Properties

Label 2184.2.a.v.1.2
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 1.2
Root \(-2.52690\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.152457 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.152457 q^{5} -1.00000 q^{7} +1.00000 q^{9} +0.385245 q^{11} +1.00000 q^{13} +0.152457 q^{15} +7.43905 q^{17} -7.20627 q^{19} +1.00000 q^{21} +2.90135 q^{23} -4.97676 q^{25} -1.00000 q^{27} -5.20627 q^{29} +1.76721 q^{31} -0.385245 q^{33} +0.152457 q^{35} +7.43905 q^{37} -1.00000 q^{39} +7.05381 q^{41} +2.90135 q^{43} -0.152457 q^{45} -3.59151 q^{47} +1.00000 q^{49} -7.43905 q^{51} -10.9502 q^{53} -0.0587335 q^{55} +7.20627 q^{57} -5.82430 q^{59} +12.5145 q^{61} -1.00000 q^{63} -0.152457 q^{65} +9.80270 q^{67} -2.90135 q^{69} +5.82430 q^{71} +3.09865 q^{73} +4.97676 q^{75} -0.385245 q^{77} +12.6453 q^{79} +1.00000 q^{81} +11.8964 q^{83} -1.13414 q^{85} +5.20627 q^{87} -5.59151 q^{89} -1.00000 q^{91} -1.76721 q^{93} +1.09865 q^{95} +18.9502 q^{97} +0.385245 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) 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
\(5\) −0.152457 −0.0681810 −0.0340905 0.999419i \(-0.510853\pi\)
−0.0340905 + 0.999419i \(0.510853\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.385245 0.116156 0.0580779 0.998312i \(-0.481503\pi\)
0.0580779 + 0.998312i \(0.481503\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 0.152457 0.0393643
\(16\) 0 0
\(17\) 7.43905 1.80424 0.902118 0.431490i \(-0.142012\pi\)
0.902118 + 0.431490i \(0.142012\pi\)
\(18\) 0 0
\(19\) −7.20627 −1.65323 −0.826615 0.562767i \(-0.809737\pi\)
−0.826615 + 0.562767i \(0.809737\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.90135 0.604974 0.302487 0.953154i \(-0.402183\pi\)
0.302487 + 0.953154i \(0.402183\pi\)
\(24\) 0 0
\(25\) −4.97676 −0.995351
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.20627 −0.966779 −0.483390 0.875405i \(-0.660595\pi\)
−0.483390 + 0.875405i \(0.660595\pi\)
\(30\) 0 0
\(31\) 1.76721 0.317401 0.158700 0.987327i \(-0.449270\pi\)
0.158700 + 0.987327i \(0.449270\pi\)
\(32\) 0 0
\(33\) −0.385245 −0.0670626
\(34\) 0 0
\(35\) 0.152457 0.0257700
\(36\) 0 0
\(37\) 7.43905 1.22297 0.611486 0.791255i \(-0.290572\pi\)
0.611486 + 0.791255i \(0.290572\pi\)
\(38\) 0 0
\(39\) −1.00000 −0.160128
\(40\) 0 0
\(41\) 7.05381 1.10162 0.550810 0.834631i \(-0.314319\pi\)
0.550810 + 0.834631i \(0.314319\pi\)
\(42\) 0 0
\(43\) 2.90135 0.442452 0.221226 0.975223i \(-0.428994\pi\)
0.221226 + 0.975223i \(0.428994\pi\)
\(44\) 0 0
\(45\) −0.152457 −0.0227270
\(46\) 0 0
\(47\) −3.59151 −0.523876 −0.261938 0.965085i \(-0.584362\pi\)
−0.261938 + 0.965085i \(0.584362\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −7.43905 −1.04168
\(52\) 0 0
\(53\) −10.9502 −1.50413 −0.752065 0.659089i \(-0.770942\pi\)
−0.752065 + 0.659089i \(0.770942\pi\)
\(54\) 0 0
\(55\) −0.0587335 −0.00791963
\(56\) 0 0
\(57\) 7.20627 0.954493
\(58\) 0 0
\(59\) −5.82430 −0.758259 −0.379130 0.925344i \(-0.623777\pi\)
−0.379130 + 0.925344i \(0.623777\pi\)
\(60\) 0 0
\(61\) 12.5145 1.60231 0.801156 0.598455i \(-0.204219\pi\)
0.801156 + 0.598455i \(0.204219\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −0.152457 −0.0189100
\(66\) 0 0
\(67\) 9.80270 1.19759 0.598795 0.800902i \(-0.295646\pi\)
0.598795 + 0.800902i \(0.295646\pi\)
\(68\) 0 0
\(69\) −2.90135 −0.349282
\(70\) 0 0
\(71\) 5.82430 0.691217 0.345609 0.938379i \(-0.387672\pi\)
0.345609 + 0.938379i \(0.387672\pi\)
\(72\) 0 0
\(73\) 3.09865 0.362669 0.181335 0.983421i \(-0.441958\pi\)
0.181335 + 0.983421i \(0.441958\pi\)
\(74\) 0 0
\(75\) 4.97676 0.574666
\(76\) 0 0
\(77\) −0.385245 −0.0439028
\(78\) 0 0
\(79\) 12.6453 1.42271 0.711355 0.702833i \(-0.248082\pi\)
