Properties

Label 2184.2.a.v.1.3
Level $2184$
Weight $2$
Character 2184.1
Self dual yes
Analytic conductor $17.439$
Analytic rank $0$
Dimension $4$
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] = [2184,2,Mod(1,2184)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2184, 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("2184.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2184 = 2^{3} \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2184.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: \(17.4393278014\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.138892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 2x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.44214\) of defining polynomial
Character \(\chi\) \(=\) 2184.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.69903 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.69903 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.84830 q^{11} +1.00000 q^{13} -1.69903 q^{15} +0.964026 q^{17} +6.58330 q^{19} +1.00000 q^{21} -7.18524 q^{23} -2.11329 q^{25} -1.00000 q^{27} +8.58330 q^{29} -5.54733 q^{31} -5.84830 q^{33} -1.69903 q^{35} +0.964026 q^{37} -1.00000 q^{39} -4.88427 q^{41} -7.18524 q^{43} +1.69903 q^{45} +4.73501 q^{47} +1.00000 q^{49} -0.964026 q^{51} +13.0173 q^{53} +9.93644 q^{55} -6.58330 q^{57} -4.81232 q^{59} +13.2626 q^{61} -1.00000 q^{63} +1.69903 q^{65} -10.3705 q^{67} +7.18524 q^{69} +4.81232 q^{71} +13.1852 q^{73} +2.11329 q^{75} -5.84830 q^{77} -7.61928 q^{79} +1.00000 q^{81} -0.133069 q^{83} +1.63791 q^{85} -8.58330 q^{87} +2.73501 q^{89} -1.00000 q^{91} +5.54733 q^{93} +11.1852 q^{95} -5.01734 q^{97} +5.84830 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9} - 3 q^{11} + 4 q^{13} - 2 q^{15} + 3 q^{17} - 4 q^{19} + 4 q^{21} - 8 q^{23} + 14 q^{25} - 4 q^{27} + 4 q^{29} + 9 q^{31} + 3 q^{33} - 2 q^{35} + 3 q^{37} - 4 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45} + 15 q^{47} + 4 q^{49} - 3 q^{51} + 13 q^{53} + 7 q^{55} + 4 q^{57} + 8 q^{59} + 9 q^{61} - 4 q^{63} + 2 q^{65} + 8 q^{69} - 8 q^{71} + 32 q^{73} - 14 q^{75} + 3 q^{77} - q^{79} + 4 q^{81} + 13 q^{83} + 17 q^{85} - 4 q^{87} + 7 q^{89} - 4 q^{91} - 9 q^{93} + 24 q^{95} + 19 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(5\) 1.69903 0.759830 0.379915 0.925021i \(-0.375953\pi\)
0.379915 + 0.925021i \(0.375953\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.84830 1.76333 0.881664 0.471878i \(-0.156424\pi\)
0.881664 + 0.471878i \(0.156424\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −1.69903 −0.438688
\(16\) 0 0
\(17\) 0.964026 0.233811 0.116905 0.993143i \(-0.462703\pi\)
0.116905 + 0.993143i \(0.462703\pi\)
\(18\) 0 0
\(19\) 6.58330 1.51031 0.755157 0.655544i \(-0.227561\pi\)
0.755157 + 0.655544i \(0.227561\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −7.18524 −1.49823 −0.749113 0.662442i \(-0.769520\pi\)
−0.749113 + 0.662442i \(0.769520\pi\)
\(24\) 0 0
\(25\) −2.11329 −0.422658
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.58330 1.59388 0.796940 0.604059i \(-0.206451\pi\)
0.796940 + 0.604059i \(0.206451\pi\)
\(30\) 0 0
\(31\) −5.54733 −0.996330 −0.498165 0.867082i \(-0.665993\pi\)
−0.498165 + 0.867082i \(0.665993\pi\)
\(32\) 0 0
\(33\) −5.84830 −1.01806
\(34\) 0 0
\(35\) −1.69903 −0.287189
\(36\) 0 0
\(37\) 0.964026 0.158485 0.0792425 0.996855i \(-0.474750\pi\)
0.0792425 + 0.996855i \(0.474750\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) −4.88427 −0.762795 −0.381397 0.924411i \(-0.624557\pi\)
−0.381397 + 0.924411i \(0.624557\pi\)
\(42\) 0 0
\(43\) −7.18524 −1.09574 −0.547869 0.836564i \(-0.684561\pi\)
−0.547869 + 0.836564i \(0.684561\pi\)
\(44\) 0 0
\(45\) 1.69903 0.253277
\(46\) 0 0
\(47\) 4.73501 0.690672 0.345336 0.938479i \(-0.387765\pi\)
0.345336 + 0.938479i \(0.387765\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.964026 −0.134991
\(52\) 0 0
\(53\) 13.0173 1.78807 0.894035 0.447998i \(-0.147863\pi\)
0.894035 + 0.447998i \(0.147863\pi\)
\(54\) 0 0
\(55\) 9.93644 1.33983
\(56\) 0 0
\(57\) −6.58330 −0.871980
\(58\) 0 0
\(59\) −4.81232 −0.626511 −0.313256 0.949669i \(-0.601420\pi\)
−0.313256 + 0.949669i \(0.601420\pi\)
\(60\) 0 0
