Properties

Label 3192.2.a.r.1.1
Level $3192$
Weight $2$
Character 3192.1
Self dual yes
Analytic conductor $25.488$
Analytic rank $1$
Dimension $2$
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] = [3192,2,Mod(1,3192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3192.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3192 = 2^{3} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3192.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: \(25.4882483252\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3192.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -3.41421 q^{5} -1.00000 q^{7} +1.00000 q^{9} -2.82843 q^{11} +5.65685 q^{13} -3.41421 q^{15} +2.24264 q^{17} +1.00000 q^{19} -1.00000 q^{21} -8.48528 q^{23} +6.65685 q^{25} +1.00000 q^{27} +10.2426 q^{29} +8.82843 q^{31} -2.82843 q^{33} +3.41421 q^{35} -6.00000 q^{37} +5.65685 q^{39} -8.82843 q^{41} -8.82843 q^{43} -3.41421 q^{45} -10.5858 q^{47} +1.00000 q^{49} +2.24264 q^{51} -5.07107 q^{53} +9.65685 q^{55} +1.00000 q^{57} -2.34315 q^{59} +2.00000 q^{61} -1.00000 q^{63} -19.3137 q^{65} -9.17157 q^{67} -8.48528 q^{69} +12.7279 q^{71} -8.82843 q^{73} +6.65685 q^{75} +2.82843 q^{77} +2.34315 q^{79} +1.00000 q^{81} -12.2426 q^{83} -7.65685 q^{85} +10.2426 q^{87} -16.8284 q^{89} -5.65685 q^{91} +8.82843 q^{93} -3.41421 q^{95} -0.343146 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 4 q^{5} - 2 q^{7} + 2 q^{9} - 4 q^{15} - 4 q^{17} + 2 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 12 q^{29} + 12 q^{31} + 4 q^{35} - 12 q^{37} - 12 q^{41} - 12 q^{43} - 4 q^{45} - 24 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{55} + 2 q^{57} - 16 q^{59} + 4 q^{61} - 2 q^{63} - 16 q^{65} - 24 q^{67} - 12 q^{73} + 2 q^{75} + 16 q^{79} + 2 q^{81} - 16 q^{83} - 4 q^{85} + 12 q^{87} - 28 q^{89} + 12 q^{93} - 4 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.82843 −0.852803 −0.426401 0.904534i \(-0.640219\pi\)
−0.426401 + 0.904534i \(0.640219\pi\)
\(12\) 0 0
\(13\) 5.65685 1.56893 0.784465 0.620174i \(-0.212938\pi\)
0.784465 + 0.620174i \(0.212938\pi\)
\(14\) 0 0
\(15\) −3.41421 −0.881546
\(16\) 0 0
\(17\) 2.24264 0.543920 0.271960 0.962309i \(-0.412328\pi\)
0.271960 + 0.962309i \(0.412328\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −8.48528 −1.76930 −0.884652 0.466252i \(-0.845604\pi\)
−0.884652 + 0.466252i \(0.845604\pi\)
\(24\) 0 0
\(25\) 6.65685 1.33137
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 10.2426 1.90201 0.951005 0.309175i \(-0.100053\pi\)
0.951005 + 0.309175i \(0.100053\pi\)
\(30\) 0 0
\(31\) 8.82843 1.58563 0.792816 0.609461i \(-0.208614\pi\)
0.792816 + 0.609461i \(0.208614\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 3.41421 0.577107
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) −8.82843 −1.37877 −0.689384 0.724396i \(-0.742119\pi\)
−0.689384 + 0.724396i \(0.742119\pi\)
\(42\) 0 0
\(43\) −8.82843 −1.34632 −0.673161 0.739496i \(-0.735064\pi\)
−0.673161 + 0.739496i \(0.735064\pi\)
\(44\) 0 0
\(45\) −3.41421 −0.508961
\(46\) 0 0
\(47\) −10.5858 −1.54410 −0.772048 0.635564i \(-0.780767\pi\)
−0.772048 + 0.635564i \(0.780767\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.24264 0.314033
\(52\) 0 0
\(53\) −5.07107 −0.696565 −0.348282 0.937390i \(-0.613235\pi\)
−0.348282 + 0.937390i \(0.613235\pi\)
\(54\) 0 0
\(55\) 9.65685 1.30213
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −2.34315 −0.305052 −0.152526 0.988299i \(-0.548741\pi\)
−0.152526 + 0.988299i \(0.548741\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −19.3137 −2.39557
\(66\) 0 0
\(67\) −9.17157 −1.12049 −0.560243 0.828328i \(-0.689292\pi\)
−0.560243 + 0.828328i \(0.689292\pi\)
\(68\) 0 0
\(69\) −8.48528 −1.02151
\(70\) 0 0
\(71\) 12.7279 1.51053 0.755263 0.655422i \(-0.227509\pi\)
0.755263 + 0.655422i \(0.227509\pi\)
\(72\) 0 0
\(73\) −8.82843 −1.03329 −0.516645 0.856200i \(-0.672819\pi\)
−0.516645 + 0.856200i \(0.672819\pi\)
\(74\) 0 0
\(75\) 6.65685 0.768667
\(76\) 0 0
\(77\) 2.82843 0.322329
\(78\) 0 0