0.711355 + 0.702833i \(0.248082\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.8964 1.30580 0.652901 0.757443i \(-0.273552\pi\)
0.652901 + 0.757443i \(0.273552\pi\)
\(84\) 0 0
\(85\) −1.13414 −0.123015
\(86\) 0 0
\(87\) 5.20627 0.558170
\(88\) 0 0
\(89\) −5.59151 −0.592699 −0.296350 0.955080i \(-0.595769\pi\)
−0.296350 + 0.955080i \(0.595769\pi\)
\(90\) 0 0
\(91\) −1.00000 −0.104828
\(92\) 0 0
\(93\) −1.76721 −0.183251
\(94\) 0 0
\(95\) 1.09865 0.112719
\(96\) 0 0
\(97\) 18.9502 1.92410 0.962052 0.272865i \(-0.0879711\pi\)
0.962052 + 0.272865i \(0.0879711\pi\)
\(98\) 0 0
\(99\) 0.385245 0.0387186
\(100\) 0 0
\(101\) 11.3803 1.13238 0.566192 0.824273i \(-0.308416\pi\)
0.566192 + 0.824273i \(0.308416\pi\)
\(102\) 0 0
\(103\) 5.13414 0.505882 0.252941 0.967482i \(-0.418602\pi\)
0.252941 + 0.967482i \(0.418602\pi\)
\(104\) 0 0
\(105\) −0.152457 −0.0148783
\(106\) 0 0
\(107\) −18.8781 −1.82502 −0.912508 0.409059i \(-0.865857\pi\)
−0.912508 + 0.409059i \(0.865857\pi\)
\(108\) 0 0
\(109\) 19.7440 1.89113 0.945565 0.325434i \(-0.105511\pi\)
0.945565 + 0.325434i \(0.105511\pi\)
\(110\) 0 0
\(111\) −7.43905 −0.706084
\(112\) 0 0
\(113\) 0.842618 0.0792668 0.0396334 0.999214i \(-0.487381\pi\)
0.0396334 + 0.999214i \(0.487381\pi\)
\(114\) 0 0
\(115\) −0.442333 −0.0412477
\(116\) 0 0
\(117\) 1.00000 0.0924500
\(118\) 0 0
\(119\) −7.43905 −0.681937
\(120\) 0 0
\(121\) −10.8516 −0.986508
\(122\) 0 0
\(123\) −7.05381 −0.636021
\(124\) 0 0
\(125\) 1.52103 0.136045
\(126\) 0 0
\(127\) −20.9857 −1.86218 −0.931091 0.364787i \(-0.881142\pi\)
−0.931091 + 0.364787i \(0.881142\pi\)
\(128\) 0 0
\(129\) −2.90135 −0.255450
\(130\) 0 0
\(131\) 9.13414 0.798053 0.399027 0.916939i \(-0.369348\pi\)
0.399027 + 0.916939i \(0.369348\pi\)
\(132\) 0 0
\(133\) 7.20627 0.624863
\(134\) 0 0
\(135\) 0.152457 0.0131214
\(136\) 0 0
\(137\) 22.6005 1.93089 0.965445 0.260608i \(-0.0839231\pi\)
0.965445 + 0.260608i \(0.0839231\pi\)
\(138\) 0 0
\(139\) −19.7929 −1.67881 −0.839404 0.543508i \(-0.817096\pi\)
−0.839404 + 0.543508i \(0.817096\pi\)
\(140\) 0 0
\(141\) 3.59151 0.302460
\(142\) 0 0
\(143\) 0.385245 0.0322158
\(144\) 0 0
\(145\) 0.793734 0.0659160
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −12.5948 −1.03181 −0.515903 0.856647i \(-0.672543\pi\)
−0.515903 + 0.856647i \(0.672543\pi\)
\(150\) 0 0
\(151\) 21.1341 1.71987 0.859936 0.510402i \(-0.170503\pi\)
0.859936 + 0.510402i \(0.170503\pi\)
\(152\) 0 0
\(153\) 7.43905 0.601412
\(154\) 0 0
\(155\) −0.269425 −0.0216407
\(156\) 0 0
\(157\) 15.4391 1.23217 0.616085 0.787679i \(-0.288718\pi\)
0.616085 + 0.787679i \(0.288718\pi\)
\(158\) 0 0
\(159\) 10.9502 0.868410
\(160\) 0 0
\(161\) −2.90135 −0.228659
\(162\) 0 0
\(163\) −3.22951 −0.252955 −0.126477 0.991969i \(-0.540367\pi\)
−0.126477 + 0.991969i \(0.540367\pi\)
\(164\) 0 0
\(165\) 0.0587335 0.00457240
\(166\) 0 0
\(167\) 17.3355 1.34146 0.670730 0.741702i \(-0.265981\pi\)
0.670730 + 0.741702i \(0.265981\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −7.20627 −0.551077
\(172\) 0 0
\(173\) 8.57319 0.651808 0.325904 0.945403i \(-0.394331\pi\)
0.325904 + 0.945403i \(0.394331\pi\)
\(174\) 0 0
\(175\) 4.97676 0.376207
\(176\) 0 0
\(177\) 5.82430 0.437781
\(178\) 0 0
\(179\) 1.15738 0.0865068 0.0432534 0.999064i \(-0.486228\pi\)
0.0432534 + 0.999064i \(0.486228\pi\)
\(180\) 0 0
\(181\) 6.60983 0.491305 0.245652 0.969358i \(-0.420998\pi\)
0.245652 + 0.969358i \(0.420998\pi\)
\(182\) 0 0
\(183\) −12.5145 −0.925095
\(184\) 0 0
\(185\) −1.13414 −0.0833836
\(186\) 0 0
\(187\) 2.86586 0.209573
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −20.0122 −1.44804 −0.724018 0.689781i \(-0.757707\pi\)