\(61\) 13.2626 1.69810 0.849048 0.528315i \(-0.177176\pi\)
0.849048 + 0.528315i \(0.177176\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 1.69903 0.210739
\(66\) 0 0
\(67\) −10.3705 −1.26696 −0.633478 0.773761i \(-0.718373\pi\)
−0.633478 + 0.773761i \(0.718373\pi\)
\(68\) 0 0
\(69\) 7.18524 0.865001
\(70\) 0 0
\(71\) 4.81232 0.571118 0.285559 0.958361i \(-0.407821\pi\)
0.285559 + 0.958361i \(0.407821\pi\)
\(72\) 0 0
\(73\) 13.1852 1.54322 0.771608 0.636099i \(-0.219453\pi\)
0.771608 + 0.636099i \(0.219453\pi\)
\(74\) 0 0
\(75\) 2.11329 0.244022
\(76\) 0 0
\(77\) −5.84830 −0.666475
\(78\) 0 0
\(79\) −7.61928 −0.857236 −0.428618 0.903486i \(-0.640999\pi\)
−0.428618 + 0.903486i \(0.640999\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.133069 −0.0146062 −0.00730310 0.999973i \(-0.502325\pi\)
−0.00730310 + 0.999973i \(0.502325\pi\)
\(84\) 0 0
\(85\) 1.63791 0.177656
\(86\) 0 0
\(87\) −8.58330 −0.920227
\(88\) 0 0
\(89\) 2.73501 0.289910 0.144955 0.989438i \(-0.453696\pi\)
0.144955 + 0.989438i \(0.453696\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) 5.54733 0.575231
\(94\) 0 0
\(95\) 11.1852 1.14758
\(96\) 0 0
\(97\) −5.01734 −0.509434 −0.254717 0.967016i \(-0.581982\pi\)
−0.254717 + 0.967016i \(0.581982\pi\)
\(98\) 0 0
\(99\) 5.84830 0.587776
\(100\) 0 0
\(101\) 14.9005 1.48265 0.741326 0.671145i \(-0.234197\pi\)
0.741326 + 0.671145i \(0.234197\pi\)
\(102\) 0 0
\(103\) 2.36209 0.232744 0.116372 0.993206i \(-0.462874\pi\)
0.116372 + 0.993206i \(0.462874\pi\)
\(104\) 0 0
\(105\) 1.69903 0.165809
\(106\) 0 0
\(107\) −5.92805 −0.573086 −0.286543 0.958067i \(-0.592506\pi\)
−0.286543 + 0.958067i \(0.592506\pi\)
\(108\) 0 0
\(109\) 9.56596 0.916253 0.458127 0.888887i \(-0.348521\pi\)
0.458127 + 0.888887i \(0.348521\pi\)
\(110\) 0 0
\(111\) −0.964026 −0.0915013
\(112\) 0 0
\(113\) 0.751202 0.0706672 0.0353336 0.999376i \(-0.488751\pi\)
0.0353336 + 0.999376i \(0.488751\pi\)
\(114\) 0 0
\(115\) −12.2079 −1.13840
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −0.964026 −0.0883721
\(120\) 0 0
\(121\) 23.2026 2.10933
\(122\) 0 0
\(123\) 4.88427 0.440400
\(124\) 0 0
\(125\) −12.0857 −1.08098
\(126\) 0 0
\(127\) 15.8405 1.40562 0.702808 0.711379i \(-0.251929\pi\)
0.702808 + 0.711379i \(0.251929\pi\)
\(128\) 0 0
\(129\) 7.18524 0.632625
\(130\) 0 0
\(131\) 6.36209 0.555858 0.277929 0.960602i \(-0.410352\pi\)
0.277929 + 0.960602i \(0.410352\pi\)
\(132\) 0 0
\(133\) −6.58330 −0.570845
\(134\) 0 0
\(135\) −1.69903 −0.146229
\(136\) 0 0
\(137\) −19.6888 −1.68213 −0.841063 0.540937i \(-0.818070\pi\)
−0.841063 + 0.540937i \(0.818070\pi\)
\(138\) 0 0
\(139\) 4.26614 0.361849 0.180925 0.983497i \(-0.442091\pi\)
0.180925 + 0.983497i \(0.442091\pi\)
\(140\) 0 0
\(141\) −4.73501 −0.398759
\(142\) 0 0
\(143\) 5.84830 0.489059
\(144\) 0 0
\(145\) 14.5833 1.21108
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −22.5089 −1.84400 −0.922001 0.387187i \(-0.873447\pi\)
−0.922001 + 0.387187i \(0.873447\pi\)
\(150\) 0 0
\(151\) 18.3621 1.49429 0.747143 0.664664i \(-0.231425\pi\)
0.747143 + 0.664664i \(0.231425\pi\)
\(152\) 0 0
\(153\) 0.964026 0.0779369
\(154\) 0 0
\(155\) −9.42509 −0.757041
\(156\) 0 0
\(157\) 8.96403 0.715407 0.357704 0.933835i \(-0.383560\pi\)
0.357704 + 0.933835i \(0.383560\pi\)
\(158\) 0 0
\(159\) −13.0173 −1.03234
\(160\) 0 0
\(161\) 7.18524 0.566276
\(162\) 0 0
\(163\) 7.69659 0.602844 0.301422 0.953491i \(-0.402539\pi\)
0.301422 + 0.953491i \(0.402539\pi\)
\(164\) 0 0
\(165\) −9.93644 −0.773551
\(166\) 0 0
\(167\) −1.16904 −0.0904632 −0.0452316 0.998977i \(-0.514403\pi\)
−0.0452316 + 0.998977i \(0.514403\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 6.58330 0.503438
\(172\) 0 0
\(173\) −0.673885 −0.0512345 −0.0256173 0.999672i \(-0.508155\pi\)
−0.0256173 + 0.999672i \(0.508155\pi\)
\(174\) 0 0
\(175\) 2.11329 0.159750
\(176\) 0 0
\(177\) 4.81232 0.361716
\(178\) 0 0
\(179\) 1.24880 0.0933395 0.0466698 0.998910i \(-0.485139\pi\)
0.0466698 + 0.998910i \(0.485139\pi\)
\(180\) 0 0
\(181\) −0.796126 −0.0591756 −0.0295878 0.999562i \(-0.509419\pi\)
−0.0295878 + 0.999562i \(0.509419\pi\)
\(182\) 0 0
\(183\) −13.2626 −0.980396
\(184\) 0 0
