Properties

Label 3192.2.a.u.1.1
Level $3192$
Weight $2$
Character 3192.1
Self dual yes
Analytic conductor $25.488$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.12489\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.64002 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.64002 q^{5} -1.00000 q^{7} +1.00000 q^{9} -6.24977 q^{11} +4.00000 q^{13} +2.64002 q^{15} -2.64002 q^{17} -1.00000 q^{19} +1.00000 q^{21} -6.24977 q^{23} +1.96972 q^{25} -1.00000 q^{27} +3.60975 q^{29} -8.24977 q^{31} +6.24977 q^{33} +2.64002 q^{35} +2.00000 q^{37} -4.00000 q^{39} -5.21949 q^{41} +4.24977 q^{43} -2.64002 q^{45} +1.60975 q^{47} +1.00000 q^{49} +2.64002 q^{51} -12.8898 q^{53} +16.4995 q^{55} +1.00000 q^{57} +1.93945 q^{59} +0.0605522 q^{61} -1.00000 q^{63} -10.5601 q^{65} +6.24977 q^{67} +6.24977 q^{69} +0.329700 q^{71} -1.21949 q^{73} -1.96972 q^{75} +6.24977 q^{77} +0.969724 q^{79} +1.00000 q^{81} +3.67030 q^{83} +6.96972 q^{85} -3.60975 q^{87} +9.21949 q^{89} -4.00000 q^{91} +8.24977 q^{93} +2.64002 q^{95} +9.03028 q^{97} -6.24977 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 2 q^{11} + 12 q^{13} - 3 q^{19} + 3 q^{21} - 2 q^{23} + 5 q^{25} - 3 q^{27} + 2 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} - 12 q^{39} + 2 q^{41} - 4 q^{43} - 4 q^{47} + 3 q^{49} - 14 q^{53} + 16 q^{55} + 3 q^{57} + 4 q^{59} + 2 q^{61} - 3 q^{63} + 2 q^{67} + 2 q^{69} + 8 q^{71} + 14 q^{73} - 5 q^{75} + 2 q^{77} + 2 q^{79} + 3 q^{81} + 4 q^{83} + 20 q^{85} - 2 q^{87} + 10 q^{89} - 12 q^{91} + 8 q^{93} + 28 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.64002 −1.18065 −0.590327 0.807164i \(-0.701001\pi\)
−0.590327 + 0.807164i \(0.701001\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −6.24977 −1.88438 −0.942188 0.335084i \(-0.891235\pi\)
−0.942188 + 0.335084i \(0.891235\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 2.64002 0.681651
\(16\) 0 0
\(17\) −2.64002 −0.640300 −0.320150 0.947367i \(-0.603733\pi\)
−0.320150 + 0.947367i \(0.603733\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −6.24977 −1.30317 −0.651584 0.758577i \(-0.725895\pi\)
−0.651584 + 0.758577i \(0.725895\pi\)
\(24\) 0 0
\(25\) 1.96972 0.393945
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 3.60975 0.670313 0.335157 0.942162i \(-0.391211\pi\)
0.335157 + 0.942162i \(0.391211\pi\)
\(30\) 0 0
\(31\) −8.24977 −1.48170 −0.740851 0.671669i \(-0.765578\pi\)
−0.740851 + 0.671669i \(0.765578\pi\)
\(32\) 0 0
\(33\) 6.24977 1.08795
\(34\) 0 0
\(35\) 2.64002 0.446245
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −5.21949 −0.815148 −0.407574 0.913172i \(-0.633625\pi\)
−0.407574 + 0.913172i \(0.633625\pi\)
\(42\) 0 0
\(43\) 4.24977 0.648084 0.324042 0.946043i \(-0.394958\pi\)
0.324042 + 0.946043i \(0.394958\pi\)
\(44\) 0 0
\(45\) −2.64002 −0.393551
\(46\) 0 0
\(47\) 1.60975 0.234806 0.117403 0.993084i \(-0.462543\pi\)
0.117403 + 0.993084i \(0.462543\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.64002 0.369677
\(52\) 0 0
\(53\) −12.8898 −1.77055 −0.885275 0.465068i \(-0.846030\pi\)
−0.885275 + 0.465068i \(0.846030\pi\)
\(54\) 0 0
\(55\) 16.4995 2.22480
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) 1.93945 0.252495 0.126247 0.991999i \(-0.459707\pi\)
0.126247 + 0.991999i \(0.459707\pi\)
\(60\) 0 0
\(61\) 0.0605522 0.00775291 0.00387646 0.999992i \(-0.498766\pi\)
0.00387646 + 0.999992i \(0.498766\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −10.5601 −1.30982
\(66\) 0 0
\(67\) 6.24977 0.763531 0.381766 0.924259i \(-0.375316\pi\)
0.381766 + 0.924259i \(0.375316\pi\)
\(68\) 0 0
\(69\) 6.24977 0.752384
\(70\) 0 0
\(71\) 0.329700 0.0391282 0.0195641 0.999809i \(-0.493772\pi\)
0.0195641 + 0.999809i \(0.493772\pi\)
\(72\) 0 0
\(73\) −1.21949 −0.142731 −0.0713655 0.997450i \(-0.522736\pi\)
−0.0713655 + 0.997450i \(0.522736\pi\)
\(74\) 0 0
\(75\) −1.96972 −0.227444
\(76\) 0 0
\(77\) 6.24977 0.712227
\(78\) 0 0
\(79\) 0.969724 0.109102 0.0545512 0.998511i \(-0.482627\pi\)
0.0545512 + 0.998511i \(0.482627\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.67030 0.402868 0.201434 0.979502i \(-0.435440\pi\)