−0.724018 + 0.689781i \(0.757707\pi\)
\(192\) 0 0
\(193\) −0.878108 −0.0632076 −0.0316038 0.999500i \(-0.510061\pi\)
−0.0316038 + 0.999500i \(0.510061\pi\)
\(194\) 0 0
\(195\) 0.152457 0.0109177
\(196\) 0 0
\(197\) −0.336362 −0.0239648 −0.0119824 0.999928i \(-0.503814\pi\)
−0.0119824 + 0.999928i \(0.503814\pi\)
\(198\) 0 0
\(199\) 5.90463 0.418568 0.209284 0.977855i \(-0.432887\pi\)
0.209284 + 0.977855i \(0.432887\pi\)
\(200\) 0 0
\(201\) −9.80270 −0.691429
\(202\) 0 0
\(203\) 5.20627 0.365408
\(204\) 0 0
\(205\) −1.07541 −0.0751096
\(206\) 0 0
\(207\) 2.90135 0.201658
\(208\) 0 0
\(209\) −2.77618 −0.192032
\(210\) 0 0
\(211\) 3.67184 0.252780 0.126390 0.991981i \(-0.459661\pi\)
0.126390 + 0.991981i \(0.459661\pi\)
\(212\) 0 0
\(213\) −5.82430 −0.399074
\(214\) 0 0
\(215\) −0.442333 −0.0301668
\(216\) 0 0
\(217\) −1.76721 −0.119966
\(218\) 0 0
\(219\) −3.09865 −0.209387
\(220\) 0 0
\(221\) 7.43905 0.500405
\(222\) 0 0
\(223\) −12.6453 −0.846793 −0.423397 0.905944i \(-0.639162\pi\)
−0.423397 + 0.905944i \(0.639162\pi\)
\(224\) 0 0
\(225\) −4.97676 −0.331784
\(226\) 0 0
\(227\) 6.59479 0.437712 0.218856 0.975757i \(-0.429768\pi\)
0.218856 + 0.975757i \(0.429768\pi\)
\(228\) 0 0
\(229\) −2.60983 −0.172462 −0.0862312 0.996275i \(-0.527482\pi\)
−0.0862312 + 0.996275i \(0.527482\pi\)
\(230\) 0 0
\(231\) 0.385245 0.0253473
\(232\) 0 0
\(233\) −1.35468 −0.0887480 −0.0443740 0.999015i \(-0.514129\pi\)
−0.0443740 + 0.999015i \(0.514129\pi\)
\(234\) 0 0
\(235\) 0.547553 0.0357184
\(236\) 0 0
\(237\) −12.6453 −0.821402
\(238\) 0 0
\(239\) 14.4341 0.933666 0.466833 0.884345i \(-0.345395\pi\)
0.466833 + 0.884345i \(0.345395\pi\)
\(240\) 0 0
\(241\) 17.7306 1.14213 0.571063 0.820906i \(-0.306531\pi\)
0.571063 + 0.820906i \(0.306531\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.152457 −0.00974015
\(246\) 0 0
\(247\) −7.20627 −0.458524
\(248\) 0 0
\(249\) −11.8964 −0.753905
\(250\) 0 0
\(251\) 4.05873 0.256185 0.128092 0.991762i \(-0.459115\pi\)
0.128092 + 0.991762i \(0.459115\pi\)
\(252\) 0 0
\(253\) 1.11773 0.0702712
\(254\) 0 0
\(255\) 1.13414 0.0710225
\(256\) 0 0
\(257\) −30.2584 −1.88747 −0.943734 0.330704i \(-0.892714\pi\)
−0.943734 + 0.330704i \(0.892714\pi\)
\(258\) 0 0
\(259\) −7.43905 −0.462240
\(260\) 0 0
\(261\) −5.20627 −0.322260
\(262\) 0 0
\(263\) 12.9502 0.798546 0.399273 0.916832i \(-0.369263\pi\)
0.399273 + 0.916832i \(0.369263\pi\)
\(264\) 0 0
\(265\) 1.66944 0.102553
\(266\) 0 0
\(267\) 5.59151 0.342195
\(268\) 0 0
\(269\) 17.7929 1.08485 0.542425 0.840104i \(-0.317506\pi\)
0.542425 + 0.840104i \(0.317506\pi\)
\(270\) 0 0
\(271\) 17.9470 1.09020 0.545100 0.838371i \(-0.316492\pi\)
0.545100 + 0.838371i \(0.316492\pi\)
\(272\) 0 0
\(273\) 1.00000 0.0605228
\(274\) 0 0
\(275\) −1.91727 −0.115616
\(276\) 0 0
\(277\) −5.87483 −0.352984 −0.176492 0.984302i \(-0.556475\pi\)
−0.176492 + 0.984302i \(0.556475\pi\)
\(278\) 0 0
\(279\) 1.76721 0.105800
\(280\) 0 0
\(281\) −26.5417 −1.58335 −0.791674 0.610944i \(-0.790790\pi\)
−0.791674 + 0.610944i \(0.790790\pi\)
\(282\) 0 0
\(283\) 17.7562 1.05550 0.527749 0.849401i \(-0.323036\pi\)
0.527749 + 0.849401i \(0.323036\pi\)
\(284\) 0 0
\(285\) −1.09865 −0.0650783
\(286\) 0 0
\(287\) −7.05381 −0.416373
\(288\) 0 0
\(289\) 38.3395 2.25527
\(290\) 0 0
\(291\) −18.9502 −1.11088
\(292\) 0 0
\(293\) −14.0139 −0.818700 −0.409350 0.912377i \(-0.634245\pi\)
−0.409350 + 0.912377i \(0.634245\pi\)
\(294\) 0 0
\(295\) 0.887958 0.0516989
\(296\) 0 0
\(297\) −0.385245 −0.0223542
\(298\) 0 0
\(299\) 2.90135 0.167789
\(300\) 0 0
\(301\) −2.90135 −0.167231
\(302\) 0 0
\(303\) −11.3803 −0.653782
\(304\) 0 0
\(305\) −1.90792 −0.109247
\(306\) 0 0