\(79\) 2.34315 0.263624 0.131812 0.991275i \(-0.457920\pi\)
0.131812 + 0.991275i \(0.457920\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.2426 −1.34380 −0.671902 0.740640i \(-0.734522\pi\)
−0.671902 + 0.740640i \(0.734522\pi\)
\(84\) 0 0
\(85\) −7.65685 −0.830502
\(86\) 0 0
\(87\) 10.2426 1.09813
\(88\) 0 0
\(89\) −16.8284 −1.78381 −0.891905 0.452223i \(-0.850631\pi\)
−0.891905 + 0.452223i \(0.850631\pi\)
\(90\) 0 0
\(91\) −5.65685 −0.592999
\(92\) 0 0
\(93\) 8.82843 0.915465
\(94\) 0 0
\(95\) −3.41421 −0.350291
\(96\) 0 0
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) 0 0
\(99\) −2.82843 −0.284268
\(100\) 0 0
\(101\) 9.07107 0.902605 0.451302 0.892371i \(-0.350960\pi\)
0.451302 + 0.892371i \(0.350960\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 3.41421 0.333193
\(106\) 0 0
\(107\) −16.7279 −1.61715 −0.808575 0.588394i \(-0.799761\pi\)
−0.808575 + 0.588394i \(0.799761\pi\)
\(108\) 0 0
\(109\) −16.1421 −1.54614 −0.773068 0.634323i \(-0.781279\pi\)
−0.773068 + 0.634323i \(0.781279\pi\)
\(110\) 0 0
\(111\) −6.00000 −0.569495
\(112\) 0 0
\(113\) 8.58579 0.807683 0.403841 0.914829i \(-0.367675\pi\)
0.403841 + 0.914829i \(0.367675\pi\)
\(114\) 0 0
\(115\) 28.9706 2.70152
\(116\) 0 0
\(117\) 5.65685 0.522976
\(118\) 0 0
\(119\) −2.24264 −0.205583
\(120\) 0 0
\(121\) −3.00000 −0.272727
\(122\) 0 0
\(123\) −8.82843 −0.796032
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) −8.48528 −0.752947 −0.376473 0.926427i \(-0.622863\pi\)
−0.376473 + 0.926427i \(0.622863\pi\)
\(128\) 0 0
\(129\) −8.82843 −0.777300
\(130\) 0 0
\(131\) 9.89949 0.864923 0.432461 0.901652i \(-0.357645\pi\)
0.432461 + 0.901652i \(0.357645\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −3.41421 −0.293849
\(136\) 0 0
\(137\) −11.6569 −0.995912 −0.497956 0.867202i \(-0.665916\pi\)
−0.497956 + 0.867202i \(0.665916\pi\)
\(138\) 0 0
\(139\) −12.4853 −1.05899 −0.529494 0.848314i \(-0.677618\pi\)
−0.529494 + 0.848314i \(0.677618\pi\)
\(140\) 0 0
\(141\) −10.5858 −0.891484
\(142\) 0 0
\(143\) −16.0000 −1.33799
\(144\) 0 0
\(145\) −34.9706 −2.90415
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 18.4853 1.51437 0.757187 0.653199i \(-0.226573\pi\)
0.757187 + 0.653199i \(0.226573\pi\)
\(150\) 0 0
\(151\) 8.48528 0.690522 0.345261 0.938507i \(-0.387790\pi\)
0.345261 + 0.938507i \(0.387790\pi\)
\(152\) 0 0
\(153\) 2.24264 0.181307
\(154\) 0 0
\(155\) −30.1421 −2.42107
\(156\) 0 0
\(157\) −7.65685 −0.611083 −0.305542 0.952179i \(-0.598838\pi\)
−0.305542 + 0.952179i \(0.598838\pi\)
\(158\) 0 0
\(159\) −5.07107 −0.402162
\(160\) 0 0
\(161\) 8.48528 0.668734
\(162\) 0 0
\(163\) −1.51472 −0.118642 −0.0593210 0.998239i \(-0.518894\pi\)
−0.0593210 + 0.998239i \(0.518894\pi\)
\(164\) 0 0
\(165\) 9.65685 0.751785
\(166\) 0 0
\(167\) −22.8284 −1.76652 −0.883258 0.468887i \(-0.844655\pi\)
−0.883258 + 0.468887i \(0.844655\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −3.17157 −0.241130 −0.120565 0.992705i \(-0.538471\pi\)
−0.120565 + 0.992705i \(0.538471\pi\)
\(174\) 0 0
\(175\) −6.65685 −0.503211
\(176\) 0 0
\(177\) −2.34315 −0.176122
\(178\) 0 0
\(179\) 3.07107 0.229542 0.114771 0.993392i \(-0.463387\pi\)
0.114771 + 0.993392i \(0.463387\pi\)
\(180\) 0 0
\(181\) −9.31371 −0.692283 −0.346141 0.938182i \(-0.612508\pi\)
−0.346141 + 0.938182i \(0.612508\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 20.4853 1.50611
\(186\) 0 0
\(187\) −6.34315 −0.463857
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −14.8284 −1.07295 −0.536474 0.843917i \(-0.680244\pi\)
−0.536474 + 0.843917i \(0.680244\pi\)
\(192\) 0 0
\(193\) −0.343146 −0.0247002 −0.0123501 0.999924i \(-0.503931\pi\)
−0.0123501 + 0.999924i \(0.503931\pi\)
\(194\) 0 0
\(195\) −19.3137 −1.38308
\(196\) 0 0
\(197\) 14.9706 1.06661 0.533304 0.845924i \(-0.320950\pi\)
0.533304 + 0.845924i \(0.320950\pi\)
\(198\) 0 0
\(199\) −14.1421 −1.00251 −0.501255 0.865300i \(-0.667128\pi\)
−0.501255 + 0.865300i \(0.667128\pi\)
\(200\) 0 0
\(201\) −9.17157 −0.646913
\(202\) 0 0
\(203\) −10.2426 −0.718892
\(204\) 0 0