0.201434 + 0.979502i \(0.435440\pi\)
\(84\) 0 0
\(85\) 6.96972 0.755973
\(86\) 0 0
\(87\) −3.60975 −0.387006
\(88\) 0 0
\(89\) 9.21949 0.977264 0.488632 0.872490i \(-0.337496\pi\)
0.488632 + 0.872490i \(0.337496\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 0 0
\(93\) 8.24977 0.855461
\(94\) 0 0
\(95\) 2.64002 0.270861
\(96\) 0 0
\(97\) 9.03028 0.916886 0.458443 0.888724i \(-0.348407\pi\)
0.458443 + 0.888724i \(0.348407\pi\)
\(98\) 0 0
\(99\) −6.24977 −0.628126
\(100\) 0 0
\(101\) 2.64002 0.262692 0.131346 0.991337i \(-0.458070\pi\)
0.131346 + 0.991337i \(0.458070\pi\)
\(102\) 0 0
\(103\) −1.93945 −0.191099 −0.0955497 0.995425i \(-0.530461\pi\)
−0.0955497 + 0.995425i \(0.530461\pi\)
\(104\) 0 0
\(105\) −2.64002 −0.257640
\(106\) 0 0
\(107\) −8.82924 −0.853555 −0.426778 0.904357i \(-0.640351\pi\)
−0.426778 + 0.904357i \(0.640351\pi\)
\(108\) 0 0
\(109\) 7.28005 0.697302 0.348651 0.937253i \(-0.386640\pi\)
0.348651 + 0.937253i \(0.386640\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −4.88979 −0.459993 −0.229997 0.973191i \(-0.573872\pi\)
−0.229997 + 0.973191i \(0.573872\pi\)
\(114\) 0 0
\(115\) 16.4995 1.53859
\(116\) 0 0
\(117\) 4.00000 0.369800
\(118\) 0 0
\(119\) 2.64002 0.242011
\(120\) 0 0
\(121\) 28.0596 2.55088
\(122\) 0 0
\(123\) 5.21949 0.470626
\(124\) 0 0
\(125\) 8.00000 0.715542
\(126\) 0 0
\(127\) −14.7493 −1.30879 −0.654395 0.756153i \(-0.727077\pi\)
−0.654395 + 0.756153i \(0.727077\pi\)
\(128\) 0 0
\(129\) −4.24977 −0.374171
\(130\) 0 0
\(131\) 20.9503 1.83044 0.915220 0.402954i \(-0.132017\pi\)
0.915220 + 0.402954i \(0.132017\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) 2.64002 0.227217
\(136\) 0 0
\(137\) 6.49954 0.555293 0.277647 0.960683i \(-0.410446\pi\)
0.277647 + 0.960683i \(0.410446\pi\)
\(138\) 0 0
\(139\) 7.21949 0.612350 0.306175 0.951975i \(-0.400951\pi\)
0.306175 + 0.951975i \(0.400951\pi\)
\(140\) 0 0
\(141\) −1.60975 −0.135565
\(142\) 0 0
\(143\) −24.9991 −2.09053
\(144\) 0 0
\(145\) −9.52982 −0.791408
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 21.8401 1.78921 0.894607 0.446854i \(-0.147456\pi\)
0.894607 + 0.446854i \(0.147456\pi\)
\(150\) 0 0
\(151\) −2.24977 −0.183084 −0.0915419 0.995801i \(-0.529180\pi\)
−0.0915419 + 0.995801i \(0.529180\pi\)
\(152\) 0 0
\(153\) −2.64002 −0.213433
\(154\) 0 0
\(155\) 21.7796 1.74938
\(156\) 0 0
\(157\) −6.49954 −0.518720 −0.259360 0.965781i \(-0.583512\pi\)
−0.259360 + 0.965781i \(0.583512\pi\)
\(158\) 0 0
\(159\) 12.8898 1.02223
\(160\) 0 0
\(161\) 6.24977 0.492551
\(162\) 0 0
\(163\) −14.8099 −1.16000 −0.579999 0.814617i \(-0.696947\pi\)
−0.579999 + 0.814617i \(0.696947\pi\)
\(164\) 0 0
\(165\) −16.4995 −1.28449
\(166\) 0 0
\(167\) 19.7190 1.52590 0.762952 0.646455i \(-0.223749\pi\)
0.762952 + 0.646455i \(0.223749\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −3.28005 −0.249377 −0.124689 0.992196i \(-0.539793\pi\)
−0.124689 + 0.992196i \(0.539793\pi\)
\(174\) 0 0
\(175\) −1.96972 −0.148897
\(176\) 0 0
\(177\) −1.93945 −0.145778
\(178\) 0 0
\(179\) 19.3893 1.44923 0.724614 0.689155i \(-0.242018\pi\)
0.724614 + 0.689155i \(0.242018\pi\)
\(180\) 0 0
\(181\) 6.47018 0.480925 0.240462 0.970658i \(-0.422701\pi\)
0.240462 + 0.970658i \(0.422701\pi\)
\(182\) 0 0
\(183\) −0.0605522 −0.00447615
\(184\) 0 0
\(185\) −5.28005 −0.388197
\(186\) 0 0
\(187\) 16.4995 1.20657
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −8.31032 −0.601314 −0.300657 0.953732i \(-0.597206\pi\)
−0.300657 + 0.953732i \(0.597206\pi\)
\(192\) 0 0
\(193\) −8.56009 −0.616169 −0.308085 0.951359i \(-0.599688\pi\)
−0.308085 + 0.951359i \(0.599688\pi\)
\(194\) 0 0
\(195\) 10.5601 0.756224
\(196\) 0 0
\(197\) 14.0000 0.997459 0.498729 0.866758i \(-0.333800\pi\)
0.498729 + 0.866758i \(0.333800\pi\)
\(198\) 0 0
\(199\) −11.2195 −0.795329 −0.397664 0.917531i \(-0.630179\pi\)
−0.397664 + 0.917531i \(0.630179\pi\)
\(200\) 0 0
\(201\) −6.24977 −0.440825
\(202\) 0 0
\(203\) −3.60975 −0.253355
\(204\) 0 0
\(205\) 13.7796 0.962408
\(206\) 0 0
\(207\) −6.24977 −0.434389
\(208\) 0 0