\(307\) 22.4912 1.28364 0.641821 0.766855i \(-0.278179\pi\)
0.641821 + 0.766855i \(0.278179\pi\)
\(308\) 0 0
\(309\) −5.13414 −0.292071
\(310\) 0 0
\(311\) −33.1300 −1.87863 −0.939314 0.343058i \(-0.888537\pi\)
−0.939314 + 0.343058i \(0.888537\pi\)
\(312\) 0 0
\(313\) 19.3371 1.09300 0.546500 0.837459i \(-0.315960\pi\)
0.546500 + 0.837459i \(0.315960\pi\)
\(314\) 0 0
\(315\) 0.152457 0.00859000
\(316\) 0 0
\(317\) −16.0860 −0.903481 −0.451740 0.892149i \(-0.649197\pi\)
−0.451740 + 0.892149i \(0.649197\pi\)
\(318\) 0 0
\(319\) −2.00569 −0.112297
\(320\) 0 0
\(321\) 18.8781 1.05367
\(322\) 0 0
\(323\) −53.6078 −2.98282
\(324\) 0 0
\(325\) −4.97676 −0.276061
\(326\) 0 0
\(327\) −19.7440 −1.09184
\(328\) 0 0
\(329\) 3.59151 0.198006
\(330\) 0 0
\(331\) −27.9535 −1.53646 −0.768232 0.640172i \(-0.778863\pi\)
−0.768232 + 0.640172i \(0.778863\pi\)
\(332\) 0 0
\(333\) 7.43905 0.407658
\(334\) 0 0
\(335\) −1.49449 −0.0816530
\(336\) 0 0
\(337\) 7.31388 0.398413 0.199206 0.979958i \(-0.436164\pi\)
0.199206 + 0.979958i \(0.436164\pi\)
\(338\) 0 0
\(339\) −0.842618 −0.0457647
\(340\) 0 0
\(341\) 0.680810 0.0368679
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0.442333 0.0238144
\(346\) 0 0
\(347\) −26.1076 −1.40153 −0.700765 0.713392i \(-0.747158\pi\)
−0.700765 + 0.713392i \(0.747158\pi\)
\(348\) 0 0
\(349\) −3.76721 −0.201654 −0.100827 0.994904i \(-0.532149\pi\)
−0.100827 + 0.994904i \(0.532149\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) 0 0
\(353\) −11.8243 −0.629344 −0.314672 0.949200i \(-0.601895\pi\)
−0.314672 + 0.949200i \(0.601895\pi\)
\(354\) 0 0
\(355\) −0.887958 −0.0471279
\(356\) 0 0
\(357\) 7.43905 0.393716
\(358\) 0 0
\(359\) 14.4341 0.761804 0.380902 0.924615i \(-0.375613\pi\)
0.380902 + 0.924615i \(0.375613\pi\)
\(360\) 0 0
\(361\) 32.9303 1.73317
\(362\) 0 0
\(363\) 10.8516 0.569561
\(364\) 0 0
\(365\) −0.472412 −0.0247272
\(366\) 0 0
\(367\) −13.9634 −0.728882 −0.364441 0.931227i \(-0.618740\pi\)
−0.364441 + 0.931227i \(0.618740\pi\)
\(368\) 0 0
\(369\) 7.05381 0.367207
\(370\) 0 0
\(371\) 10.9502 0.568508
\(372\) 0 0
\(373\) −31.0289 −1.60662 −0.803308 0.595563i \(-0.796929\pi\)
−0.803308 + 0.595563i \(0.796929\pi\)
\(374\) 0 0
\(375\) −1.52103 −0.0785457
\(376\) 0 0
\(377\) −5.20627 −0.268136
\(378\) 0 0
\(379\) −25.9634 −1.33365 −0.666824 0.745215i \(-0.732347\pi\)
−0.666824 + 0.745215i \(0.732347\pi\)
\(380\) 0 0
\(381\) 20.9857 1.07513
\(382\) 0 0
\(383\) 13.4175 0.685600 0.342800 0.939408i \(-0.388625\pi\)
0.342800 + 0.939408i \(0.388625\pi\)
\(384\) 0 0
\(385\) 0.0587335 0.00299334
\(386\) 0 0
\(387\) 2.90135 0.147484
\(388\) 0 0
\(389\) −2.00000 −0.101404 −0.0507020 0.998714i \(-0.516146\pi\)
−0.0507020 + 0.998714i \(0.516146\pi\)
\(390\) 0 0
\(391\) 21.5833 1.09151
\(392\) 0 0
\(393\) −9.13414 −0.460756
\(394\) 0 0
\(395\) −1.92787 −0.0970018
\(396\) 0 0
\(397\) −25.5234 −1.28098 −0.640492 0.767965i \(-0.721270\pi\)
−0.640492 + 0.767965i \(0.721270\pi\)
\(398\) 0 0
\(399\) −7.20627 −0.360765
\(400\) 0 0
\(401\) 3.00732 0.150178 0.0750892 0.997177i \(-0.476076\pi\)
0.0750892 + 0.997177i \(0.476076\pi\)
\(402\) 0 0
\(403\) 1.76721 0.0880311
\(404\) 0 0
\(405\) −0.152457 −0.00757567
\(406\) 0 0
\(407\) 2.86586 0.142055
\(408\) 0 0
\(409\) −22.8483 −1.12978 −0.564888 0.825168i \(-0.691081\pi\)
−0.564888 + 0.825168i \(0.691081\pi\)
\(410\) 0 0
\(411\) −22.6005 −1.11480
\(412\) 0 0
\(413\) 5.82430 0.286595
\(414\) 0 0
\(415\) −1.81370 −0.0890310
\(416\) 0 0
\(417\) 19.7929 0.969260
\(418\) 0 0
\(419\) 16.1565 0.789297 0.394648 0.918832i \(-0.370866\pi\)
0.394648 + 0.918832i \(0.370866\pi\)
\(420\) 0 0
\(421\) 21.7929 1.06212 0.531059 0.847335i \(-0.321794\pi\)