\(205\) 30.1421 2.10522
\(206\) 0 0
\(207\) −8.48528 −0.589768
\(208\) 0 0
\(209\) −2.82843 −0.195646
\(210\) 0 0
\(211\) 28.9706 1.99442 0.997208 0.0746754i \(-0.0237921\pi\)
0.997208 + 0.0746754i \(0.0237921\pi\)
\(212\) 0 0
\(213\) 12.7279 0.872103
\(214\) 0 0
\(215\) 30.1421 2.05568
\(216\) 0 0
\(217\) −8.82843 −0.599313
\(218\) 0 0
\(219\) −8.82843 −0.596570
\(220\) 0 0
\(221\) 12.6863 0.853372
\(222\) 0 0
\(223\) −0.828427 −0.0554756 −0.0277378 0.999615i \(-0.508830\pi\)
−0.0277378 + 0.999615i \(0.508830\pi\)
\(224\) 0 0
\(225\) 6.65685 0.443790
\(226\) 0 0
\(227\) 11.5147 0.764259 0.382129 0.924109i \(-0.375191\pi\)
0.382129 + 0.924109i \(0.375191\pi\)
\(228\) 0 0
\(229\) 10.4853 0.692887 0.346443 0.938071i \(-0.387389\pi\)
0.346443 + 0.938071i \(0.387389\pi\)
\(230\) 0 0
\(231\) 2.82843 0.186097
\(232\) 0 0
\(233\) 10.4853 0.686914 0.343457 0.939168i \(-0.388402\pi\)
0.343457 + 0.939168i \(0.388402\pi\)
\(234\) 0 0
\(235\) 36.1421 2.35765
\(236\) 0 0
\(237\) 2.34315 0.152204
\(238\) 0 0
\(239\) −2.34315 −0.151565 −0.0757827 0.997124i \(-0.524146\pi\)
−0.0757827 + 0.997124i \(0.524146\pi\)
\(240\) 0 0
\(241\) 14.0000 0.901819 0.450910 0.892570i \(-0.351100\pi\)
0.450910 + 0.892570i \(0.351100\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.41421 −0.218126
\(246\) 0 0
\(247\) 5.65685 0.359937
\(248\) 0 0
\(249\) −12.2426 −0.775846
\(250\) 0 0
\(251\) −4.24264 −0.267793 −0.133897 0.990995i \(-0.542749\pi\)
−0.133897 + 0.990995i \(0.542749\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) −7.65685 −0.479491
\(256\) 0 0
\(257\) −14.0000 −0.873296 −0.436648 0.899632i \(-0.643834\pi\)
−0.436648 + 0.899632i \(0.643834\pi\)
\(258\) 0 0
\(259\) 6.00000 0.372822
\(260\) 0 0
\(261\) 10.2426 0.634004
\(262\) 0 0
\(263\) 1.65685 0.102166 0.0510830 0.998694i \(-0.483733\pi\)
0.0510830 + 0.998694i \(0.483733\pi\)
\(264\) 0 0
\(265\) 17.3137 1.06357
\(266\) 0 0
\(267\) −16.8284 −1.02988
\(268\) 0 0
\(269\) −11.6569 −0.710731 −0.355365 0.934727i \(-0.615644\pi\)
−0.355365 + 0.934727i \(0.615644\pi\)
\(270\) 0 0
\(271\) −16.9706 −1.03089 −0.515444 0.856923i \(-0.672373\pi\)
−0.515444 + 0.856923i \(0.672373\pi\)
\(272\) 0 0
\(273\) −5.65685 −0.342368
\(274\) 0 0
\(275\) −18.8284 −1.13540
\(276\) 0 0
\(277\) −11.3137 −0.679775 −0.339887 0.940466i \(-0.610389\pi\)
−0.339887 + 0.940466i \(0.610389\pi\)
\(278\) 0 0
\(279\) 8.82843 0.528544
\(280\) 0 0
\(281\) 1.75736 0.104835 0.0524176 0.998625i \(-0.483307\pi\)
0.0524176 + 0.998625i \(0.483307\pi\)
\(282\) 0 0
\(283\) 14.1421 0.840663 0.420331 0.907371i \(-0.361914\pi\)
0.420331 + 0.907371i \(0.361914\pi\)
\(284\) 0 0
\(285\) −3.41421 −0.202241
\(286\) 0 0
\(287\) 8.82843 0.521126
\(288\) 0 0
\(289\) −11.9706 −0.704151
\(290\) 0 0
\(291\) −0.343146 −0.0201156
\(292\) 0 0
\(293\) 1.51472 0.0884908 0.0442454 0.999021i \(-0.485912\pi\)
0.0442454 + 0.999021i \(0.485912\pi\)
\(294\) 0 0
\(295\) 8.00000 0.465778
\(296\) 0 0
\(297\) −2.82843 −0.164122
\(298\) 0 0
\(299\) −48.0000 −2.77591
\(300\) 0 0
\(301\) 8.82843 0.508862
\(302\) 0 0
\(303\) 9.07107 0.521119
\(304\) 0 0
\(305\) −6.82843 −0.390995
\(306\) 0 0
\(307\) −16.1421 −0.921280 −0.460640 0.887587i \(-0.652380\pi\)
−0.460640 + 0.887587i \(0.652380\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −28.0416 −1.59009 −0.795047 0.606547i \(-0.792554\pi\)
−0.795047 + 0.606547i \(0.792554\pi\)
\(312\) 0 0
\(313\) 2.97056 0.167906 0.0839531 0.996470i \(-0.473245\pi\)
0.0839531 + 0.996470i \(0.473245\pi\)
\(314\) 0 0
\(315\) 3.41421 0.192369
\(316\) 0 0
\(317\) 27.2132 1.52845 0.764223 0.644952i \(-0.223123\pi\)
0.764223 + 0.644952i \(0.223123\pi\)
\(318\) 0 0
\(319\) −28.9706 −1.62204
\(320\) 0 0
\(321\) −16.7279 −0.933662
\(322\) 0 0
\(323\) 2.24264 0.124784
\(324\) 0 0
\(325\) 37.6569 2.08883
\(326\) 0 0
\(327\) −16.1421 −0.892662
\(328\) 0 0
\(329\) 10.5858 0.583613
\(330\) 0 0
\(331\) −14.1421 −0.777322 −0.388661 0.921381i \(-0.627062\pi\)
−0.388661 + 0.921381i \(0.627062\pi\)
\(332\) 0 0
\(333\) −6.00000 −0.328798