\(185\) 1.63791 0.120422
\(186\) 0 0
\(187\) 5.63791 0.412285
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −4.29014 −0.310424 −0.155212 0.987881i \(-0.549606\pi\)
−0.155212 + 0.987881i \(0.549606\pi\)
\(192\) 0 0
\(193\) 12.0719 0.868958 0.434479 0.900682i \(-0.356933\pi\)
0.434479 + 0.900682i \(0.356933\pi\)
\(194\) 0 0
\(195\) −1.69903 −0.121670
\(196\) 0 0
\(197\) −19.6804 −1.40217 −0.701085 0.713078i \(-0.747301\pi\)
−0.701085 + 0.713078i \(0.747301\pi\)
\(198\) 0 0
\(199\) 14.0587 0.996594 0.498297 0.867007i \(-0.333959\pi\)
0.498297 + 0.867007i \(0.333959\pi\)
\(200\) 0 0
\(201\) 10.3705 0.731477
\(202\) 0 0
\(203\) −8.58330 −0.602430
\(204\) 0 0
\(205\) −8.29853 −0.579595
\(206\) 0 0
\(207\) −7.18524 −0.499409
\(208\) 0 0
\(209\) 38.5011 2.66318
\(210\) 0 0
\(211\) 4.51135 0.310574 0.155287 0.987869i \(-0.450370\pi\)
0.155287 + 0.987869i \(0.450370\pi\)
\(212\) 0 0
\(213\) −4.81232 −0.329735
\(214\) 0 0
\(215\) −12.2079 −0.832575
\(216\) 0 0
\(217\) 5.54733 0.376577
\(218\) 0 0
\(219\) −13.1852 −0.890976
\(220\) 0 0
\(221\) 0.964026 0.0648474
\(222\) 0 0
\(223\) 7.61928 0.510225 0.255112 0.966911i \(-0.417888\pi\)
0.255112 + 0.966911i \(0.417888\pi\)
\(224\) 0 0
\(225\) −2.11329 −0.140886
\(226\) 0 0
\(227\) 16.5089 1.09574 0.547868 0.836565i \(-0.315440\pi\)
0.547868 + 0.836565i \(0.315440\pi\)
\(228\) 0 0
\(229\) 4.79613 0.316937 0.158468 0.987364i \(-0.449344\pi\)
0.158468 + 0.987364i \(0.449344\pi\)
\(230\) 0 0
\(231\) 5.84830 0.384790
\(232\) 0 0
\(233\) −21.6193 −1.41633 −0.708163 0.706049i \(-0.750476\pi\)
−0.708163 + 0.706049i \(0.750476\pi\)
\(234\) 0 0
\(235\) 8.04492 0.524793
\(236\) 0 0
\(237\) 7.61928 0.494925
\(238\) 0 0
\(239\) 6.01620 0.389155 0.194578 0.980887i \(-0.437666\pi\)
0.194578 + 0.980887i \(0.437666\pi\)
\(240\) 0 0
\(241\) 8.57491 0.552359 0.276179 0.961106i \(-0.410932\pi\)
0.276179 + 0.961106i \(0.410932\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.69903 0.108547
\(246\) 0 0
\(247\) 6.58330 0.418886
\(248\) 0 0
\(249\) 0.133069 0.00843289
\(250\) 0 0
\(251\) −5.93644 −0.374705 −0.187352 0.982293i \(-0.559991\pi\)
−0.187352 + 0.982293i \(0.559991\pi\)
\(252\) 0 0
\(253\) −42.0214 −2.64186
\(254\) 0 0
\(255\) −1.63791 −0.102570
\(256\) 0 0
\(257\) −20.8285 −1.29925 −0.649624 0.760256i \(-0.725074\pi\)
−0.649624 + 0.760256i \(0.725074\pi\)
\(258\) 0 0
\(259\) −0.964026 −0.0599017
\(260\) 0 0
\(261\) 8.58330 0.531293
\(262\) 0 0
\(263\) −11.0173 −0.679358 −0.339679 0.940541i \(-0.610318\pi\)
−0.339679 + 0.940541i \(0.610318\pi\)
\(264\) 0 0
\(265\) 22.1169 1.35863
\(266\) 0 0
\(267\) −2.73501 −0.167380
\(268\) 0 0
\(269\) −6.26614 −0.382053 −0.191027 0.981585i \(-0.561182\pi\)
−0.191027 + 0.981585i \(0.561182\pi\)
\(270\) 0 0
\(271\) −24.2613 −1.47377 −0.736883 0.676020i \(-0.763703\pi\)
−0.736883 + 0.676020i \(0.763703\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −12.3592 −0.745285
\(276\) 0 0
\(277\) 25.3159 1.52108 0.760542 0.649289i \(-0.224933\pi\)
0.760542 + 0.649289i \(0.224933\pi\)
\(278\) 0 0
\(279\) −5.54733 −0.332110
\(280\) 0 0
\(281\) 5.75235 0.343156 0.171578 0.985171i \(-0.445113\pi\)
0.171578 + 0.985171i \(0.445113\pi\)
\(282\) 0 0
\(283\) −8.14390 −0.484104 −0.242052 0.970263i \(-0.577821\pi\)
−0.242052 + 0.970263i \(0.577821\pi\)
\(284\) 0 0
\(285\) −11.1852 −0.662556
\(286\) 0 0
\(287\) 4.88427 0.288309
\(288\) 0 0
\(289\) −16.0707 −0.945333
\(290\) 0 0
\(291\) 5.01734 0.294122
\(292\) 0 0
\(293\) 18.0060 1.05192 0.525959 0.850510i \(-0.323706\pi\)
0.525959 + 0.850510i \(0.323706\pi\)
\(294\) 0 0
\(295\) −8.17629 −0.476042
\(296\) 0 0
\(297\) −5.84830 −0.339353
\(298\) 0 0
\(299\) −7.18524 −0.415533
\(300\) 0 0
\(301\) 7.18524 0.414150
\(302\) 0 0
\(303\) −14.9005 −0.856010
\(304\) 0 0
\(305\) 22.5335 1.29026
\(306\) 0 0
\(307\) 20.3758 1.16291 0.581456 0.813578i \(-0.302483\pi\)
0.581456 + 0.813578i \(0.302483\pi\)
\(308\) 0 0
\(309\) −2.36209 −0.134375
\(310\) 0 0
\(311\) 25.7313 1.45909 0.729543 0.683935i \(-0.239733\pi\)
0.729543 + 0.683935i \(0.239733\pi\)
\(312\) 0 0
\(313\) −15.4651 −0.874141 −0.437071 0.899427i \(-0.643984\pi\)
−0.437071 + 0.899427i \(0.643984\pi\)
\(314\) 0 0
\(315\) −1.69903 −0.0957296