0.531059 + 0.847335i \(0.321794\pi\)
\(422\) 0 0
\(423\) −3.59151 −0.174625
\(424\) 0 0
\(425\) −37.0224 −1.79585
\(426\) 0 0
\(427\) −12.5145 −0.605617
\(428\) 0 0
\(429\) −0.385245 −0.0185998
\(430\) 0 0
\(431\) −6.80194 −0.327638 −0.163819 0.986490i \(-0.552381\pi\)
−0.163819 + 0.986490i \(0.552381\pi\)
\(432\) 0 0
\(433\) 6.26172 0.300919 0.150460 0.988616i \(-0.451925\pi\)
0.150460 + 0.988616i \(0.451925\pi\)
\(434\) 0 0
\(435\) −0.793734 −0.0380566
\(436\) 0 0
\(437\) −20.9079 −1.00016
\(438\) 0 0
\(439\) −21.0877 −1.00646 −0.503229 0.864153i \(-0.667855\pi\)
−0.503229 + 0.864153i \(0.667855\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −6.39345 −0.303762 −0.151881 0.988399i \(-0.548533\pi\)
−0.151881 + 0.988399i \(0.548533\pi\)
\(444\) 0 0
\(445\) 0.852467 0.0404108
\(446\) 0 0
\(447\) 12.5948 0.595713
\(448\) 0 0
\(449\) 4.49286 0.212031 0.106016 0.994364i \(-0.466191\pi\)
0.106016 + 0.994364i \(0.466191\pi\)
\(450\) 0 0
\(451\) 2.71745 0.127960
\(452\) 0 0
\(453\) −21.1341 −0.992968
\(454\) 0 0
\(455\) 0.152457 0.00714731
\(456\) 0 0
\(457\) −31.7398 −1.48473 −0.742363 0.669998i \(-0.766295\pi\)
−0.742363 + 0.669998i \(0.766295\pi\)
\(458\) 0 0
\(459\) −7.43905 −0.347225
\(460\) 0 0
\(461\) 3.76886 0.175533 0.0877666 0.996141i \(-0.472027\pi\)
0.0877666 + 0.996141i \(0.472027\pi\)
\(462\) 0 0
\(463\) 18.1664 0.844262 0.422131 0.906535i \(-0.361282\pi\)
0.422131 + 0.906535i \(0.361282\pi\)
\(464\) 0 0
\(465\) 0.269425 0.0124943
\(466\) 0 0
\(467\) −14.0057 −0.648106 −0.324053 0.946039i \(-0.605046\pi\)
−0.324053 + 0.946039i \(0.605046\pi\)
\(468\) 0 0
\(469\) −9.80270 −0.452647
\(470\) 0 0
\(471\) −15.4391 −0.711394
\(472\) 0 0
\(473\) 1.11773 0.0513934
\(474\) 0 0
\(475\) 35.8638 1.64555
\(476\) 0 0
\(477\) −10.9502 −0.501377
\(478\) 0 0
\(479\) 36.1704 1.65267 0.826334 0.563181i \(-0.190423\pi\)
0.826334 + 0.563181i \(0.190423\pi\)
\(480\) 0 0
\(481\) 7.43905 0.339192
\(482\) 0 0
\(483\) 2.90135 0.132016
\(484\) 0 0
\(485\) −2.88910 −0.131187
\(486\) 0 0
\(487\) 5.03221 0.228031 0.114016 0.993479i \(-0.463629\pi\)
0.114016 + 0.993479i \(0.463629\pi\)
\(488\) 0 0
\(489\) 3.22951 0.146043
\(490\) 0 0
\(491\) 10.8781 0.490922 0.245461 0.969406i \(-0.421061\pi\)
0.245461 + 0.969406i \(0.421061\pi\)
\(492\) 0 0
\(493\) −38.7297 −1.74430
\(494\) 0 0
\(495\) −0.0587335 −0.00263988
\(496\) 0 0
\(497\) −5.82430 −0.261256
\(498\) 0 0
\(499\) −12.0000 −0.537194 −0.268597 0.963253i \(-0.586560\pi\)
−0.268597 + 0.963253i \(0.586560\pi\)
\(500\) 0 0
\(501\) −17.3355 −0.774492
\(502\) 0 0
\(503\) 13.8459 0.617358 0.308679 0.951166i \(-0.400113\pi\)
0.308679 + 0.951166i \(0.400113\pi\)
\(504\) 0 0
\(505\) −1.73501 −0.0772071
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −37.1814 −1.64804 −0.824018 0.566564i \(-0.808272\pi\)
−0.824018 + 0.566564i \(0.808272\pi\)
\(510\) 0 0
\(511\) −3.09865 −0.137076
\(512\) 0 0
\(513\) 7.20627 0.318164
\(514\) 0 0
\(515\) −0.782738 −0.0344915
\(516\) 0 0
\(517\) −1.38361 −0.0608512
\(518\) 0 0
\(519\) −8.57319 −0.376321
\(520\) 0 0
\(521\) 43.4936 1.90549 0.952745 0.303771i \(-0.0982456\pi\)
0.952745 + 0.303771i \(0.0982456\pi\)
\(522\) 0 0
\(523\) −25.9634 −1.13530 −0.567649 0.823270i \(-0.692147\pi\)
−0.567649 + 0.823270i \(0.692147\pi\)
\(524\) 0 0
\(525\) −4.97676 −0.217203
\(526\) 0 0
\(527\) 13.1464 0.572666
\(528\) 0 0
\(529\) −14.5822 −0.634007
\(530\) 0 0
\(531\) −5.82430 −0.252753
\(532\) 0 0
\(533\) 7.05381 0.305534
\(534\) 0 0
\(535\) 2.87811 0.124431
\(536\) 0 0
\(537\) −1.15738 −0.0499447
\(538\) 0 0
\(539\) 0.385245 0.0165937
\(540\) 0 0
\(541\) −14.1565 −0.608636 −0.304318 0.952571i \(-0.598428\pi\)
−0.304318 + 0.952571i \(0.598428\pi\)
\(542\) 0 0