\(209\) 6.24977 0.432306
\(210\) 0 0
\(211\) −18.0899 −1.24536 −0.622680 0.782476i \(-0.713956\pi\)
−0.622680 + 0.782476i \(0.713956\pi\)
\(212\) 0 0
\(213\) −0.329700 −0.0225907
\(214\) 0 0
\(215\) −11.2195 −0.765163
\(216\) 0 0
\(217\) 8.24977 0.560031
\(218\) 0 0
\(219\) 1.21949 0.0824058
\(220\) 0 0
\(221\) −10.5601 −0.710349
\(222\) 0 0
\(223\) 6.31032 0.422570 0.211285 0.977424i \(-0.432235\pi\)
0.211285 + 0.977424i \(0.432235\pi\)
\(224\) 0 0
\(225\) 1.96972 0.131315
\(226\) 0 0
\(227\) 7.21949 0.479175 0.239587 0.970875i \(-0.422988\pi\)
0.239587 + 0.970875i \(0.422988\pi\)
\(228\) 0 0
\(229\) 21.8401 1.44324 0.721619 0.692291i \(-0.243398\pi\)
0.721619 + 0.692291i \(0.243398\pi\)
\(230\) 0 0
\(231\) −6.24977 −0.411205
\(232\) 0 0
\(233\) −21.7190 −1.42286 −0.711431 0.702756i \(-0.751952\pi\)
−0.711431 + 0.702756i \(0.751952\pi\)
\(234\) 0 0
\(235\) −4.24977 −0.277224
\(236\) 0 0
\(237\) −0.969724 −0.0629903
\(238\) 0 0
\(239\) −28.0294 −1.81307 −0.906534 0.422132i \(-0.861282\pi\)
−0.906534 + 0.422132i \(0.861282\pi\)
\(240\) 0 0
\(241\) −5.52982 −0.356207 −0.178103 0.984012i \(-0.556996\pi\)
−0.178103 + 0.984012i \(0.556996\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.64002 −0.168665
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) −3.67030 −0.232596
\(250\) 0 0
\(251\) −12.9503 −0.817419 −0.408709 0.912665i \(-0.634021\pi\)
−0.408709 + 0.912665i \(0.634021\pi\)
\(252\) 0 0
\(253\) 39.0596 2.45566
\(254\) 0 0
\(255\) −6.96972 −0.436461
\(256\) 0 0
\(257\) −6.49954 −0.405430 −0.202715 0.979238i \(-0.564977\pi\)
−0.202715 + 0.979238i \(0.564977\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 3.60975 0.223438
\(262\) 0 0
\(263\) 28.5289 1.75917 0.879584 0.475744i \(-0.157821\pi\)
0.879584 + 0.475744i \(0.157821\pi\)
\(264\) 0 0
\(265\) 34.0294 2.09041
\(266\) 0 0
\(267\) −9.21949 −0.564224
\(268\) 0 0
\(269\) 24.9385 1.52053 0.760265 0.649614i \(-0.225069\pi\)
0.760265 + 0.649614i \(0.225069\pi\)
\(270\) 0 0
\(271\) 14.4390 0.877106 0.438553 0.898705i \(-0.355491\pi\)
0.438553 + 0.898705i \(0.355491\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −12.3103 −0.742340
\(276\) 0 0
\(277\) 25.5904 1.53758 0.768788 0.639504i \(-0.220860\pi\)
0.768788 + 0.639504i \(0.220860\pi\)
\(278\) 0 0
\(279\) −8.24977 −0.493901
\(280\) 0 0
\(281\) −26.1698 −1.56116 −0.780581 0.625055i \(-0.785077\pi\)
−0.780581 + 0.625055i \(0.785077\pi\)
\(282\) 0 0
\(283\) 17.2800 1.02719 0.513596 0.858032i \(-0.328313\pi\)
0.513596 + 0.858032i \(0.328313\pi\)
\(284\) 0 0
\(285\) −2.64002 −0.156381
\(286\) 0 0
\(287\) 5.21949 0.308097
\(288\) 0 0
\(289\) −10.0303 −0.590016
\(290\) 0 0
\(291\) −9.03028 −0.529364
\(292\) 0 0
\(293\) 11.4012 0.666062 0.333031 0.942916i \(-0.391929\pi\)
0.333031 + 0.942916i \(0.391929\pi\)
\(294\) 0 0
\(295\) −5.12019 −0.298109
\(296\) 0 0
\(297\) 6.24977 0.362648
\(298\) 0 0
\(299\) −24.9991 −1.44573
\(300\) 0 0
\(301\) −4.24977 −0.244953
\(302\) 0 0
\(303\) −2.64002 −0.151665
\(304\) 0 0
\(305\) −0.159859 −0.00915351
\(306\) 0 0
\(307\) −12.7493 −0.727642 −0.363821 0.931469i \(-0.618528\pi\)
−0.363821 + 0.931469i \(0.618528\pi\)
\(308\) 0 0
\(309\) 1.93945 0.110331
\(310\) 0 0
\(311\) 18.8898 1.07114 0.535571 0.844490i \(-0.320096\pi\)
0.535571 + 0.844490i \(0.320096\pi\)
\(312\) 0 0
\(313\) 16.4390 0.929187 0.464593 0.885524i \(-0.346201\pi\)
0.464593 + 0.885524i \(0.346201\pi\)
\(314\) 0 0
\(315\) 2.64002 0.148748
\(316\) 0 0
\(317\) 4.23039 0.237603 0.118801 0.992918i \(-0.462095\pi\)
0.118801 + 0.992918i \(0.462095\pi\)
\(318\) 0 0
\(319\) −22.5601 −1.26312
\(320\) 0 0
\(321\) 8.82924 0.492800
\(322\) 0 0
\(323\) 2.64002 0.146895
\(324\) 0 0
\(325\) 7.87890 0.437042
\(326\) 0 0
\(327\) −7.28005 −0.402588
\(328\) 0 0
\(329\) −1.60975 −0.0887482
\(330\) 0 0
\(331\) −20.1892 −1.10970 −0.554850 0.831950i \(-0.687224\pi\)
−0.554850 + 0.831950i \(0.687224\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) −16.4995 −0.901466
\(336\) 0 0
\(337\) 23.1589 1.26155 0.630774 0.775967i \(-0.282738\pi\)
0.630774 + 0.775967i \(0.282738\pi\)