\(334\) 0 0
\(335\) 31.3137 1.71085
\(336\) 0 0
\(337\) −8.82843 −0.480915 −0.240458 0.970660i \(-0.577297\pi\)
−0.240458 + 0.970660i \(0.577297\pi\)
\(338\) 0 0
\(339\) 8.58579 0.466316
\(340\) 0 0
\(341\) −24.9706 −1.35223
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 28.9706 1.55972
\(346\) 0 0
\(347\) 20.9706 1.12576 0.562879 0.826539i \(-0.309694\pi\)
0.562879 + 0.826539i \(0.309694\pi\)
\(348\) 0 0
\(349\) 29.3137 1.56913 0.784563 0.620049i \(-0.212887\pi\)
0.784563 + 0.620049i \(0.212887\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) 0.100505 0.00534934 0.00267467 0.999996i \(-0.499149\pi\)
0.00267467 + 0.999996i \(0.499149\pi\)
\(354\) 0 0
\(355\) −43.4558 −2.30640
\(356\) 0 0
\(357\) −2.24264 −0.118693
\(358\) 0 0
\(359\) 4.48528 0.236724 0.118362 0.992971i \(-0.462236\pi\)
0.118362 + 0.992971i \(0.462236\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.00000 −0.157459
\(364\) 0 0
\(365\) 30.1421 1.57771
\(366\) 0 0
\(367\) 5.85786 0.305778 0.152889 0.988243i \(-0.451142\pi\)
0.152889 + 0.988243i \(0.451142\pi\)
\(368\) 0 0
\(369\) −8.82843 −0.459590
\(370\) 0 0
\(371\) 5.07107 0.263277
\(372\) 0 0
\(373\) 24.3431 1.26044 0.630220 0.776416i \(-0.282965\pi\)
0.630220 + 0.776416i \(0.282965\pi\)
\(374\) 0 0
\(375\) −5.65685 −0.292119
\(376\) 0 0
\(377\) 57.9411 2.98412
\(378\) 0 0
\(379\) 14.8284 0.761685 0.380843 0.924640i \(-0.375634\pi\)
0.380843 + 0.924640i \(0.375634\pi\)
\(380\) 0 0
\(381\) −8.48528 −0.434714
\(382\) 0 0
\(383\) −20.4853 −1.04675 −0.523374 0.852103i \(-0.675327\pi\)
−0.523374 + 0.852103i \(0.675327\pi\)
\(384\) 0 0
\(385\) −9.65685 −0.492159
\(386\) 0 0
\(387\) −8.82843 −0.448774
\(388\) 0 0
\(389\) 35.4558 1.79768 0.898841 0.438274i \(-0.144410\pi\)
0.898841 + 0.438274i \(0.144410\pi\)
\(390\) 0 0
\(391\) −19.0294 −0.962360
\(392\) 0 0
\(393\) 9.89949 0.499363
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 1.31371 0.0659331 0.0329666 0.999456i \(-0.489505\pi\)
0.0329666 + 0.999456i \(0.489505\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −12.3848 −0.618466 −0.309233 0.950986i \(-0.600072\pi\)
−0.309233 + 0.950986i \(0.600072\pi\)
\(402\) 0 0
\(403\) 49.9411 2.48774
\(404\) 0 0
\(405\) −3.41421 −0.169654
\(406\) 0 0
\(407\) 16.9706 0.841200
\(408\) 0 0
\(409\) −18.6274 −0.921066 −0.460533 0.887642i \(-0.652342\pi\)
−0.460533 + 0.887642i \(0.652342\pi\)
\(410\) 0 0
\(411\) −11.6569 −0.574990
\(412\) 0 0
\(413\) 2.34315 0.115299
\(414\) 0 0
\(415\) 41.7990 2.05183
\(416\) 0 0
\(417\) −12.4853 −0.611407
\(418\) 0 0
\(419\) −4.72792 −0.230974 −0.115487 0.993309i \(-0.536843\pi\)
−0.115487 + 0.993309i \(0.536843\pi\)
\(420\) 0 0
\(421\) −1.51472 −0.0738229 −0.0369114 0.999319i \(-0.511752\pi\)
−0.0369114 + 0.999319i \(0.511752\pi\)
\(422\) 0 0
\(423\) −10.5858 −0.514699
\(424\) 0 0
\(425\) 14.9289 0.724160
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) 0 0
\(429\) −16.0000 −0.772487
\(430\) 0 0
\(431\) −33.4142 −1.60951 −0.804753 0.593610i \(-0.797702\pi\)
−0.804753 + 0.593610i \(0.797702\pi\)
\(432\) 0 0
\(433\) 4.62742 0.222379 0.111190 0.993799i \(-0.464534\pi\)
0.111190 + 0.993799i \(0.464534\pi\)
\(434\) 0 0
\(435\) −34.9706 −1.67671
\(436\) 0 0
\(437\) −8.48528 −0.405906
\(438\) 0 0
\(439\) −41.7990 −1.99496 −0.997478 0.0709697i \(-0.977391\pi\)
−0.997478 + 0.0709697i \(0.977391\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 0.686292 0.0326067 0.0163033 0.999867i \(-0.494810\pi\)
0.0163033 + 0.999867i \(0.494810\pi\)
\(444\) 0 0
\(445\) 57.4558 2.72367
\(446\) 0 0
\(447\) 18.4853 0.874324
\(448\) 0 0
\(449\) 16.8701 0.796147 0.398074 0.917353i \(-0.369679\pi\)
0.398074 + 0.917353i \(0.369679\pi\)
\(450\) 0 0
\(451\) 24.9706 1.17582
\(452\) 0 0
\(453\) 8.48528 0.398673
\(454\) 0 0
\(455\) 19.3137 0.905441
\(456\) 0 0
\(457\) −27.3137 −1.27768 −0.638841 0.769339i \(-0.720586\pi\)
−0.638841 + 0.769339i \(0.720586\pi\)
\(458\) 0 0
\(459\) 2.24264 0.104678
\(460\) 0 0
\(461\) −38.5269 −1.79438 −0.897189 0.441648i \(-0.854394\pi\)