\(543\) −6.60983 −0.283655
\(544\) 0 0
\(545\) −3.01011 −0.128939
\(546\) 0 0
\(547\) −14.3338 −0.612871 −0.306436 0.951891i \(-0.599136\pi\)
−0.306436 + 0.951891i \(0.599136\pi\)
\(548\) 0 0
\(549\) 12.5145 0.534104
\(550\) 0 0
\(551\) 37.5177 1.59831
\(552\) 0 0
\(553\) −12.6453 −0.537734
\(554\) 0 0
\(555\) 1.13414 0.0481415
\(556\) 0 0
\(557\) −38.4953 −1.63110 −0.815548 0.578689i \(-0.803564\pi\)
−0.815548 + 0.578689i \(0.803564\pi\)
\(558\) 0 0
\(559\) 2.90135 0.122714
\(560\) 0 0
\(561\) −2.86586 −0.120997
\(562\) 0 0
\(563\) −39.9494 −1.68366 −0.841832 0.539739i \(-0.818523\pi\)
−0.841832 + 0.539739i \(0.818523\pi\)
\(564\) 0 0
\(565\) −0.128463 −0.00540449
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 2.83606 0.118894 0.0594469 0.998231i \(-0.481066\pi\)
0.0594469 + 0.998231i \(0.481066\pi\)
\(570\) 0 0
\(571\) −15.9890 −0.669119 −0.334559 0.942375i \(-0.608588\pi\)
−0.334559 + 0.942375i \(0.608588\pi\)
\(572\) 0 0
\(573\) 20.0122 0.836024
\(574\) 0 0
\(575\) −14.4393 −0.602161
\(576\) 0 0
\(577\) 15.2826 0.636221 0.318111 0.948054i \(-0.396952\pi\)
0.318111 + 0.948054i \(0.396952\pi\)
\(578\) 0 0
\(579\) 0.878108 0.0364929
\(580\) 0 0
\(581\) −11.8964 −0.493547
\(582\) 0 0
\(583\) −4.21853 −0.174714
\(584\) 0 0
\(585\) −0.152457 −0.00630334
\(586\) 0 0
\(587\) −44.5340 −1.83812 −0.919058 0.394122i \(-0.871049\pi\)
−0.919058 + 0.394122i \(0.871049\pi\)
\(588\) 0 0
\(589\) −12.7350 −0.524737
\(590\) 0 0
\(591\) 0.336362 0.0138361
\(592\) 0 0
\(593\) −23.9063 −0.981713 −0.490857 0.871240i \(-0.663316\pi\)
−0.490857 + 0.871240i \(0.663316\pi\)
\(594\) 0 0
\(595\) 1.13414 0.0464952
\(596\) 0 0
\(597\) −5.90463 −0.241660
\(598\) 0 0
\(599\) 23.9303 0.977764 0.488882 0.872350i \(-0.337405\pi\)
0.488882 + 0.872350i \(0.337405\pi\)
\(600\) 0 0
\(601\) 6.51206 0.265633 0.132816 0.991141i \(-0.457598\pi\)
0.132816 + 0.991141i \(0.457598\pi\)
\(602\) 0 0
\(603\) 9.80270 0.399197
\(604\) 0 0
\(605\) 1.65440 0.0672611
\(606\) 0 0
\(607\) 40.5756 1.64691 0.823456 0.567380i \(-0.192043\pi\)
0.823456 + 0.567380i \(0.192043\pi\)
\(608\) 0 0
\(609\) −5.20627 −0.210969
\(610\) 0 0
\(611\) −3.59151 −0.145297
\(612\) 0 0
\(613\) 39.1422 1.58094 0.790470 0.612501i \(-0.209836\pi\)
0.790470 + 0.612501i \(0.209836\pi\)
\(614\) 0 0
\(615\) 1.07541 0.0433645
\(616\) 0 0
\(617\) −31.7191 −1.27696 −0.638481 0.769638i \(-0.720437\pi\)
−0.638481 + 0.769638i \(0.720437\pi\)
\(618\) 0 0
\(619\) 0.246181 0.00989486 0.00494743 0.999988i \(-0.498425\pi\)
0.00494743 + 0.999988i \(0.498425\pi\)
\(620\) 0 0
\(621\) −2.90135 −0.116427
\(622\) 0 0
\(623\) 5.59151 0.224019
\(624\) 0 0
\(625\) 24.6519 0.986076
\(626\) 0 0
\(627\) 2.77618 0.110870
\(628\) 0 0
\(629\) 55.3395 2.20653
\(630\) 0 0
\(631\) 24.0587 0.957763 0.478882 0.877880i \(-0.341042\pi\)
0.478882 + 0.877880i \(0.341042\pi\)
\(632\) 0 0
\(633\) −3.67184 −0.145943
\(634\) 0 0
\(635\) 3.19943 0.126965
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 0 0
\(639\) 5.82430 0.230406
\(640\) 0 0
\(641\) −18.1962 −0.718705 −0.359352 0.933202i \(-0.617002\pi\)
−0.359352 + 0.933202i \(0.617002\pi\)
\(642\) 0 0
\(643\) −21.3926 −0.843641 −0.421820 0.906679i \(-0.638609\pi\)
−0.421820 + 0.906679i \(0.638609\pi\)
\(644\) 0 0
\(645\) 0.442333 0.0174168
\(646\) 0 0
\(647\) 19.8557 0.780610 0.390305 0.920686i \(-0.372370\pi\)
0.390305 + 0.920686i \(0.372370\pi\)
\(648\) 0 0
\(649\) −2.24378 −0.0880762
\(650\) 0 0
\(651\) 1.76721 0.0692625
\(652\) 0 0
\(653\) −34.4182 −1.34689 −0.673445 0.739238i \(-0.735186\pi\)
−0.673445 + 0.739238i \(0.735186\pi\)
\(654\) 0 0
\(655\) −1.39257 −0.0544121
\(656\) 0 0
\(657\) 3.09865 0.120890
\(658\) 0 0