\(338\) 0 0
\(339\) 4.88979 0.265577
\(340\) 0 0
\(341\) 51.5592 2.79209
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −16.4995 −0.888305
\(346\) 0 0
\(347\) −21.4693 −1.15253 −0.576265 0.817263i \(-0.695490\pi\)
−0.576265 + 0.817263i \(0.695490\pi\)
\(348\) 0 0
\(349\) −14.9991 −0.802883 −0.401441 0.915885i \(-0.631491\pi\)
−0.401441 + 0.915885i \(0.631491\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −34.4802 −1.83519 −0.917597 0.397512i \(-0.869874\pi\)
−0.917597 + 0.397512i \(0.869874\pi\)
\(354\) 0 0
\(355\) −0.870417 −0.0461969
\(356\) 0 0
\(357\) −2.64002 −0.139725
\(358\) 0 0
\(359\) −8.68876 −0.458575 −0.229288 0.973359i \(-0.573640\pi\)
−0.229288 + 0.973359i \(0.573640\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −28.0596 −1.47275
\(364\) 0 0
\(365\) 3.21949 0.168516
\(366\) 0 0
\(367\) 27.7190 1.44692 0.723461 0.690365i \(-0.242550\pi\)
0.723461 + 0.690365i \(0.242550\pi\)
\(368\) 0 0
\(369\) −5.21949 −0.271716
\(370\) 0 0
\(371\) 12.8898 0.669205
\(372\) 0 0
\(373\) 16.5601 0.857449 0.428725 0.903435i \(-0.358963\pi\)
0.428725 + 0.903435i \(0.358963\pi\)
\(374\) 0 0
\(375\) −8.00000 −0.413118
\(376\) 0 0
\(377\) 14.4390 0.743646
\(378\) 0 0
\(379\) −8.80986 −0.452532 −0.226266 0.974066i \(-0.572652\pi\)
−0.226266 + 0.974066i \(0.572652\pi\)
\(380\) 0 0
\(381\) 14.7493 0.755630
\(382\) 0 0
\(383\) 30.2791 1.54719 0.773596 0.633680i \(-0.218456\pi\)
0.773596 + 0.633680i \(0.218456\pi\)
\(384\) 0 0
\(385\) −16.4995 −0.840895
\(386\) 0 0
\(387\) 4.24977 0.216028
\(388\) 0 0
\(389\) −34.2186 −1.73495 −0.867475 0.497480i \(-0.834259\pi\)
−0.867475 + 0.497480i \(0.834259\pi\)
\(390\) 0 0
\(391\) 16.4995 0.834418
\(392\) 0 0
\(393\) −20.9503 −1.05681
\(394\) 0 0
\(395\) −2.56009 −0.128812
\(396\) 0 0
\(397\) 12.0606 0.605302 0.302651 0.953101i \(-0.402128\pi\)
0.302651 + 0.953101i \(0.402128\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −24.8898 −1.24294 −0.621469 0.783439i \(-0.713464\pi\)
−0.621469 + 0.783439i \(0.713464\pi\)
\(402\) 0 0
\(403\) −32.9991 −1.64380
\(404\) 0 0
\(405\) −2.64002 −0.131184
\(406\) 0 0
\(407\) −12.4995 −0.619579
\(408\) 0 0
\(409\) 18.5601 0.917738 0.458869 0.888504i \(-0.348255\pi\)
0.458869 + 0.888504i \(0.348255\pi\)
\(410\) 0 0
\(411\) −6.49954 −0.320599
\(412\) 0 0
\(413\) −1.93945 −0.0954340
\(414\) 0 0
\(415\) −9.68968 −0.475648
\(416\) 0 0
\(417\) −7.21949 −0.353540
\(418\) 0 0
\(419\) −13.6097 −0.664880 −0.332440 0.943124i \(-0.607872\pi\)
−0.332440 + 0.943124i \(0.607872\pi\)
\(420\) 0 0
\(421\) −11.6585 −0.568200 −0.284100 0.958795i \(-0.591695\pi\)
−0.284100 + 0.958795i \(0.591695\pi\)
\(422\) 0 0
\(423\) 1.60975 0.0782686
\(424\) 0 0
\(425\) −5.20012 −0.252243
\(426\) 0 0
\(427\) −0.0605522 −0.00293033
\(428\) 0 0
\(429\) 24.9991 1.20697
\(430\) 0 0
\(431\) −22.7687 −1.09673 −0.548365 0.836239i \(-0.684749\pi\)
−0.548365 + 0.836239i \(0.684749\pi\)
\(432\) 0 0
\(433\) −36.5895 −1.75838 −0.879188 0.476474i \(-0.841915\pi\)
−0.879188 + 0.476474i \(0.841915\pi\)
\(434\) 0 0
\(435\) 9.52982 0.456920
\(436\) 0 0
\(437\) 6.24977 0.298967
\(438\) 0 0
\(439\) −10.0681 −0.480525 −0.240262 0.970708i \(-0.577234\pi\)
−0.240262 + 0.970708i \(0.577234\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −23.6509 −1.12369 −0.561845 0.827243i \(-0.689908\pi\)
−0.561845 + 0.827243i \(0.689908\pi\)
\(444\) 0 0
\(445\) −24.3397 −1.15381
\(446\) 0 0
\(447\) −21.8401 −1.03300
\(448\) 0 0
\(449\) 10.2909 0.485660 0.242830 0.970069i \(-0.421924\pi\)
0.242830 + 0.970069i \(0.421924\pi\)
\(450\) 0 0
\(451\) 32.6206 1.53605
\(452\) 0 0
\(453\) 2.24977 0.105703
\(454\) 0 0
\(455\) 10.5601 0.495065
\(456\) 0 0
\(457\) 26.5895 1.24380 0.621901 0.783096i \(-0.286361\pi\)
0.621901 + 0.783096i \(0.286361\pi\)
\(458\) 0 0
\(459\) 2.64002 0.123226
\(460\) 0 0
\(461\) 13.2001 0.614791 0.307395 0.951582i \(-0.400543\pi\)
0.307395 + 0.951582i \(0.400543\pi\)
\(462\) 0 0
\(463\) 36.8780 1.71387 0.856933 0.515429i \(-0.172367\pi\)
0.856933 + 0.515429i \(0.172367\pi\)
\(464\) 0 0
\(465\) −21.7796 −1.01000
\(466\) 0 0