\(316\) 0 0
\(317\) 26.9513 1.51374 0.756869 0.653566i \(-0.226728\pi\)
0.756869 + 0.653566i \(0.226728\pi\)
\(318\) 0 0
\(319\) 50.1977 2.81053
\(320\) 0 0
\(321\) 5.92805 0.330872
\(322\) 0 0
\(323\) 6.34648 0.353127
\(324\) 0 0
\(325\) −2.11329 −0.117224
\(326\) 0 0
\(327\) −9.56596 −0.528999
\(328\) 0 0
\(329\) −4.73501 −0.261049
\(330\) 0 0
\(331\) −22.2266 −1.22168 −0.610842 0.791753i \(-0.709169\pi\)
−0.610842 + 0.791753i \(0.709169\pi\)
\(332\) 0 0
\(333\) 0.964026 0.0528283
\(334\) 0 0
\(335\) −17.6198 −0.962671
\(336\) 0 0
\(337\) −30.3518 −1.65337 −0.826685 0.562665i \(-0.809776\pi\)
−0.826685 + 0.562665i \(0.809776\pi\)
\(338\) 0 0
\(339\) −0.751202 −0.0407997
\(340\) 0 0
\(341\) −32.4424 −1.75686
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 12.2079 0.657254
\(346\) 0 0
\(347\) −2.23146 −0.119791 −0.0598955 0.998205i \(-0.519077\pi\)
−0.0598955 + 0.998205i \(0.519077\pi\)
\(348\) 0 0
\(349\) 3.54733 0.189884 0.0949421 0.995483i \(-0.469733\pi\)
0.0949421 + 0.995483i \(0.469733\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −10.8123 −0.575482 −0.287741 0.957708i \(-0.592904\pi\)
−0.287741 + 0.957708i \(0.592904\pi\)
\(354\) 0 0
\(355\) 8.17629 0.433952
\(356\) 0 0
\(357\) 0.964026 0.0510217
\(358\) 0 0
\(359\) 6.01620 0.317523 0.158761 0.987317i \(-0.449250\pi\)
0.158761 + 0.987317i \(0.449250\pi\)
\(360\) 0 0
\(361\) 24.3399 1.28105
\(362\) 0 0
\(363\) −23.2026 −1.21782
\(364\) 0 0
\(365\) 22.4021 1.17258
\(366\) 0 0
\(367\) −12.1222 −0.632776 −0.316388 0.948630i \(-0.602470\pi\)
−0.316388 + 0.948630i \(0.602470\pi\)
\(368\) 0 0
\(369\) −4.88427 −0.254265
\(370\) 0 0
\(371\) −13.0173 −0.675827
\(372\) 0 0
\(373\) −32.5251 −1.68409 −0.842043 0.539410i \(-0.818647\pi\)
−0.842043 + 0.539410i \(0.818647\pi\)
\(374\) 0 0
\(375\) 12.0857 0.624103
\(376\) 0 0
\(377\) 8.58330 0.442063
\(378\) 0 0
\(379\) −24.1222 −1.23908 −0.619538 0.784967i \(-0.712680\pi\)
−0.619538 + 0.784967i \(0.712680\pi\)
\(380\) 0 0
\(381\) −15.8405 −0.811533
\(382\) 0 0
\(383\) −12.2188 −0.624350 −0.312175 0.950025i \(-0.601058\pi\)
−0.312175 + 0.950025i \(0.601058\pi\)
\(384\) 0 0
\(385\) −9.93644 −0.506408
\(386\) 0 0
\(387\) −7.18524 −0.365246
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) −6.92676 −0.350301
\(392\) 0 0
\(393\) −6.36209 −0.320925
\(394\) 0 0
\(395\) −12.9454 −0.651353
\(396\) 0 0
\(397\) 7.69122 0.386011 0.193006 0.981198i \(-0.438176\pi\)
0.193006 + 0.981198i \(0.438176\pi\)
\(398\) 0 0
\(399\) 6.58330 0.329577
\(400\) 0 0
\(401\) −14.6577 −0.731970 −0.365985 0.930621i \(-0.619268\pi\)
−0.365985 + 0.930621i \(0.619268\pi\)
\(402\) 0 0
\(403\) −5.54733 −0.276332
\(404\) 0 0
\(405\) 1.69903 0.0844256
\(406\) 0 0
\(407\) 5.63791 0.279461
\(408\) 0 0
\(409\) 29.4465 1.45604 0.728018 0.685558i \(-0.240442\pi\)
0.728018 + 0.685558i \(0.240442\pi\)
\(410\) 0 0
\(411\) 19.6888 0.971176
\(412\) 0 0
\(413\) 4.81232 0.236799
\(414\) 0 0
\(415\) −0.226088 −0.0110982
\(416\) 0 0
\(417\) −4.26614 −0.208914
\(418\) 0 0
\(419\) −21.6006 −1.05526 −0.527630 0.849474i \(-0.676919\pi\)
−0.527630 + 0.849474i \(0.676919\pi\)
\(420\) 0 0
\(421\) −2.26614 −0.110445 −0.0552224 0.998474i \(-0.517587\pi\)
−0.0552224 + 0.998474i \(0.517587\pi\)
\(422\) 0 0
\(423\) 4.73501 0.230224
\(424\) 0 0
\(425\) −2.03727 −0.0988220
\(426\) 0 0
\(427\) −13.2626 −0.641820
\(428\) 0 0
\(429\) −5.84830 −0.282358
\(430\) 0 0
\(431\) −40.7751 −1.96407 −0.982033 0.188711i \(-0.939569\pi\)
−0.982033 + 0.188711i \(0.939569\pi\)
\(432\) 0 0
\(433\) −35.7637 −1.71869 −0.859346 0.511394i \(-0.829129\pi\)
−0.859346 + 0.511394i \(0.829129\pi\)
\(434\) 0 0
\(435\) −14.5833 −0.699216
\(436\) 0 0
\(437\) −47.3026 −2.26279
\(438\) 0 0
\(439\) −12.5887 −0.600825 −0.300412 0.953809i \(-0.597124\pi\)
−0.300412 + 0.953809i \(0.597124\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −32.0400 −1.52227 −0.761134 0.648594i \(-0.775357\pi\)
−0.761134 + 0.648594i \(0.775357\pi\)
\(444\) 0 0
\(445\) 4.64686 0.220282
\(446\) 0 0
\(447\) 22.5089 1.06464
\(448\) 0 0
\(449\) −13.9202 −0.656937 −0.328468 0.944515i \(-0.606532\pi\)
−0.328468 + 0.944515i \(0.606532\pi\)
\(450\) 0 0
\(451\) −28.5647 −1.34506
