(929\) 52.9575 1.73748 0.868740 0.495268i \(-0.164930\pi\)
0.868740 + 0.495268i \(0.164930\pi\)
\(930\) 0 0
\(931\) −22.0929 −0.724067
\(932\) 0 0
\(933\) −28.1734 −0.922355
\(934\) 0 0
\(935\) −5.27055 −0.172365
\(936\) 0 0
\(937\) 17.6093 0.575270 0.287635 0.957740i \(-0.407131\pi\)
0.287635 + 0.957740i \(0.407131\pi\)
\(938\) 0 0
\(939\) 62.5361 2.04079
\(940\) 0 0
\(941\) −21.4295 −0.698583 −0.349292 0.937014i \(-0.613578\pi\)
−0.349292 + 0.937014i \(0.613578\pi\)
\(942\) 0 0
\(943\) 11.7735 0.383397
\(944\) 0 0
\(945\) −35.9933 −1.17086
\(946\) 0 0
\(947\) −35.2679 −1.14605 −0.573027 0.819537i \(-0.694231\pi\)
−0.573027 + 0.819537i \(0.694231\pi\)
\(948\) 0 0
\(949\) 22.1282 0.718311
\(950\) 0 0
\(951\) −48.5414 −1.57406
\(952\) 0 0
\(953\) 36.9565 1.19714 0.598570 0.801071i \(-0.295736\pi\)
0.598570 + 0.801071i \(0.295736\pi\)
\(954\) 0 0
\(955\) −9.14797 −0.296021
\(956\) 0 0
\(957\) −11.4965 −0.371629
\(958\) 0 0
\(959\) 9.36283 0.302341
\(960\) 0 0
\(961\) 61.4590 1.98255
\(962\) 0 0
\(963\) 22.9498 0.739546
\(964\) 0 0
\(965\) 21.6049 0.695486
\(966\) 0 0
\(967\) 18.3859 0.591250 0.295625 0.955304i \(-0.404472\pi\)
0.295625 + 0.955304i \(0.404472\pi\)
\(968\) 0 0
\(969\) −16.5185 −0.530651
\(970\) 0 0
\(971\) −0.173527 −0.00556874 −0.00278437 0.999996i \(-0.500886\pi\)
−0.00278437 + 0.999996i \(0.500886\pi\)
\(972\) 0 0
\(973\) 50.0947 1.60596
\(974\) 0 0
\(975\) −6.06841 −0.194345
\(976\) 0 0
\(977\) 9.78931 0.313188 0.156594 0.987663i \(-0.449949\pi\)
0.156594 + 0.987663i \(0.449949\pi\)
\(978\) 0 0
\(979\) −7.50474 −0.239852
\(980\) 0 0
\(981\) 39.5652 1.26322
\(982\) 0 0
\(983\) 47.3555 1.51040 0.755202 0.655492i \(-0.227539\pi\)
0.755202 + 0.655492i \(0.227539\pi\)
\(984\) 0 0
\(985\) 14.5944 0.465018
\(986\) 0 0
\(987\) 101.818 3.24091
\(988\) 0 0
\(989\) 84.1938 2.67721
\(990\) 0 0
\(991\) −34.1753 −1.08562 −0.542808 0.839857i \(-0.682639\pi\)
−0.542808 + 0.839857i \(0.682639\pi\)
\(992\) 0 0
\(993\) −34.2272 −1.08617
\(994\) 0 0
\(995\) −5.19815 −0.164793
\(996\) 0 0
\(997\) 53.5988 1.69749 0.848746 0.528800i \(-0.177358\pi\)
0.848746 + 0.528800i \(0.177358\pi\)
\(998\) 0 0
\(999\) −4.92529 −0.155829
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 2188.2.a.d.1.3 19
4.3 odd 2 8752.2.a.t.1.17 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2188.2.a.d.1.3 19 1.1 even 1 trivial
8752.2.a.t.1.17 19 4.3 odd 2