\(467\) 18.7687 0.868511 0.434256 0.900790i \(-0.357011\pi\)
0.434256 + 0.900790i \(0.357011\pi\)
\(468\) 0 0
\(469\) −6.24977 −0.288588
\(470\) 0 0
\(471\) 6.49954 0.299483
\(472\) 0 0
\(473\) −26.5601 −1.22123
\(474\) 0 0
\(475\) −1.96972 −0.0903771
\(476\) 0 0
\(477\) −12.8898 −0.590183
\(478\) 0 0
\(479\) 9.32878 0.426243 0.213122 0.977026i \(-0.431637\pi\)
0.213122 + 0.977026i \(0.431637\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) −6.24977 −0.284374
\(484\) 0 0
\(485\) −23.8401 −1.08253
\(486\) 0 0
\(487\) −25.4693 −1.15412 −0.577061 0.816701i \(-0.695801\pi\)
−0.577061 + 0.816701i \(0.695801\pi\)
\(488\) 0 0
\(489\) 14.8099 0.669725
\(490\) 0 0
\(491\) 7.65092 0.345281 0.172641 0.984985i \(-0.444770\pi\)
0.172641 + 0.984985i \(0.444770\pi\)
\(492\) 0 0
\(493\) −9.52982 −0.429201
\(494\) 0 0
\(495\) 16.4995 0.741599
\(496\) 0 0
\(497\) −0.329700 −0.0147891
\(498\) 0 0
\(499\) −36.6206 −1.63937 −0.819683 0.572818i \(-0.805850\pi\)
−0.819683 + 0.572818i \(0.805850\pi\)
\(500\) 0 0
\(501\) −19.7190 −0.880982
\(502\) 0 0
\(503\) −15.6703 −0.698704 −0.349352 0.936992i \(-0.613598\pi\)
−0.349352 + 0.936992i \(0.613598\pi\)
\(504\) 0 0
\(505\) −6.96972 −0.310149
\(506\) 0 0
\(507\) −3.00000 −0.133235
\(508\) 0 0
\(509\) −10.6206 −0.470752 −0.235376 0.971904i \(-0.575632\pi\)
−0.235376 + 0.971904i \(0.575632\pi\)
\(510\) 0 0
\(511\) 1.21949 0.0539473
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 5.12019 0.225622
\(516\) 0 0
\(517\) −10.0606 −0.442463
\(518\) 0 0
\(519\) 3.28005 0.143978
\(520\) 0 0
\(521\) −34.0000 −1.48957 −0.744784 0.667306i \(-0.767447\pi\)
−0.744784 + 0.667306i \(0.767447\pi\)
\(522\) 0 0
\(523\) −0.870417 −0.0380607 −0.0190303 0.999819i \(-0.506058\pi\)
−0.0190303 + 0.999819i \(0.506058\pi\)
\(524\) 0 0
\(525\) 1.96972 0.0859658
\(526\) 0 0
\(527\) 21.7796 0.948734
\(528\) 0 0
\(529\) 16.0596 0.698245
\(530\) 0 0
\(531\) 1.93945 0.0841649
\(532\) 0 0
\(533\) −20.8780 −0.904326
\(534\) 0 0
\(535\) 23.3094 1.00775
\(536\) 0 0
\(537\) −19.3893 −0.836712
\(538\) 0 0
\(539\) −6.24977 −0.269197
\(540\) 0 0
\(541\) −2.12110 −0.0911934 −0.0455967 0.998960i \(-0.514519\pi\)
−0.0455967 + 0.998960i \(0.514519\pi\)
\(542\) 0 0
\(543\) −6.47018 −0.277662
\(544\) 0 0
\(545\) −19.2195 −0.823273
\(546\) 0 0
\(547\) −25.8477 −1.10517 −0.552584 0.833457i \(-0.686358\pi\)
−0.552584 + 0.833457i \(0.686358\pi\)
\(548\) 0 0
\(549\) 0.0605522 0.00258430
\(550\) 0 0
\(551\) −3.60975 −0.153780
\(552\) 0 0
\(553\) −0.969724 −0.0412369
\(554\) 0 0
\(555\) 5.28005 0.224126
\(556\) 0 0
\(557\) 26.2186 1.11092 0.555458 0.831544i \(-0.312543\pi\)
0.555458 + 0.831544i \(0.312543\pi\)
\(558\) 0 0
\(559\) 16.9991 0.718985
\(560\) 0 0
\(561\) −16.4995 −0.696611
\(562\) 0 0
\(563\) 35.8401 1.51048 0.755241 0.655447i \(-0.227520\pi\)
0.755241 + 0.655447i \(0.227520\pi\)
\(564\) 0 0
\(565\) 12.9092 0.543093
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 24.4490 1.02495 0.512477 0.858701i \(-0.328728\pi\)
0.512477 + 0.858701i \(0.328728\pi\)
\(570\) 0 0
\(571\) −22.5601 −0.944111 −0.472055 0.881569i \(-0.656488\pi\)
−0.472055 + 0.881569i \(0.656488\pi\)
\(572\) 0 0
\(573\) 8.31032 0.347169
\(574\) 0 0
\(575\) −12.3103 −0.513376
\(576\) 0 0
\(577\) −24.5601 −1.02245 −0.511225 0.859447i \(-0.670808\pi\)
−0.511225 + 0.859447i \(0.670808\pi\)
\(578\) 0 0
\(579\) 8.56009 0.355745
\(580\) 0 0
\(581\) −3.67030 −0.152270
\(582\) 0 0
\(583\) 80.5583 3.33638
\(584\) 0 0
\(585\) −10.5601 −0.436606
\(586\) 0 0
\(587\) −35.7678 −1.47629 −0.738147 0.674640i \(-0.764299\pi\)
−0.738147 + 0.674640i \(0.764299\pi\)
\(588\) 0 0
\(589\) 8.24977 0.339926
\(590\) 0 0
\(591\) −14.0000 −0.575883
\(592\) 0 0
\(593\) 0.0411748 0.00169085 0.000845423 1.00000i \(-0.499731\pi\)
0.000845423 1.00000i \(0.499731\pi\)
\(594\) 0 0
\(595\) −6.96972 −0.285731
\(596\) 0 0
\(597\) 11.2195 0.459183
\(598\) 0 0
\(599\) 35.8889 1.46638 0.733190 0.680024i \(-0.238031\pi\)
0.733190 + 0.680024i \(0.238031\pi\)
\(600\) 0 0
\(601\) −9.52982 −0.388729 −0.194365 0.980929i \(-0.562265\pi\)
−0.194365 + 0.980929i \(0.562265\pi\)