−0.897189 + 0.441648i \(0.854394\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) 0 0
\(465\) −30.1421 −1.39781
\(466\) 0 0
\(467\) 9.89949 0.458094 0.229047 0.973415i \(-0.426439\pi\)
0.229047 + 0.973415i \(0.426439\pi\)
\(468\) 0 0
\(469\) 9.17157 0.423504
\(470\) 0 0
\(471\) −7.65685 −0.352809
\(472\) 0 0
\(473\) 24.9706 1.14815
\(474\) 0 0
\(475\) 6.65685 0.305437
\(476\) 0 0
\(477\) −5.07107 −0.232188
\(478\) 0 0
\(479\) −0.727922 −0.0332596 −0.0166298 0.999862i \(-0.505294\pi\)
−0.0166298 + 0.999862i \(0.505294\pi\)
\(480\) 0 0
\(481\) −33.9411 −1.54758
\(482\) 0 0
\(483\) 8.48528 0.386094
\(484\) 0 0
\(485\) 1.17157 0.0531984
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) −1.51472 −0.0684979
\(490\) 0 0
\(491\) −9.65685 −0.435808 −0.217904 0.975970i \(-0.569922\pi\)
−0.217904 + 0.975970i \(0.569922\pi\)
\(492\) 0 0
\(493\) 22.9706 1.03454
\(494\) 0 0
\(495\) 9.65685 0.434043
\(496\) 0 0
\(497\) −12.7279 −0.570925
\(498\) 0 0
\(499\) 43.5980 1.95171 0.975857 0.218411i \(-0.0700874\pi\)
0.975857 + 0.218411i \(0.0700874\pi\)
\(500\) 0 0
\(501\) −22.8284 −1.01990
\(502\) 0 0
\(503\) 42.1838 1.88088 0.940441 0.339958i \(-0.110413\pi\)
0.940441 + 0.339958i \(0.110413\pi\)
\(504\) 0 0
\(505\) −30.9706 −1.37817
\(506\) 0 0
\(507\) 19.0000 0.843820
\(508\) 0 0
\(509\) −14.9706 −0.663559 −0.331779 0.943357i \(-0.607649\pi\)
−0.331779 + 0.943357i \(0.607649\pi\)
\(510\) 0 0
\(511\) 8.82843 0.390547
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) −27.3137 −1.20359
\(516\) 0 0
\(517\) 29.9411 1.31681
\(518\) 0 0
\(519\) −3.17157 −0.139217
\(520\) 0 0
\(521\) −38.9706 −1.70733 −0.853666 0.520821i \(-0.825626\pi\)
−0.853666 + 0.520821i \(0.825626\pi\)
\(522\) 0 0
\(523\) −0.142136 −0.00621516 −0.00310758 0.999995i \(-0.500989\pi\)
−0.00310758 + 0.999995i \(0.500989\pi\)
\(524\) 0 0
\(525\) −6.65685 −0.290529
\(526\) 0 0
\(527\) 19.7990 0.862458
\(528\) 0 0
\(529\) 49.0000 2.13043
\(530\) 0 0
\(531\) −2.34315 −0.101684
\(532\) 0 0
\(533\) −49.9411 −2.16319
\(534\) 0 0
\(535\) 57.1127 2.46920
\(536\) 0 0
\(537\) 3.07107 0.132526
\(538\) 0 0
\(539\) −2.82843 −0.121829
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) −9.31371 −0.399689
\(544\) 0 0
\(545\) 55.1127 2.36077
\(546\) 0 0
\(547\) −35.5980 −1.52206 −0.761030 0.648717i \(-0.775306\pi\)
−0.761030 + 0.648717i \(0.775306\pi\)
\(548\) 0 0
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 10.2426 0.436351
\(552\) 0 0
\(553\) −2.34315 −0.0996407
\(554\) 0 0
\(555\) 20.4853 0.869552
\(556\) 0 0
\(557\) 3.17157 0.134384 0.0671919 0.997740i \(-0.478596\pi\)
0.0671919 + 0.997740i \(0.478596\pi\)
\(558\) 0 0
\(559\) −49.9411 −2.11228
\(560\) 0 0
\(561\) −6.34315 −0.267808
\(562\) 0 0
\(563\) −24.4853 −1.03193 −0.515966 0.856609i \(-0.672567\pi\)
−0.515966 + 0.856609i \(0.672567\pi\)
\(564\) 0 0
\(565\) −29.3137 −1.23324
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −18.5269 −0.776689 −0.388344 0.921514i \(-0.626953\pi\)
−0.388344 + 0.921514i \(0.626953\pi\)
\(570\) 0 0
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 0 0
\(573\) −14.8284 −0.619466
\(574\) 0 0
\(575\) −56.4853 −2.35560
\(576\) 0 0
\(577\) 4.34315 0.180808 0.0904038 0.995905i \(-0.471184\pi\)
0.0904038 + 0.995905i \(0.471184\pi\)
\(578\) 0 0
\(579\) −0.343146 −0.0142607
\(580\) 0 0
\(581\) 12.2426 0.507910
\(582\) 0 0
\(583\) 14.3431 0.594032
\(584\) 0 0
\(585\) −19.3137 −0.798524
\(586\) 0 0
\(587\) −0.928932 −0.0383411 −0.0191706 0.999816i \(-0.506103\pi\)
−0.0191706 + 0.999816i \(0.506103\pi\)
\(588\) 0 0
\(589\) 8.82843 0.363769
\(590\) 0 0
\(591\) 14.9706 0.615807
\(592\) 0 0
\(593\) −48.3848 −1.98692 −0.993462 0.114161i \(-0.963582\pi\)
−0.993462 + 0.114161i \(0.963582\pi\)
\(594\) 0 0
\(595\) 7.65685 0.313900
\(596\) 0 0
\(597\) −14.1421 −0.578799
\(598\) 0 0
\(599\) 22.1005 0.903002 0.451501 0.892271i \(-0.350889\pi\)
0.451501 + 0.892271i \(0.350889\pi\)
\(600\) 0 0
\(601\) −9.31371 −0.379914 −0.189957 0.981792i \(-0.560835\pi\)
−0.189957 + 0.981792i \(0.560835\pi\)
\(602\) 0 0