\(452\) 0 0
\(453\) −18.3621 −0.862726
\(454\) 0 0
\(455\) −1.69903 −0.0796518
\(456\) 0 0
\(457\) 34.5274 1.61512 0.807562 0.589783i \(-0.200787\pi\)
0.807562 + 0.589783i \(0.200787\pi\)
\(458\) 0 0
\(459\) −0.964026 −0.0449969
\(460\) 0 0
\(461\) −19.8434 −0.924200 −0.462100 0.886828i \(-0.652904\pi\)
−0.462100 + 0.886828i \(0.652904\pi\)
\(462\) 0 0
\(463\) −15.7050 −0.729873 −0.364936 0.931033i \(-0.618909\pi\)
−0.364936 + 0.931033i \(0.618909\pi\)
\(464\) 0 0
\(465\) 9.42509 0.437078
\(466\) 0 0
\(467\) 38.1977 1.76758 0.883789 0.467885i \(-0.154984\pi\)
0.883789 + 0.467885i \(0.154984\pi\)
\(468\) 0 0
\(469\) 10.3705 0.478864
\(470\) 0 0
\(471\) −8.96403 −0.413041
\(472\) 0 0
\(473\) −42.0214 −1.93215
\(474\) 0 0
\(475\) −13.9124 −0.638346
\(476\) 0 0
\(477\) 13.0173 0.596023
\(478\) 0 0
\(479\) −33.6066 −1.53552 −0.767762 0.640735i \(-0.778630\pi\)
−0.767762 + 0.640735i \(0.778630\pi\)
\(480\) 0 0
\(481\) 0.964026 0.0439558
\(482\) 0 0
\(483\) −7.18524 −0.326940
\(484\) 0 0
\(485\) −8.52462 −0.387083
\(486\) 0 0
\(487\) −26.0671 −1.18121 −0.590606 0.806960i \(-0.701111\pi\)
−0.590606 + 0.806960i \(0.701111\pi\)
\(488\) 0 0
\(489\) −7.69659 −0.348052
\(490\) 0 0
\(491\) −2.07195 −0.0935057 −0.0467529 0.998906i \(-0.514887\pi\)
−0.0467529 + 0.998906i \(0.514887\pi\)
\(492\) 0 0
\(493\) 8.27453 0.372666
\(494\) 0 0
\(495\) 9.93644 0.446610
\(496\) 0 0
\(497\) −4.81232 −0.215862
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) 1.16904 0.0522290
\(502\) 0 0
\(503\) 31.9951 1.42659 0.713296 0.700863i \(-0.247202\pi\)
0.713296 + 0.700863i \(0.247202\pi\)
\(504\) 0 0
\(505\) 25.3164 1.12656
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −36.8261 −1.63229 −0.816144 0.577849i \(-0.803892\pi\)
−0.816144 + 0.577849i \(0.803892\pi\)
\(510\) 0 0
\(511\) −13.1852 −0.583281
\(512\) 0 0
\(513\) −6.58330 −0.290660
\(514\) 0 0
\(515\) 4.01326 0.176846
\(516\) 0 0
\(517\) 27.6917 1.21788
\(518\) 0 0
\(519\) 0.673885 0.0295803
\(520\) 0 0
\(521\) −29.0658 −1.27339 −0.636697 0.771114i \(-0.719700\pi\)
−0.636697 + 0.771114i \(0.719700\pi\)
\(522\) 0 0
\(523\) −24.1222 −1.05479 −0.527396 0.849620i \(-0.676832\pi\)
−0.527396 + 0.849620i \(0.676832\pi\)
\(524\) 0 0
\(525\) −2.11329 −0.0922316
\(526\) 0 0
\(527\) −5.34777 −0.232952
\(528\) 0 0
\(529\) 28.6277 1.24468
\(530\) 0 0
\(531\) −4.81232 −0.208837
\(532\) 0 0
\(533\) −4.88427 −0.211561
\(534\) 0 0
\(535\) −10.0719 −0.435448
\(536\) 0 0
\(537\) −1.24880 −0.0538896
\(538\) 0 0
\(539\) 5.84830 0.251904
\(540\) 0 0
\(541\) 23.6006 1.01467 0.507335 0.861749i \(-0.330631\pi\)
0.507335 + 0.861749i \(0.330631\pi\)
\(542\) 0 0
\(543\) 0.796126 0.0341651
\(544\) 0 0
\(545\) 16.2529 0.696197
\(546\) 0 0
\(547\) 38.7091 1.65508 0.827540 0.561407i \(-0.189740\pi\)
0.827540 + 0.561407i \(0.189740\pi\)
\(548\) 0 0
\(549\) 13.2626 0.566032
\(550\) 0 0
\(551\) 56.5065 2.40726
\(552\) 0 0
\(553\) 7.61928 0.324005
\(554\) 0 0
\(555\) −1.63791 −0.0695254
\(556\) 0 0
\(557\) −0.474238 −0.0200941 −0.0100470 0.999950i \(-0.503198\pi\)
−0.0100470 + 0.999950i \(0.503198\pi\)
\(558\) 0 0
\(559\) −7.18524 −0.303903
\(560\) 0 0
\(561\) −5.63791 −0.238033
\(562\) 0 0
\(563\) 21.8668 0.921575 0.460787 0.887511i \(-0.347567\pi\)
0.460787 + 0.887511i \(0.347567\pi\)
\(564\) 0 0
\(565\) 1.27632 0.0536950
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −33.7366 −1.41431 −0.707157 0.707057i \(-0.750023\pi\)
−0.707157 + 0.707057i \(0.750023\pi\)
\(570\) 0 0
\(571\) 2.59657 0.108663 0.0543315 0.998523i \(-0.482697\pi\)
0.0543315 + 0.998523i \(0.482697\pi\)
\(572\) 0 0
\(573\) 4.29014 0.179223
\(574\) 0 0
\(575\) 15.1845 0.633238
\(576\) 0 0
\(577\) 46.5647 1.93851 0.969256 0.246053i \(-0.0791339\pi\)
0.969256 + 0.246053i \(0.0791339\pi\)
\(578\) 0 0
\(579\) −12.0719 −0.501693
\(580\) 0 0
\(581\) 0.133069 0.00552062
\(582\) 0 0
\(583\) 76.1293 3.15295
\(584\) 0 0
\(585\) 1.69903 0.0702463
\(586\) 0 0
\(587\) 38.9411 1.60727 0.803636 0.595122i \(-0.202896\pi\)
0.803636 + 0.595122i \(0.202896\pi\)
\(588\) 0 0
\(589\) −36.5197 −1.50477
\(590\) 0 0
\(591\) 19.6804 0.809543
\(592\) 0 0
\(593\) −15.7626 −0.647292 −0.323646 0.946178i \(-0.604909\pi\)