\(602\) 0 0
\(603\) 6.24977 0.254510
\(604\) 0 0
\(605\) −74.0781 −3.01170
\(606\) 0 0
\(607\) 5.12019 0.207822 0.103911 0.994587i \(-0.466864\pi\)
0.103911 + 0.994587i \(0.466864\pi\)
\(608\) 0 0
\(609\) 3.60975 0.146274
\(610\) 0 0
\(611\) 6.43899 0.260494
\(612\) 0 0
\(613\) −1.21193 −0.0489495 −0.0244747 0.999700i \(-0.507791\pi\)
−0.0244747 + 0.999700i \(0.507791\pi\)
\(614\) 0 0
\(615\) −13.7796 −0.555647
\(616\) 0 0
\(617\) 30.9991 1.24798 0.623988 0.781434i \(-0.285511\pi\)
0.623988 + 0.781434i \(0.285511\pi\)
\(618\) 0 0
\(619\) 3.34060 0.134270 0.0671350 0.997744i \(-0.478614\pi\)
0.0671350 + 0.997744i \(0.478614\pi\)
\(620\) 0 0
\(621\) 6.24977 0.250795
\(622\) 0 0
\(623\) −9.21949 −0.369371
\(624\) 0 0
\(625\) −30.9688 −1.23875
\(626\) 0 0
\(627\) −6.24977 −0.249592
\(628\) 0 0
\(629\) −5.28005 −0.210529
\(630\) 0 0
\(631\) 35.8089 1.42553 0.712766 0.701402i \(-0.247442\pi\)
0.712766 + 0.701402i \(0.247442\pi\)
\(632\) 0 0
\(633\) 18.0899 0.719009
\(634\) 0 0
\(635\) 38.9385 1.54523
\(636\) 0 0
\(637\) 4.00000 0.158486
\(638\) 0 0
\(639\) 0.329700 0.0130427
\(640\) 0 0
\(641\) −4.55011 −0.179719 −0.0898593 0.995954i \(-0.528642\pi\)
−0.0898593 + 0.995954i \(0.528642\pi\)
\(642\) 0 0
\(643\) 44.8392 1.76829 0.884143 0.467216i \(-0.154743\pi\)
0.884143 + 0.467216i \(0.154743\pi\)
\(644\) 0 0
\(645\) 11.2195 0.441767
\(646\) 0 0
\(647\) −9.32878 −0.366752 −0.183376 0.983043i \(-0.558703\pi\)
−0.183376 + 0.983043i \(0.558703\pi\)
\(648\) 0 0
\(649\) −12.1211 −0.475795
\(650\) 0 0
\(651\) −8.24977 −0.323334
\(652\) 0 0
\(653\) −13.1807 −0.515802 −0.257901 0.966171i \(-0.583031\pi\)
−0.257901 + 0.966171i \(0.583031\pi\)
\(654\) 0 0
\(655\) −55.3094 −2.16112
\(656\) 0 0
\(657\) −1.21949 −0.0475770
\(658\) 0 0
\(659\) 29.2900 1.14098 0.570489 0.821305i \(-0.306754\pi\)
0.570489 + 0.821305i \(0.306754\pi\)
\(660\) 0 0
\(661\) 31.4381 1.22280 0.611400 0.791322i \(-0.290607\pi\)
0.611400 + 0.791322i \(0.290607\pi\)
\(662\) 0 0
\(663\) 10.5601 0.410120
\(664\) 0 0
\(665\) −2.64002 −0.102376
\(666\) 0 0
\(667\) −22.5601 −0.873530
\(668\) 0 0
\(669\) −6.31032 −0.243971
\(670\) 0 0
\(671\) −0.378437 −0.0146094
\(672\) 0 0
\(673\) 16.0606 0.619089 0.309544 0.950885i \(-0.399823\pi\)
0.309544 + 0.950885i \(0.399823\pi\)
\(674\) 0 0
\(675\) −1.96972 −0.0758147
\(676\) 0 0
\(677\) 30.4995 1.17219 0.586096 0.810241i \(-0.300664\pi\)
0.586096 + 0.810241i \(0.300664\pi\)
\(678\) 0 0
\(679\) −9.03028 −0.346550
\(680\) 0 0
\(681\) −7.21949 −0.276652
\(682\) 0 0
\(683\) −18.1093 −0.692933 −0.346466 0.938062i \(-0.612619\pi\)
−0.346466 + 0.938062i \(0.612619\pi\)
\(684\) 0 0
\(685\) −17.1589 −0.655609
\(686\) 0 0
\(687\) −21.8401 −0.833253
\(688\) 0 0
\(689\) −51.5592 −1.96425
\(690\) 0 0
\(691\) −6.31789 −0.240344 −0.120172 0.992753i \(-0.538345\pi\)
−0.120172 + 0.992753i \(0.538345\pi\)
\(692\) 0 0
\(693\) 6.24977 0.237409
\(694\) 0 0
\(695\) −19.0596 −0.722973
\(696\) 0 0
\(697\) 13.7796 0.521939
\(698\) 0 0
\(699\) 21.7190 0.821489
\(700\) 0 0
\(701\) −19.4399 −0.734235 −0.367118 0.930175i \(-0.619655\pi\)
−0.367118 + 0.930175i \(0.619655\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 4.24977 0.160056
\(706\) 0 0
\(707\) −2.64002 −0.0992883
\(708\) 0 0
\(709\) −25.8477 −0.970731 −0.485365 0.874311i \(-0.661313\pi\)
−0.485365 + 0.874311i \(0.661313\pi\)
\(710\) 0 0
\(711\) 0.969724 0.0363675
\(712\) 0 0
\(713\) 51.5592 1.93091
\(714\) 0 0
\(715\) 65.9982 2.46819
\(716\) 0 0
\(717\) 28.0294 1.04678
\(718\) 0 0
\(719\) −8.32970 −0.310645 −0.155323 0.987864i \(-0.549642\pi\)
−0.155323 + 0.987864i \(0.549642\pi\)
\(720\) 0 0
\(721\) 1.93945 0.0722288
\(722\) 0 0
\(723\) 5.52982 0.205656
\(724\) 0 0
\(725\) 7.11021 0.264066
\(726\) 0 0
\(727\) −29.9394 −1.11039 −0.555196 0.831719i \(-0.687357\pi\)
−0.555196 + 0.831719i \(0.687357\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −11.2195 −0.414968
\(732\) 0 0
\(733\) 12.7200 0.469822 0.234911 0.972017i \(-0.424520\pi\)
0.234911 + 0.972017i \(0.424520\pi\)
\(734\) 0 0
\(735\) 2.64002 0.0973787
\(736\) 0 0
\(737\) −39.0596 −1.43878