\(603\) −9.17157 −0.373495
\(604\) 0 0
\(605\) 10.2426 0.416423
\(606\) 0 0
\(607\) 25.9411 1.05292 0.526459 0.850201i \(-0.323519\pi\)
0.526459 + 0.850201i \(0.323519\pi\)
\(608\) 0 0
\(609\) −10.2426 −0.415053
\(610\) 0 0
\(611\) −59.8823 −2.42258
\(612\) 0 0
\(613\) −12.2843 −0.496157 −0.248079 0.968740i \(-0.579799\pi\)
−0.248079 + 0.968740i \(0.579799\pi\)
\(614\) 0 0
\(615\) 30.1421 1.21545
\(616\) 0 0
\(617\) −35.9411 −1.44694 −0.723468 0.690358i \(-0.757453\pi\)
−0.723468 + 0.690358i \(0.757453\pi\)
\(618\) 0 0
\(619\) 11.5147 0.462816 0.231408 0.972857i \(-0.425667\pi\)
0.231408 + 0.972857i \(0.425667\pi\)
\(620\) 0 0
\(621\) −8.48528 −0.340503
\(622\) 0 0
\(623\) 16.8284 0.674217
\(624\) 0 0
\(625\) −13.9706 −0.558823
\(626\) 0 0
\(627\) −2.82843 −0.112956
\(628\) 0 0
\(629\) −13.4558 −0.536520
\(630\) 0 0
\(631\) 15.4558 0.615287 0.307644 0.951502i \(-0.400460\pi\)
0.307644 + 0.951502i \(0.400460\pi\)
\(632\) 0 0
\(633\) 28.9706 1.15148
\(634\) 0 0
\(635\) 28.9706 1.14966
\(636\) 0 0
\(637\) 5.65685 0.224133
\(638\) 0 0
\(639\) 12.7279 0.503509
\(640\) 0 0
\(641\) 47.6985 1.88398 0.941988 0.335645i \(-0.108954\pi\)
0.941988 + 0.335645i \(0.108954\pi\)
\(642\) 0 0
\(643\) 21.1716 0.834925 0.417463 0.908694i \(-0.362919\pi\)
0.417463 + 0.908694i \(0.362919\pi\)
\(644\) 0 0
\(645\) 30.1421 1.18685
\(646\) 0 0
\(647\) 0.727922 0.0286176 0.0143088 0.999898i \(-0.495445\pi\)
0.0143088 + 0.999898i \(0.495445\pi\)
\(648\) 0 0
\(649\) 6.62742 0.260149
\(650\) 0 0
\(651\) −8.82843 −0.346013
\(652\) 0 0
\(653\) 37.5980 1.47132 0.735661 0.677350i \(-0.236872\pi\)
0.735661 + 0.677350i \(0.236872\pi\)
\(654\) 0 0
\(655\) −33.7990 −1.32064
\(656\) 0 0
\(657\) −8.82843 −0.344430
\(658\) 0 0
\(659\) 16.2426 0.632723 0.316362 0.948639i \(-0.397539\pi\)
0.316362 + 0.948639i \(0.397539\pi\)
\(660\) 0 0
\(661\) 12.0000 0.466746 0.233373 0.972387i \(-0.425024\pi\)
0.233373 + 0.972387i \(0.425024\pi\)
\(662\) 0 0
\(663\) 12.6863 0.492695
\(664\) 0 0
\(665\) 3.41421 0.132398
\(666\) 0 0
\(667\) −86.9117 −3.36523
\(668\) 0 0
\(669\) −0.828427 −0.0320288
\(670\) 0 0
\(671\) −5.65685 −0.218380
\(672\) 0 0
\(673\) 21.3137 0.821583 0.410792 0.911729i \(-0.365252\pi\)
0.410792 + 0.911729i \(0.365252\pi\)
\(674\) 0 0
\(675\) 6.65685 0.256222
\(676\) 0 0
\(677\) 39.9411 1.53506 0.767531 0.641012i \(-0.221485\pi\)
0.767531 + 0.641012i \(0.221485\pi\)
\(678\) 0 0
\(679\) 0.343146 0.0131687
\(680\) 0 0
\(681\) 11.5147 0.441245
\(682\) 0 0
\(683\) −21.8995 −0.837961 −0.418980 0.907995i \(-0.637612\pi\)
−0.418980 + 0.907995i \(0.637612\pi\)
\(684\) 0 0
\(685\) 39.7990 1.52064
\(686\) 0 0
\(687\) 10.4853 0.400038
\(688\) 0 0
\(689\) −28.6863 −1.09286
\(690\) 0 0
\(691\) 28.9706 1.10209 0.551046 0.834475i \(-0.314229\pi\)
0.551046 + 0.834475i \(0.314229\pi\)
\(692\) 0 0
\(693\) 2.82843 0.107443
\(694\) 0 0
\(695\) 42.6274 1.61695
\(696\) 0 0
\(697\) −19.7990 −0.749940
\(698\) 0 0
\(699\) 10.4853 0.396590
\(700\) 0 0
\(701\) 50.0000 1.88847 0.944237 0.329267i \(-0.106802\pi\)
0.944237 + 0.329267i \(0.106802\pi\)
\(702\) 0 0
\(703\) −6.00000 −0.226294
\(704\) 0 0
\(705\) 36.1421 1.36119
\(706\) 0 0
\(707\) −9.07107 −0.341153
\(708\) 0 0
\(709\) −1.65685 −0.0622245 −0.0311122 0.999516i \(-0.509905\pi\)
−0.0311122 + 0.999516i \(0.509905\pi\)
\(710\) 0 0
\(711\) 2.34315 0.0878748
\(712\) 0 0
\(713\) −74.9117 −2.80546
\(714\) 0 0
\(715\) 54.6274 2.04295
\(716\) 0 0
\(717\) −2.34315 −0.0875064
\(718\) 0 0
\(719\) 6.38478 0.238112 0.119056 0.992888i \(-0.462013\pi\)
0.119056 + 0.992888i \(0.462013\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 14.0000 0.520666
\(724\) 0 0
\(725\) 68.1838 2.53228
\(726\) 0 0
\(727\) −26.6274 −0.987556 −0.493778 0.869588i \(-0.664385\pi\)
−0.493778 + 0.869588i \(0.664385\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −19.7990 −0.732292
\(732\) 0 0
\(733\) −20.8284 −0.769316 −0.384658 0.923059i \(-0.625681\pi\)
−0.384658 + 0.923059i \(0.625681\pi\)
\(734\) 0 0
\(735\) −3.41421 −0.125935
\(736\) 0 0
\(737\) 25.9411 0.955554