−0.323646 + 0.946178i \(0.604909\pi\)
\(594\) 0 0
\(595\) −1.63791 −0.0671478
\(596\) 0 0
\(597\) −14.0587 −0.575384
\(598\) 0 0
\(599\) 15.3399 0.626770 0.313385 0.949626i \(-0.398537\pi\)
0.313385 + 0.949626i \(0.398537\pi\)
\(600\) 0 0
\(601\) 26.8681 1.09597 0.547986 0.836488i \(-0.315395\pi\)
0.547986 + 0.836488i \(0.315395\pi\)
\(602\) 0 0
\(603\) −10.3705 −0.422319
\(604\) 0 0
\(605\) 39.4219 1.60273
\(606\) 0 0
\(607\) 11.7206 0.475724 0.237862 0.971299i \(-0.423553\pi\)
0.237862 + 0.971299i \(0.423553\pi\)
\(608\) 0 0
\(609\) 8.58330 0.347813
\(610\) 0 0
\(611\) 4.73501 0.191558
\(612\) 0 0
\(613\) −35.4411 −1.43145 −0.715727 0.698380i \(-0.753904\pi\)
−0.715727 + 0.698380i \(0.753904\pi\)
\(614\) 0 0
\(615\) 8.29853 0.334629
\(616\) 0 0
\(617\) −34.9753 −1.40805 −0.704027 0.710173i \(-0.748617\pi\)
−0.704027 + 0.710173i \(0.748617\pi\)
\(618\) 0 0
\(619\) 6.53838 0.262800 0.131400 0.991329i \(-0.458053\pi\)
0.131400 + 0.991329i \(0.458053\pi\)
\(620\) 0 0
\(621\) 7.18524 0.288334
\(622\) 0 0
\(623\) −2.73501 −0.109576
\(624\) 0 0
\(625\) −9.96754 −0.398702
\(626\) 0 0
\(627\) −38.5011 −1.53759
\(628\) 0 0
\(629\) 0.929346 0.0370555
\(630\) 0 0
\(631\) 14.0636 0.559861 0.279931 0.960020i \(-0.409689\pi\)
0.279931 + 0.960020i \(0.409689\pi\)
\(632\) 0 0
\(633\) −4.51135 −0.179310
\(634\) 0 0
\(635\) 26.9135 1.06803
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 4.81232 0.190373
\(640\) 0 0
\(641\) −23.6696 −0.934892 −0.467446 0.884022i \(-0.654826\pi\)
−0.467446 + 0.884022i \(0.654826\pi\)
\(642\) 0 0
\(643\) −9.19061 −0.362442 −0.181221 0.983442i \(-0.558005\pi\)
−0.181221 + 0.983442i \(0.558005\pi\)
\(644\) 0 0
\(645\) 12.2079 0.480688
\(646\) 0 0
\(647\) 41.8908 1.64690 0.823448 0.567391i \(-0.192047\pi\)
0.823448 + 0.567391i \(0.192047\pi\)
\(648\) 0 0
\(649\) −28.1439 −1.10474
\(650\) 0 0
\(651\) −5.54733 −0.217417
\(652\) 0 0
\(653\) 45.3643 1.77524 0.887621 0.460574i \(-0.152356\pi\)
0.887621 + 0.460574i \(0.152356\pi\)
\(654\) 0 0
\(655\) 10.8094 0.422358
\(656\) 0 0
\(657\) 13.1852 0.514405
\(658\) 0 0
\(659\) −39.4549 −1.53694 −0.768472 0.639883i \(-0.778983\pi\)
−0.768472 + 0.639883i \(0.778983\pi\)
\(660\) 0 0
\(661\) −7.39269 −0.287542 −0.143771 0.989611i \(-0.545923\pi\)
−0.143771 + 0.989611i \(0.545923\pi\)
\(662\) 0 0
\(663\) −0.964026 −0.0374397
\(664\) 0 0
\(665\) −11.1852 −0.433745
\(666\) 0 0
\(667\) −61.6731 −2.38799
\(668\) 0 0
\(669\) −7.61928 −0.294578
\(670\) 0 0
\(671\) 77.5634 2.99430
\(672\) 0 0
\(673\) −6.67086 −0.257143 −0.128571 0.991700i \(-0.541039\pi\)
−0.128571 + 0.991700i \(0.541039\pi\)
\(674\) 0 0
\(675\) 2.11329 0.0813406
\(676\) 0 0
\(677\) −0.940022 −0.0361280 −0.0180640 0.999837i \(-0.505750\pi\)
−0.0180640 + 0.999837i \(0.505750\pi\)
\(678\) 0 0
\(679\) 5.01734 0.192548
\(680\) 0 0
\(681\) −16.5089 −0.632623
\(682\) 0 0
\(683\) −22.1144 −0.846185 −0.423093 0.906086i \(-0.639056\pi\)
−0.423093 + 0.906086i \(0.639056\pi\)
\(684\) 0 0
\(685\) −33.4519 −1.27813
\(686\) 0 0
\(687\) −4.79613 −0.182984
\(688\) 0 0
\(689\) 13.0173 0.495921
\(690\) 0 0
\(691\) −47.1515 −1.79373 −0.896864 0.442307i \(-0.854160\pi\)
−0.896864 + 0.442307i \(0.854160\pi\)
\(692\) 0 0
\(693\) −5.84830 −0.222158
\(694\) 0 0
\(695\) 7.24830 0.274944
\(696\) 0 0
\(697\) −4.70856 −0.178350
\(698\) 0 0
\(699\) 21.6193 0.817716
\(700\) 0 0
\(701\) 49.9129 1.88519 0.942593 0.333945i \(-0.108380\pi\)
0.942593 + 0.333945i \(0.108380\pi\)
\(702\) 0 0
\(703\) 6.34648 0.239362
\(704\) 0 0
\(705\) −8.04492 −0.302989
\(706\) 0 0
\(707\) −14.9005 −0.560390
\(708\) 0 0
\(709\) −40.8979 −1.53595 −0.767976 0.640479i \(-0.778736\pi\)
−0.767976 + 0.640479i \(0.778736\pi\)
\(710\) 0 0
\(711\) −7.61928 −0.285745
\(712\) 0 0
\(713\) 39.8589 1.49273
\(714\) 0 0
\(715\) 9.93644 0.371602
\(716\) 0 0
\(717\) −6.01620 −0.224679
\(718\) 0 0
\(719\) −13.3429 −0.497606 −0.248803 0.968554i \(-0.580037\pi\)
−0.248803 + 0.968554i \(0.580037\pi\)
\(720\) 0 0
\(721\) −2.36209 −0.0879688
\(722\) 0 0
\(723\) −8.57491 −0.318904
\(724\) 0 0
\(725\) −18.1390 −0.673666
\(726\) 0 0
\(727\) −24.9089 −0.923818 −0.461909 0.886927i \(-0.652835\pi\)
−0.461909 + 0.886927i \(0.652835\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −6.92676 −0.256195