\(738\) 0 0
\(739\) 27.8089 1.02297 0.511484 0.859293i \(-0.329096\pi\)
0.511484 + 0.859293i \(0.329096\pi\)
\(740\) 0 0
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 1.60975 0.0590559 0.0295280 0.999564i \(-0.490600\pi\)
0.0295280 + 0.999564i \(0.490600\pi\)
\(744\) 0 0
\(745\) −57.6585 −2.11244
\(746\) 0 0
\(747\) 3.67030 0.134289
\(748\) 0 0
\(749\) 8.82924 0.322613
\(750\) 0 0
\(751\) −39.7484 −1.45044 −0.725220 0.688517i \(-0.758262\pi\)
−0.725220 + 0.688517i \(0.758262\pi\)
\(752\) 0 0
\(753\) 12.9503 0.471937
\(754\) 0 0
\(755\) 5.93945 0.216159
\(756\) 0 0
\(757\) −11.4986 −0.417925 −0.208962 0.977924i \(-0.567009\pi\)
−0.208962 + 0.977924i \(0.567009\pi\)
\(758\) 0 0
\(759\) −39.0596 −1.41777
\(760\) 0 0
\(761\) −19.7602 −0.716307 −0.358154 0.933663i \(-0.616594\pi\)
−0.358154 + 0.933663i \(0.616594\pi\)
\(762\) 0 0
\(763\) −7.28005 −0.263555
\(764\) 0 0
\(765\) 6.96972 0.251991
\(766\) 0 0
\(767\) 7.75779 0.280118
\(768\) 0 0
\(769\) 33.4381 1.20581 0.602904 0.797814i \(-0.294010\pi\)
0.602904 + 0.797814i \(0.294010\pi\)
\(770\) 0 0
\(771\) 6.49954 0.234075
\(772\) 0 0
\(773\) −12.8411 −0.461861 −0.230930 0.972970i \(-0.574177\pi\)
−0.230930 + 0.972970i \(0.574177\pi\)
\(774\) 0 0
\(775\) −16.2498 −0.583709
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 0 0
\(779\) 5.21949 0.187008
\(780\) 0 0
\(781\) −2.06055 −0.0737324
\(782\) 0 0
\(783\) −3.60975 −0.129002
\(784\) 0 0
\(785\) 17.1589 0.612429
\(786\) 0 0
\(787\) 0.870417 0.0310270 0.0155135 0.999880i \(-0.495062\pi\)
0.0155135 + 0.999880i \(0.495062\pi\)
\(788\) 0 0
\(789\) −28.5289 −1.01566
\(790\) 0 0
\(791\) 4.88979 0.173861
\(792\) 0 0
\(793\) 0.242209 0.00860109
\(794\) 0 0
\(795\) −34.0294 −1.20690
\(796\) 0 0
\(797\) 31.7796 1.12569 0.562845 0.826562i \(-0.309707\pi\)
0.562845 + 0.826562i \(0.309707\pi\)
\(798\) 0 0
\(799\) −4.24977 −0.150346
\(800\) 0 0
\(801\) 9.21949 0.325755
\(802\) 0 0
\(803\) 7.62156 0.268959
\(804\) 0 0
\(805\) −16.4995 −0.581532
\(806\) 0 0
\(807\) −24.9385 −0.877878
\(808\) 0 0
\(809\) −43.9394 −1.54483 −0.772414 0.635119i \(-0.780951\pi\)
−0.772414 + 0.635119i \(0.780951\pi\)
\(810\) 0 0
\(811\) 3.37935 0.118665 0.0593326 0.998238i \(-0.481103\pi\)
0.0593326 + 0.998238i \(0.481103\pi\)
\(812\) 0 0
\(813\) −14.4390 −0.506397
\(814\) 0 0
\(815\) 39.0984 1.36956
\(816\) 0 0
\(817\) −4.24977 −0.148681
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 0 0
\(821\) −8.59885 −0.300102 −0.150051 0.988678i \(-0.547944\pi\)
−0.150051 + 0.988678i \(0.547944\pi\)
\(822\) 0 0
\(823\) −12.0076 −0.418557 −0.209279 0.977856i \(-0.567112\pi\)
−0.209279 + 0.977856i \(0.567112\pi\)
\(824\) 0 0
\(825\) 12.3103 0.428590
\(826\) 0 0
\(827\) 27.3893 0.952421 0.476210 0.879331i \(-0.342010\pi\)
0.476210 + 0.879331i \(0.342010\pi\)
\(828\) 0 0
\(829\) 0.909172 0.0315768 0.0157884 0.999875i \(-0.494974\pi\)
0.0157884 + 0.999875i \(0.494974\pi\)
\(830\) 0 0
\(831\) −25.5904 −0.887720
\(832\) 0 0
\(833\) −2.64002 −0.0914714
\(834\) 0 0
\(835\) −52.0587 −1.80157
\(836\) 0 0
\(837\) 8.24977 0.285154
\(838\) 0 0
\(839\) −40.1798 −1.38716 −0.693581 0.720379i \(-0.743968\pi\)
−0.693581 + 0.720379i \(0.743968\pi\)
\(840\) 0 0
\(841\) −15.9697 −0.550680
\(842\) 0 0
\(843\) 26.1698 0.901337
\(844\) 0 0
\(845\) −7.92007 −0.272459
\(846\) 0 0
\(847\) −28.0596 −0.964140
\(848\) 0 0
\(849\) −17.2800 −0.593050
\(850\) 0 0
\(851\) −12.4995 −0.428479
\(852\) 0 0
\(853\) 2.37844 0.0814361 0.0407181 0.999171i \(-0.487035\pi\)
0.0407181 + 0.999171i \(0.487035\pi\)
\(854\) 0 0
\(855\) 2.64002 0.0902869
\(856\) 0 0
\(857\) −39.2800 −1.34178 −0.670890 0.741556i \(-0.734088\pi\)
−0.670890 + 0.741556i \(0.734088\pi\)
\(858\) 0 0
\(859\) −45.7408 −1.56066 −0.780329 0.625370i \(-0.784948\pi\)
−0.780329 + 0.625370i \(0.784948\pi\)
\(860\) 0 0
\(861\) −5.21949 −0.177880
\(862\) 0 0
\(863\) −1.60975 −0.0547964 −0.0273982 0.999625i \(-0.508722\pi\)
−0.0273982 + 0.999625i \(0.508722\pi\)
\(864\) 0 0
\(865\) 8.65940 0.294428
\(866\) 0 0
\(867\) 10.0303 0.340646
\(868\) 0 0
\(869\) −6.06055 −0.205590
\(870\) 0 0