\(738\) 0 0
\(739\) −37.7990 −1.39046 −0.695229 0.718788i \(-0.744697\pi\)
−0.695229 + 0.718788i \(0.744697\pi\)
\(740\) 0 0
\(741\) 5.65685 0.207810
\(742\) 0 0
\(743\) −39.0711 −1.43338 −0.716689 0.697393i \(-0.754343\pi\)
−0.716689 + 0.697393i \(0.754343\pi\)
\(744\) 0 0
\(745\) −63.1127 −2.31227
\(746\) 0 0
\(747\) −12.2426 −0.447935
\(748\) 0 0
\(749\) 16.7279 0.611225
\(750\) 0 0
\(751\) 12.2010 0.445221 0.222611 0.974907i \(-0.428542\pi\)
0.222611 + 0.974907i \(0.428542\pi\)
\(752\) 0 0
\(753\) −4.24264 −0.154610
\(754\) 0 0
\(755\) −28.9706 −1.05435
\(756\) 0 0
\(757\) 4.34315 0.157854 0.0789272 0.996880i \(-0.474851\pi\)
0.0789272 + 0.996880i \(0.474851\pi\)
\(758\) 0 0
\(759\) 24.0000 0.871145
\(760\) 0 0
\(761\) 47.2132 1.71148 0.855739 0.517408i \(-0.173103\pi\)
0.855739 + 0.517408i \(0.173103\pi\)
\(762\) 0 0
\(763\) 16.1421 0.584385
\(764\) 0 0
\(765\) −7.65685 −0.276834
\(766\) 0 0
\(767\) −13.2548 −0.478604
\(768\) 0 0
\(769\) 1.02944 0.0371225 0.0185612 0.999828i \(-0.494091\pi\)
0.0185612 + 0.999828i \(0.494091\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) −13.5147 −0.486091 −0.243045 0.970015i \(-0.578146\pi\)
−0.243045 + 0.970015i \(0.578146\pi\)
\(774\) 0 0
\(775\) 58.7696 2.11106
\(776\) 0 0
\(777\) 6.00000 0.215249
\(778\) 0 0
\(779\) −8.82843 −0.316311
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) 10.2426 0.366042
\(784\) 0 0
\(785\) 26.1421 0.933053
\(786\) 0 0
\(787\) 16.1421 0.575405 0.287702 0.957720i \(-0.407109\pi\)
0.287702 + 0.957720i \(0.407109\pi\)
\(788\) 0 0
\(789\) 1.65685 0.0589856
\(790\) 0 0
\(791\) −8.58579 −0.305275
\(792\) 0 0
\(793\) 11.3137 0.401762
\(794\) 0 0
\(795\) 17.3137 0.614054
\(796\) 0 0
\(797\) 6.48528 0.229720 0.114860 0.993382i \(-0.463358\pi\)
0.114860 + 0.993382i \(0.463358\pi\)
\(798\) 0 0
\(799\) −23.7401 −0.839865
\(800\) 0 0
\(801\) −16.8284 −0.594603
\(802\) 0 0
\(803\) 24.9706 0.881192
\(804\) 0 0
\(805\) −28.9706 −1.02108
\(806\) 0 0
\(807\) −11.6569 −0.410341
\(808\) 0 0
\(809\) −26.9706 −0.948234 −0.474117 0.880462i \(-0.657233\pi\)
−0.474117 + 0.880462i \(0.657233\pi\)
\(810\) 0 0
\(811\) −44.9706 −1.57913 −0.789565 0.613667i \(-0.789694\pi\)
−0.789565 + 0.613667i \(0.789694\pi\)
\(812\) 0 0
\(813\) −16.9706 −0.595184
\(814\) 0 0
\(815\) 5.17157 0.181152
\(816\) 0 0
\(817\) −8.82843 −0.308868
\(818\) 0 0
\(819\) −5.65685 −0.197666
\(820\) 0 0
\(821\) −16.8284 −0.587316 −0.293658 0.955911i \(-0.594873\pi\)
−0.293658 + 0.955911i \(0.594873\pi\)
\(822\) 0 0
\(823\) 44.1421 1.53870 0.769349 0.638829i \(-0.220581\pi\)
0.769349 + 0.638829i \(0.220581\pi\)
\(824\) 0 0
\(825\) −18.8284 −0.655522
\(826\) 0 0
\(827\) −21.8995 −0.761520 −0.380760 0.924674i \(-0.624338\pi\)
−0.380760 + 0.924674i \(0.624338\pi\)
\(828\) 0 0
\(829\) 50.2843 1.74644 0.873222 0.487322i \(-0.162026\pi\)
0.873222 + 0.487322i \(0.162026\pi\)
\(830\) 0 0
\(831\) −11.3137 −0.392468
\(832\) 0 0
\(833\) 2.24264 0.0777029
\(834\) 0 0
\(835\) 77.9411 2.69726
\(836\) 0 0
\(837\) 8.82843 0.305155
\(838\) 0 0
\(839\) 41.2548 1.42428 0.712138 0.702040i \(-0.247727\pi\)
0.712138 + 0.702040i \(0.247727\pi\)
\(840\) 0 0
\(841\) 75.9117 2.61764
\(842\) 0 0
\(843\) 1.75736 0.0605267
\(844\) 0 0
\(845\) −64.8701 −2.23160
\(846\) 0 0
\(847\) 3.00000 0.103081
\(848\) 0 0
\(849\) 14.1421 0.485357
\(850\) 0 0
\(851\) 50.9117 1.74523
\(852\) 0 0
\(853\) 1.02944 0.0352473 0.0176236 0.999845i \(-0.494390\pi\)
0.0176236 + 0.999845i \(0.494390\pi\)
\(854\) 0 0
\(855\) −3.41421 −0.116764
\(856\) 0 0
\(857\) −26.4853 −0.904720 −0.452360 0.891835i \(-0.649418\pi\)
−0.452360 + 0.891835i \(0.649418\pi\)
\(858\) 0 0
\(859\) 53.6569 1.83075 0.915374 0.402604i \(-0.131895\pi\)
0.915374 + 0.402604i \(0.131895\pi\)
\(860\) 0 0
\(861\) 8.82843 0.300872
\(862\) 0 0
\(863\) 27.7574 0.944871 0.472436 0.881365i \(-0.343375\pi\)
0.472436 + 0.881365i \(0.343375\pi\)
\(864\) 0 0
\(865\) 10.8284 0.368178
\(866\) 0 0
\(867\) −11.9706 −0.406542
\(868\) 0 0
\(869\) −6.62742 −0.224820
\(870\) 0 0
\(871\) −51.8823 −1.75796