\(659\) 35.7851 1.39399 0.696996 0.717075i \(-0.254520\pi\)
0.696996 + 0.717075i \(0.254520\pi\)
\(660\) 0 0
\(661\) 18.5988 0.723411 0.361705 0.932292i \(-0.382195\pi\)
0.361705 + 0.932292i \(0.382195\pi\)
\(662\) 0 0
\(663\) −7.43905 −0.288909
\(664\) 0 0
\(665\) −1.09865 −0.0426038
\(666\) 0 0
\(667\) −15.1052 −0.584876
\(668\) 0 0
\(669\) 12.6453 0.488896
\(670\) 0 0
\(671\) 4.82114 0.186118
\(672\) 0 0
\(673\) −42.6576 −1.64433 −0.822164 0.569250i \(-0.807233\pi\)
−0.822164 + 0.569250i \(0.807233\pi\)
\(674\) 0 0
\(675\) 4.97676 0.191555
\(676\) 0 0
\(677\) 32.3660 1.24393 0.621964 0.783046i \(-0.286335\pi\)
0.621964 + 0.783046i \(0.286335\pi\)
\(678\) 0 0
\(679\) −18.9502 −0.727243
\(680\) 0 0
\(681\) −6.59479 −0.252713
\(682\) 0 0
\(683\) 7.40761 0.283444 0.141722 0.989906i \(-0.454736\pi\)
0.141722 + 0.989906i \(0.454736\pi\)
\(684\) 0 0
\(685\) −3.44561 −0.131650
\(686\) 0 0
\(687\) 2.60983 0.0995712
\(688\) 0 0
\(689\) −10.9502 −0.417171
\(690\) 0 0
\(691\) 39.0147 1.48419 0.742094 0.670296i \(-0.233833\pi\)
0.742094 + 0.670296i \(0.233833\pi\)
\(692\) 0 0
\(693\) −0.385245 −0.0146343
\(694\) 0 0
\(695\) 3.01757 0.114463
\(696\) 0 0
\(697\) 52.4737 1.98758
\(698\) 0 0
\(699\) 1.35468 0.0512387
\(700\) 0 0
\(701\) 4.27598 0.161502 0.0807508 0.996734i \(-0.474268\pi\)
0.0807508 + 0.996734i \(0.474268\pi\)
\(702\) 0 0
\(703\) −53.6078 −2.02186
\(704\) 0 0
\(705\) −0.547553 −0.0206220
\(706\) 0 0
\(707\) −11.3803 −0.428001
\(708\) 0 0
\(709\) 45.5425 1.71038 0.855192 0.518311i \(-0.173439\pi\)
0.855192 + 0.518311i \(0.173439\pi\)
\(710\) 0 0
\(711\) 12.6453 0.474237
\(712\) 0 0
\(713\) 5.12730 0.192019
\(714\) 0 0
\(715\) −0.0587335 −0.00219651
\(716\) 0 0
\(717\) −14.4341 −0.539052
\(718\) 0 0
\(719\) 23.3005 0.868962 0.434481 0.900681i \(-0.356932\pi\)
0.434481 + 0.900681i \(0.356932\pi\)
\(720\) 0 0
\(721\) −5.13414 −0.191205
\(722\) 0 0
\(723\) −17.7306 −0.659407
\(724\) 0 0
\(725\) 25.9103 0.962285
\(726\) 0 0
\(727\) 1.55652 0.0577282 0.0288641 0.999583i \(-0.490811\pi\)
0.0288641 + 0.999583i \(0.490811\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.5833 0.798287
\(732\) 0 0
\(733\) 32.6087 1.20443 0.602215 0.798334i \(-0.294285\pi\)
0.602215 + 0.798334i \(0.294285\pi\)
\(734\) 0 0
\(735\) 0.152457 0.00562348
\(736\) 0 0
\(737\) 3.77645 0.139107
\(738\) 0 0
\(739\) 28.5633 1.05072 0.525360 0.850880i \(-0.323931\pi\)
0.525360 + 0.850880i \(0.323931\pi\)
\(740\) 0 0
\(741\) 7.20627 0.264729
\(742\) 0 0
\(743\) −20.4440 −0.750017 −0.375008 0.927021i \(-0.622360\pi\)
−0.375008 + 0.927021i \(0.622360\pi\)
\(744\) 0 0
\(745\) 1.92017 0.0703496
\(746\) 0 0
\(747\) 11.8964 0.435267
\(748\) 0 0
\(749\) 18.8781 0.689791
\(750\) 0 0
\(751\) 11.8748 0.433319 0.216659 0.976247i \(-0.430484\pi\)
0.216659 + 0.976247i \(0.430484\pi\)
\(752\) 0 0
\(753\) −4.05873 −0.147908
\(754\) 0 0
\(755\) −3.22206 −0.117263
\(756\) 0 0
\(757\) −37.1720 −1.35104 −0.675520 0.737342i \(-0.736081\pi\)
−0.675520 + 0.737342i \(0.736081\pi\)
\(758\) 0 0
\(759\) −1.11773 −0.0405711
\(760\) 0 0
\(761\) −44.8920 −1.62733 −0.813667 0.581331i \(-0.802532\pi\)
−0.813667 + 0.581331i \(0.802532\pi\)
\(762\) 0 0
\(763\) −19.7440 −0.714780
\(764\) 0 0
\(765\) −1.13414 −0.0410049
\(766\) 0 0
\(767\) −5.82430 −0.210303
\(768\) 0 0
\(769\) 46.7008 1.68407 0.842036 0.539421i \(-0.181357\pi\)
0.842036 + 0.539421i \(0.181357\pi\)
\(770\) 0 0
\(771\) 30.2584 1.08973
\(772\) 0 0
\(773\) 9.56827 0.344147 0.172073 0.985084i \(-0.444953\pi\)
0.172073 + 0.985084i \(0.444953\pi\)
\(774\) 0 0
\(775\) −8.79498 −0.315925
\(776\) 0 0
\(777\) 7.43905 0.266875
\(778\) 0 0
\(779\) −50.8316 −1.82123
\(780\) 0 0
\(781\) 2.24378 0.0802889
\(782\) 0 0