\(732\) 0 0
\(733\) 10.5030 0.387936 0.193968 0.981008i \(-0.437864\pi\)
0.193968 + 0.981008i \(0.437864\pi\)
\(734\) 0 0
\(735\) −1.69903 −0.0626697
\(736\) 0 0
\(737\) −60.6496 −2.23406
\(738\) 0 0
\(739\) 15.4305 0.567619 0.283809 0.958881i \(-0.408402\pi\)
0.283809 + 0.958881i \(0.408402\pi\)
\(740\) 0 0
\(741\) −6.58330 −0.241844
\(742\) 0 0
\(743\) −15.9119 −0.583749 −0.291875 0.956457i \(-0.594279\pi\)
−0.291875 + 0.956457i \(0.594279\pi\)
\(744\) 0 0
\(745\) −38.2434 −1.40113
\(746\) 0 0
\(747\) −0.133069 −0.00486873
\(748\) 0 0
\(749\) 5.92805 0.216606
\(750\) 0 0
\(751\) −19.3159 −0.704846 −0.352423 0.935841i \(-0.614642\pi\)
−0.352423 + 0.935841i \(0.614642\pi\)
\(752\) 0 0
\(753\) 5.93644 0.216336
\(754\) 0 0
\(755\) 31.1978 1.13540
\(756\) 0 0
\(757\) −1.93342 −0.0702714 −0.0351357 0.999383i \(-0.511186\pi\)
−0.0351357 + 0.999383i \(0.511186\pi\)
\(758\) 0 0
\(759\) 42.0214 1.52528
\(760\) 0 0
\(761\) 0.0779003 0.00282388 0.00141194 0.999999i \(-0.499551\pi\)
0.00141194 + 0.999999i \(0.499551\pi\)
\(762\) 0 0
\(763\) −9.56596 −0.346311
\(764\) 0 0
\(765\) 1.63791 0.0592188
\(766\) 0 0
\(767\) −4.81232 −0.173763
\(768\) 0 0
\(769\) 49.0365 1.76830 0.884150 0.467203i \(-0.154738\pi\)
0.884150 + 0.467203i \(0.154738\pi\)
\(770\) 0 0
\(771\) 20.8285 0.750121
\(772\) 0 0
\(773\) −1.62171 −0.0583290 −0.0291645 0.999575i \(-0.509285\pi\)
−0.0291645 + 0.999575i \(0.509285\pi\)
\(774\) 0 0
\(775\) 11.7231 0.421107
\(776\) 0 0
\(777\) 0.964026 0.0345842
\(778\) 0 0
\(779\) −32.1546 −1.15206
\(780\) 0 0
\(781\) 28.1439 1.00707
\(782\) 0 0
\(783\) −8.58330 −0.306742
\(784\) 0 0
\(785\) 15.2302 0.543588
\(786\) 0 0
\(787\) 38.0880 1.35769 0.678845 0.734281i \(-0.262481\pi\)
0.678845 + 0.734281i \(0.262481\pi\)
\(788\) 0 0
\(789\) 11.0173 0.392228
\(790\) 0 0
\(791\) −0.751202 −0.0267097
\(792\) 0 0
\(793\) 13.2626 0.470967
\(794\) 0 0
\(795\) −22.1169 −0.784405
\(796\) 0 0
\(797\) −25.5203 −0.903976 −0.451988 0.892024i \(-0.649285\pi\)
−0.451988 + 0.892024i \(0.649285\pi\)
\(798\) 0 0
\(799\) 4.56467 0.161486
\(800\) 0 0
\(801\) 2.73501 0.0966367
\(802\) 0 0
\(803\) 77.1112 2.72120
\(804\) 0 0
\(805\) 12.2079 0.430274
\(806\) 0 0
\(807\) 6.26614 0.220578
\(808\) 0 0
\(809\) 9.28348 0.326390 0.163195 0.986594i \(-0.447820\pi\)
0.163195 + 0.986594i \(0.447820\pi\)
\(810\) 0 0
\(811\) 28.3069 0.993990 0.496995 0.867753i \(-0.334437\pi\)
0.496995 + 0.867753i \(0.334437\pi\)
\(812\) 0 0
\(813\) 24.2613 0.850880
\(814\) 0 0
\(815\) 13.0768 0.458059
\(816\) 0 0
\(817\) −47.3026 −1.65491
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) −3.25962 −0.113762 −0.0568808 0.998381i \(-0.518116\pi\)
−0.0568808 + 0.998381i \(0.518116\pi\)
\(822\) 0 0
\(823\) −1.73615 −0.0605183 −0.0302592 0.999542i \(-0.509633\pi\)
−0.0302592 + 0.999542i \(0.509633\pi\)
\(824\) 0 0
\(825\) 12.3592 0.430291
\(826\) 0 0
\(827\) −32.7834 −1.13999 −0.569996 0.821647i \(-0.693055\pi\)
−0.569996 + 0.821647i \(0.693055\pi\)
\(828\) 0 0
\(829\) −3.33450 −0.115812 −0.0579061 0.998322i \(-0.518442\pi\)
−0.0579061 + 0.998322i \(0.518442\pi\)
\(830\) 0 0
\(831\) −25.3159 −0.878198
\(832\) 0 0
\(833\) 0.964026 0.0334015
\(834\) 0 0
\(835\) −1.98624 −0.0687367
\(836\) 0 0
\(837\) 5.54733 0.191744
\(838\) 0 0
\(839\) 16.3916 0.565899 0.282950 0.959135i \(-0.408687\pi\)
0.282950 + 0.959135i \(0.408687\pi\)
\(840\) 0 0
\(841\) 44.6731 1.54045
\(842\) 0 0
\(843\) −5.75235 −0.198121
\(844\) 0 0
\(845\) 1.69903 0.0584485
\(846\) 0 0
\(847\) −23.2026 −0.797250
\(848\) 0 0
\(849\) 8.14390 0.279498
\(850\) 0 0
\(851\) −6.92676 −0.237446
\(852\) 0 0
\(853\) 25.3110 0.866632 0.433316 0.901242i \(-0.357343\pi\)
0.433316 + 0.901242i \(0.357343\pi\)
\(854\) 0 0
\(855\) 11.1852 0.382527
\(856\) 0 0
\(857\) 10.0324 0.342700 0.171350 0.985210i \(-0.445187\pi\)
0.171350 + 0.985210i \(0.445187\pi\)
\(858\) 0 0
\(859\) 47.6917 1.62722 0.813610 0.581411i \(-0.197499\pi\)
0.813610 + 0.581411i \(0.197499\pi\)
\(860\) 0 0
\(861\) −4.88427 −0.166456
\(862\) 0 0
\(863\) 22.7080 0.772989 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(864\) 0 0
\(865\) −1.14495 −0.0389295
\(866\) 0 0