\(871\) 24.9991 0.847062
\(872\) 0 0
\(873\) 9.03028 0.305629
\(874\) 0 0
\(875\) −8.00000 −0.270449
\(876\) 0 0
\(877\) −50.0587 −1.69036 −0.845181 0.534480i \(-0.820508\pi\)
−0.845181 + 0.534480i \(0.820508\pi\)
\(878\) 0 0
\(879\) −11.4012 −0.384551
\(880\) 0 0
\(881\) 17.0790 0.575407 0.287703 0.957720i \(-0.407108\pi\)
0.287703 + 0.957720i \(0.407108\pi\)
\(882\) 0 0
\(883\) 1.06147 0.0357213 0.0178606 0.999840i \(-0.494314\pi\)
0.0178606 + 0.999840i \(0.494314\pi\)
\(884\) 0 0
\(885\) 5.12019 0.172113
\(886\) 0 0
\(887\) 21.7796 0.731287 0.365644 0.930755i \(-0.380849\pi\)
0.365644 + 0.930755i \(0.380849\pi\)
\(888\) 0 0
\(889\) 14.7493 0.494676
\(890\) 0 0
\(891\) −6.24977 −0.209375
\(892\) 0 0
\(893\) −1.60975 −0.0538681
\(894\) 0 0
\(895\) −51.1883 −1.71104
\(896\) 0 0
\(897\) 24.9991 0.834695
\(898\) 0 0
\(899\) −29.7796 −0.993205
\(900\) 0 0
\(901\) 34.0294 1.13368
\(902\) 0 0
\(903\) 4.24977 0.141424
\(904\) 0 0
\(905\) −17.0814 −0.567806
\(906\) 0 0
\(907\) 44.9073 1.49112 0.745562 0.666436i \(-0.232181\pi\)
0.745562 + 0.666436i \(0.232181\pi\)
\(908\) 0 0
\(909\) 2.64002 0.0875641
\(910\) 0 0
\(911\) −19.8889 −0.658948 −0.329474 0.944165i \(-0.606871\pi\)
−0.329474 + 0.944165i \(0.606871\pi\)
\(912\) 0 0
\(913\) −22.9385 −0.759155
\(914\) 0 0
\(915\) 0.159859 0.00528478
\(916\) 0 0
\(917\) −20.9503 −0.691841
\(918\) 0 0
\(919\) 32.9991 1.08854 0.544270 0.838910i \(-0.316807\pi\)
0.544270 + 0.838910i \(0.316807\pi\)
\(920\) 0 0
\(921\) 12.7493 0.420104
\(922\) 0 0
\(923\) 1.31880 0.0434089
\(924\) 0 0
\(925\) 3.93945 0.129528
\(926\) 0 0
\(927\) −1.93945 −0.0636998
\(928\) 0 0
\(929\) −3.95883 −0.129885 −0.0649424 0.997889i \(-0.520686\pi\)
−0.0649424 + 0.997889i \(0.520686\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −18.8898 −0.618424
\(934\) 0 0
\(935\) −43.5592 −1.42454
\(936\) 0 0
\(937\) −35.4986 −1.15969 −0.579845 0.814727i \(-0.696887\pi\)
−0.579845 + 0.814727i \(0.696887\pi\)
\(938\) 0 0
\(939\) −16.4390 −0.536466
\(940\) 0 0
\(941\) 5.55918 0.181224 0.0906120 0.995886i \(-0.471118\pi\)
0.0906120 + 0.995886i \(0.471118\pi\)
\(942\) 0 0
\(943\) 32.6206 1.06227
\(944\) 0 0
\(945\) −2.64002 −0.0858800
\(946\) 0 0
\(947\) −19.8108 −0.643764 −0.321882 0.946780i \(-0.604315\pi\)
−0.321882 + 0.946780i \(0.604315\pi\)
\(948\) 0 0
\(949\) −4.87798 −0.158346
\(950\) 0 0
\(951\) −4.23039 −0.137180
\(952\) 0 0
\(953\) 57.7290 1.87003 0.935013 0.354613i \(-0.115387\pi\)
0.935013 + 0.354613i \(0.115387\pi\)
\(954\) 0 0
\(955\) 21.9394 0.709944
\(956\) 0 0
\(957\) 22.5601 0.729264
\(958\) 0 0
\(959\) −6.49954 −0.209881
\(960\) 0 0
\(961\) 37.0587 1.19544
\(962\) 0 0
\(963\) −8.82924 −0.284518
\(964\) 0 0
\(965\) 22.5988 0.727483
\(966\) 0 0
\(967\) 51.3094 1.65000 0.825000 0.565133i \(-0.191175\pi\)
0.825000 + 0.565133i \(0.191175\pi\)
\(968\) 0 0
\(969\) −2.64002 −0.0848098
\(970\) 0 0
\(971\) −16.6206 −0.533382 −0.266691 0.963782i \(-0.585930\pi\)
−0.266691 + 0.963782i \(0.585930\pi\)
\(972\) 0 0
\(973\) −7.21949 −0.231446
\(974\) 0 0
\(975\) −7.87890 −0.252327
\(976\) 0 0
\(977\) 34.6694 1.10917 0.554586 0.832126i \(-0.312877\pi\)
0.554586 + 0.832126i \(0.312877\pi\)
\(978\) 0 0
\(979\) −57.6197 −1.84153
\(980\) 0 0
\(981\) 7.28005 0.232434
\(982\) 0 0
\(983\) −22.1193 −0.705495 −0.352748 0.935719i \(-0.614753\pi\)
−0.352748 + 0.935719i \(0.614753\pi\)
\(984\) 0 0
\(985\) −36.9603 −1.17765
\(986\) 0 0
\(987\) 1.60975 0.0512388
\(988\) 0 0
\(989\) −26.5601 −0.844562
\(990\) 0 0
\(991\) −28.9697 −0.920254 −0.460127 0.887853i \(-0.652196\pi\)
−0.460127 + 0.887853i \(0.652196\pi\)
\(992\) 0 0
\(993\) 20.1892 0.640685
\(994\) 0 0
\(995\) 29.6197 0.939009
\(996\) 0 0
\(997\) −43.3775 −1.37378 −0.686890 0.726761i \(-0.741025\pi\)
−0.686890 + 0.726761i \(0.741025\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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.u.1.1 3
3.2 odd 2 9576.2.a.cb.1.3 3
4.3 odd 2 6384.2.a.bw.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.u.1.1 3 1.1 even 1 trivial
6384.2.a.bw.1.1 3 4.3 odd 2
9576.2.a.cb.1.3 3 3.2 odd 2