\(872\) 0 0
\(873\) −0.343146 −0.0116137
\(874\) 0 0
\(875\) 5.65685 0.191237
\(876\) 0 0
\(877\) 18.6863 0.630991 0.315496 0.948927i \(-0.397829\pi\)
0.315496 + 0.948927i \(0.397829\pi\)
\(878\) 0 0
\(879\) 1.51472 0.0510902
\(880\) 0 0
\(881\) 20.3848 0.686781 0.343390 0.939193i \(-0.388425\pi\)
0.343390 + 0.939193i \(0.388425\pi\)
\(882\) 0 0
\(883\) −44.0000 −1.48072 −0.740359 0.672212i \(-0.765344\pi\)
−0.740359 + 0.672212i \(0.765344\pi\)
\(884\) 0 0
\(885\) 8.00000 0.268917
\(886\) 0 0
\(887\) −42.4264 −1.42454 −0.712270 0.701906i \(-0.752333\pi\)
−0.712270 + 0.701906i \(0.752333\pi\)
\(888\) 0 0
\(889\) 8.48528 0.284587
\(890\) 0 0
\(891\) −2.82843 −0.0947559
\(892\) 0 0
\(893\) −10.5858 −0.354240
\(894\) 0 0
\(895\) −10.4853 −0.350484
\(896\) 0 0
\(897\) −48.0000 −1.60267
\(898\) 0 0
\(899\) 90.4264 3.01589
\(900\) 0 0
\(901\) −11.3726 −0.378876
\(902\) 0 0
\(903\) 8.82843 0.293792
\(904\) 0 0
\(905\) 31.7990 1.05703
\(906\) 0 0
\(907\) −31.5980 −1.04919 −0.524597 0.851351i \(-0.675784\pi\)
−0.524597 + 0.851351i \(0.675784\pi\)
\(908\) 0 0
\(909\) 9.07107 0.300868
\(910\) 0 0
\(911\) −16.4437 −0.544802 −0.272401 0.962184i \(-0.587818\pi\)
−0.272401 + 0.962184i \(0.587818\pi\)
\(912\) 0 0
\(913\) 34.6274 1.14600
\(914\) 0 0
\(915\) −6.82843 −0.225741
\(916\) 0 0
\(917\) −9.89949 −0.326910
\(918\) 0 0
\(919\) −24.0000 −0.791687 −0.395843 0.918318i \(-0.629548\pi\)
−0.395843 + 0.918318i \(0.629548\pi\)
\(920\) 0 0
\(921\) −16.1421 −0.531901
\(922\) 0 0
\(923\) 72.0000 2.36991
\(924\) 0 0
\(925\) −39.9411 −1.31326
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 43.2132 1.41778 0.708890 0.705319i \(-0.249196\pi\)
0.708890 + 0.705319i \(0.249196\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) −28.0416 −0.918042
\(934\) 0 0
\(935\) 21.6569 0.708255
\(936\) 0 0
\(937\) 25.0294 0.817676 0.408838 0.912607i \(-0.365934\pi\)
0.408838 + 0.912607i \(0.365934\pi\)
\(938\) 0 0
\(939\) 2.97056 0.0969407
\(940\) 0 0
\(941\) −30.2843 −0.987239 −0.493620 0.869678i \(-0.664326\pi\)
−0.493620 + 0.869678i \(0.664326\pi\)
\(942\) 0 0
\(943\) 74.9117 2.43946
\(944\) 0 0
\(945\) 3.41421 0.111064
\(946\) 0 0
\(947\) 4.48528 0.145752 0.0728760 0.997341i \(-0.476782\pi\)
0.0728760 + 0.997341i \(0.476782\pi\)
\(948\) 0 0
\(949\) −49.9411 −1.62116
\(950\) 0 0
\(951\) 27.2132 0.882449
\(952\) 0 0
\(953\) −50.0416 −1.62101 −0.810504 0.585734i \(-0.800807\pi\)
−0.810504 + 0.585734i \(0.800807\pi\)
\(954\) 0 0
\(955\) 50.6274 1.63826
\(956\) 0 0
\(957\) −28.9706 −0.936485
\(958\) 0 0
\(959\) 11.6569 0.376419
\(960\) 0 0
\(961\) 46.9411 1.51423
\(962\) 0 0
\(963\) −16.7279 −0.539050
\(964\) 0 0
\(965\) 1.17157 0.0377143
\(966\) 0 0
\(967\) −24.8284 −0.798428 −0.399214 0.916858i \(-0.630717\pi\)
−0.399214 + 0.916858i \(0.630717\pi\)
\(968\) 0 0
\(969\) 2.24264 0.0720440
\(970\) 0 0
\(971\) −12.6863 −0.407122 −0.203561 0.979062i \(-0.565252\pi\)
−0.203561 + 0.979062i \(0.565252\pi\)
\(972\) 0 0
\(973\) 12.4853 0.400260
\(974\) 0 0
\(975\) 37.6569 1.20598
\(976\) 0 0
\(977\) 61.1543 1.95650 0.978250 0.207429i \(-0.0665095\pi\)
0.978250 + 0.207429i \(0.0665095\pi\)
\(978\) 0 0
\(979\) 47.5980 1.52124
\(980\) 0 0
\(981\) −16.1421 −0.515379
\(982\) 0 0
\(983\) 28.2843 0.902128 0.451064 0.892492i \(-0.351045\pi\)
0.451064 + 0.892492i \(0.351045\pi\)
\(984\) 0 0
\(985\) −51.1127 −1.62859
\(986\) 0 0
\(987\) 10.5858 0.336949
\(988\) 0 0
\(989\) 74.9117 2.38205
\(990\) 0 0
\(991\) 24.2843 0.771415 0.385708 0.922621i \(-0.373957\pi\)
0.385708 + 0.922621i \(0.373957\pi\)
\(992\) 0 0
\(993\) −14.1421 −0.448787
\(994\) 0 0
\(995\) 48.2843 1.53071
\(996\) 0 0
\(997\) 19.6569 0.622539 0.311269 0.950322i \(-0.399246\pi\)
0.311269 + 0.950322i \(0.399246\pi\)
\(998\) 0 0
\(999\) −6.00000 −0.189832
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 3192.2.a.r.1.1 2
3.2 odd 2 9576.2.a.by.1.2 2
4.3 odd 2 6384.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.r.1.1 2 1.1 even 1 trivial
6384.2.a.bh.1.1 2 4.3 odd 2
9576.2.a.by.1.2 2 3.2 odd 2