\(867\) 16.0707 0.545788
\(868\) 0 0
\(869\) −44.5598 −1.51159
\(870\) 0 0
\(871\) −10.3705 −0.351390
\(872\) 0 0
\(873\) −5.01734 −0.169811
\(874\) 0 0
\(875\) 12.0857 0.408571
\(876\) 0 0
\(877\) −14.2985 −0.482827 −0.241414 0.970422i \(-0.577611\pi\)
−0.241414 + 0.970422i \(0.577611\pi\)
\(878\) 0 0
\(879\) −18.0060 −0.607326
\(880\) 0 0
\(881\) −8.81526 −0.296994 −0.148497 0.988913i \(-0.547443\pi\)
−0.148497 + 0.988913i \(0.547443\pi\)
\(882\) 0 0
\(883\) −39.2302 −1.32020 −0.660100 0.751178i \(-0.729486\pi\)
−0.660100 + 0.751178i \(0.729486\pi\)
\(884\) 0 0
\(885\) 8.17629 0.274843
\(886\) 0 0
\(887\) 7.08392 0.237855 0.118927 0.992903i \(-0.462054\pi\)
0.118927 + 0.992903i \(0.462054\pi\)
\(888\) 0 0
\(889\) −15.8405 −0.531273
\(890\) 0 0
\(891\) 5.84830 0.195925
\(892\) 0 0
\(893\) 31.1720 1.04313
\(894\) 0 0
\(895\) 2.12175 0.0709222
\(896\) 0 0
\(897\) 7.18524 0.239908
\(898\) 0 0
\(899\) −47.6144 −1.58803
\(900\) 0 0
\(901\) 12.5491 0.418070
\(902\) 0 0
\(903\) −7.18524 −0.239110
\(904\) 0 0
\(905\) −1.35264 −0.0449634
\(906\) 0 0
\(907\) 12.1097 0.402096 0.201048 0.979581i \(-0.435565\pi\)
0.201048 + 0.979581i \(0.435565\pi\)
\(908\) 0 0
\(909\) 14.9005 0.494217
\(910\) 0 0
\(911\) −5.37943 −0.178228 −0.0891142 0.996021i \(-0.528404\pi\)
−0.0891142 + 0.996021i \(0.528404\pi\)
\(912\) 0 0
\(913\) −0.778226 −0.0257555
\(914\) 0 0
\(915\) −22.5335 −0.744935
\(916\) 0 0
\(917\) −6.36209 −0.210095
\(918\) 0 0
\(919\) −54.0298 −1.78228 −0.891139 0.453730i \(-0.850093\pi\)
−0.891139 + 0.453730i \(0.850093\pi\)
\(920\) 0 0
\(921\) −20.3758 −0.671407
\(922\) 0 0
\(923\) 4.81232 0.158400
\(924\) 0 0
\(925\) −2.03727 −0.0669850
\(926\) 0 0
\(927\) 2.36209 0.0775812
\(928\) 0 0
\(929\) −1.41376 −0.0463841 −0.0231921 0.999731i \(-0.507383\pi\)
−0.0231921 + 0.999731i \(0.507383\pi\)
\(930\) 0 0
\(931\) 6.58330 0.215759
\(932\) 0 0
\(933\) −25.7313 −0.842404
\(934\) 0 0
\(935\) 9.57899 0.313266
\(936\) 0 0
\(937\) 34.4424 1.12519 0.562593 0.826734i \(-0.309804\pi\)
0.562593 + 0.826734i \(0.309804\pi\)
\(938\) 0 0
\(939\) 15.4651 0.504686
\(940\) 0 0
\(941\) 37.7074 1.22923 0.614613 0.788828i \(-0.289312\pi\)
0.614613 + 0.788828i \(0.289312\pi\)
\(942\) 0 0
\(943\) 35.0947 1.14284
\(944\) 0 0
\(945\) 1.69903 0.0552695
\(946\) 0 0
\(947\) −19.4573 −0.632278 −0.316139 0.948713i \(-0.602387\pi\)
−0.316139 + 0.948713i \(0.602387\pi\)
\(948\) 0 0
\(949\) 13.1852 0.428011
\(950\) 0 0
\(951\) −26.9513 −0.873957
\(952\) 0 0
\(953\) −45.1768 −1.46342 −0.731711 0.681615i \(-0.761278\pi\)
−0.731711 + 0.681615i \(0.761278\pi\)
\(954\) 0 0
\(955\) −7.28909 −0.235869
\(956\) 0 0
\(957\) −50.1977 −1.62266
\(958\) 0 0
\(959\) 19.6888 0.635784
\(960\) 0 0
\(961\) −0.227144 −0.00732722
\(962\) 0 0
\(963\) −5.92805 −0.191029
\(964\) 0 0
\(965\) 20.5106 0.660260
\(966\) 0 0
\(967\) −2.19770 −0.0706734 −0.0353367 0.999375i \(-0.511250\pi\)
−0.0353367 + 0.999375i \(0.511250\pi\)
\(968\) 0 0
\(969\) −6.34648 −0.203878
\(970\) 0 0
\(971\) −17.4279 −0.559287 −0.279643 0.960104i \(-0.590216\pi\)
−0.279643 + 0.960104i \(0.590216\pi\)
\(972\) 0 0
\(973\) −4.26614 −0.136766
\(974\) 0 0
\(975\) 2.11329 0.0676795
\(976\) 0 0
\(977\) −15.5498 −0.497481 −0.248741 0.968570i \(-0.580017\pi\)
−0.248741 + 0.968570i \(0.580017\pi\)
\(978\) 0 0
\(979\) 15.9951 0.511206
\(980\) 0 0
\(981\) 9.56596 0.305418
\(982\) 0 0
\(983\) −22.6018 −0.720885 −0.360443 0.932781i \(-0.617374\pi\)
−0.360443 + 0.932781i \(0.617374\pi\)
\(984\) 0 0
\(985\) −33.4376 −1.06541
\(986\) 0 0
\(987\) 4.73501 0.150717
\(988\) 0 0
\(989\) 51.6277 1.64166
\(990\) 0 0
\(991\) 2.67876 0.0850936 0.0425468 0.999094i \(-0.486453\pi\)
0.0425468 + 0.999094i \(0.486453\pi\)
\(992\) 0 0
\(993\) 22.2266 0.705339
\(994\) 0 0
\(995\) 23.8861 0.757242
\(996\) 0 0
\(997\) −38.8308 −1.22978 −0.614892 0.788611i \(-0.710800\pi\)
−0.614892 + 0.788611i \(0.710800\pi\)
\(998\) 0 0
\(999\) −0.964026 −0.0305004
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 2184.2.a.v.1.3 4
3.2 odd 2 6552.2.a.bs.1.2 4
4.3 odd 2 4368.2.a.bs.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.v.1.3 4 1.1 even 1 trivial
4368.2.a.bs.1.3 4 4.3 odd 2
6552.2.a.bs.1.2 4 3.2 odd 2