\(783\) 5.20627 0.186057
\(784\) 0 0
\(785\) −2.35380 −0.0840107
\(786\) 0 0
\(787\) −40.2898 −1.43617 −0.718087 0.695953i \(-0.754982\pi\)
−0.718087 + 0.695953i \(0.754982\pi\)
\(788\) 0 0
\(789\) −12.9502 −0.461041
\(790\) 0 0
\(791\) −0.842618 −0.0299600
\(792\) 0 0
\(793\) 12.5145 0.444401
\(794\) 0 0
\(795\) −1.66944 −0.0592091
\(796\) 0 0
\(797\) −23.6584 −0.838025 −0.419013 0.907980i \(-0.637624\pi\)
−0.419013 + 0.907980i \(0.637624\pi\)
\(798\) 0 0
\(799\) −26.7174 −0.945195
\(800\) 0 0
\(801\) −5.59151 −0.197566
\(802\) 0 0
\(803\) 1.19374 0.0421262
\(804\) 0 0
\(805\) 0.442333 0.0155902
\(806\) 0 0
\(807\) −17.7929 −0.626338
\(808\) 0 0
\(809\) −38.7431 −1.36213 −0.681067 0.732221i \(-0.738484\pi\)
−0.681067 + 0.732221i \(0.738484\pi\)
\(810\) 0 0
\(811\) −1.86144 −0.0653639 −0.0326819 0.999466i \(-0.510405\pi\)
−0.0326819 + 0.999466i \(0.510405\pi\)
\(812\) 0 0
\(813\) −17.9470 −0.629427
\(814\) 0 0
\(815\) 0.492363 0.0172467
\(816\) 0 0
\(817\) −20.9079 −0.731475
\(818\) 0 0
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 10.7024 0.373517 0.186758 0.982406i \(-0.440202\pi\)
0.186758 + 0.982406i \(0.440202\pi\)
\(822\) 0 0
\(823\) 38.9759 1.35861 0.679307 0.733854i \(-0.262281\pi\)
0.679307 + 0.733854i \(0.262281\pi\)
\(824\) 0 0
\(825\) 1.91727 0.0667509
\(826\) 0 0
\(827\) 24.1349 0.839253 0.419626 0.907697i \(-0.362161\pi\)
0.419626 + 0.907697i \(0.362161\pi\)
\(828\) 0 0
\(829\) 10.3636 0.359944 0.179972 0.983672i \(-0.442399\pi\)
0.179972 + 0.983672i \(0.442399\pi\)
\(830\) 0 0
\(831\) 5.87483 0.203796
\(832\) 0 0
\(833\) 7.43905 0.257748
\(834\) 0 0
\(835\) −2.64292 −0.0914621
\(836\) 0 0
\(837\) −1.76721 −0.0610838
\(838\) 0 0
\(839\) 22.7855 0.786644 0.393322 0.919401i \(-0.371326\pi\)
0.393322 + 0.919401i \(0.371326\pi\)
\(840\) 0 0
\(841\) −1.89479 −0.0653377
\(842\) 0 0
\(843\) 26.5417 0.914146
\(844\) 0 0
\(845\) −0.152457 −0.00524470
\(846\) 0 0
\(847\) 10.8516 0.372865
\(848\) 0 0
\(849\) −17.7562 −0.609392
\(850\) 0 0
\(851\) 21.5833 0.739866
\(852\) 0 0
\(853\) −24.0289 −0.822735 −0.411367 0.911470i \(-0.634949\pi\)
−0.411367 + 0.911470i \(0.634949\pi\)
\(854\) 0 0
\(855\) 1.09865 0.0375730
\(856\) 0 0
\(857\) 26.8683 0.917802 0.458901 0.888487i \(-0.348243\pi\)
0.458901 + 0.888487i \(0.348243\pi\)
\(858\) 0 0
\(859\) 18.6164 0.635183 0.317591 0.948228i \(-0.397126\pi\)
0.317591 + 0.948228i \(0.397126\pi\)
\(860\) 0 0
\(861\) 7.05381 0.240393
\(862\) 0 0
\(863\) 19.8341 0.675162 0.337581 0.941296i \(-0.390391\pi\)
0.337581 + 0.941296i \(0.390391\pi\)
\(864\) 0 0
\(865\) −1.30705 −0.0444409
\(866\) 0 0
\(867\) −38.3395 −1.30208
\(868\) 0 0
\(869\) 4.87155 0.165256
\(870\) 0 0
\(871\) 9.80270 0.332152
\(872\) 0 0
\(873\) 18.9502 0.641368
\(874\) 0 0
\(875\) −1.52103 −0.0514202
\(876\) 0 0
\(877\) −7.07541 −0.238919 −0.119460 0.992839i \(-0.538116\pi\)
−0.119460 + 0.992839i \(0.538116\pi\)
\(878\) 0 0
\(879\) 14.0139 0.472677
\(880\) 0 0
\(881\) −23.0412 −0.776277 −0.388138 0.921601i \(-0.626882\pi\)
−0.388138 + 0.921601i \(0.626882\pi\)
\(882\) 0 0
\(883\) −21.6462 −0.728453 −0.364226 0.931310i \(-0.618667\pi\)
−0.364226 + 0.931310i \(0.618667\pi\)
\(884\) 0 0
\(885\) −0.887958 −0.0298484
\(886\) 0 0
\(887\) −52.1223 −1.75009 −0.875047 0.484038i \(-0.839170\pi\)
−0.875047 + 0.484038i \(0.839170\pi\)
\(888\) 0 0
\(889\) 20.9857 0.703839
\(890\) 0 0
\(891\) 0.385245 0.0129062
\(892\) 0 0
\(893\) 25.8814 0.866088
\(894\) 0 0
\(895\) −0.176452 −0.00589812
\(896\) 0 0
\(897\) −2.90135 −0.0968733
\(898\) 0 0
\(899\) −9.20058 −0.306856
\(900\) 0 0
\(901\) −81.4594 −2.71381
\(902\) 0 0
\(903\) 2.90135 0.0965509
\(904\) 0 0
\(905\) −1.00772 −0.0334977
\(906\) 0 0
\(907\) 38.2841 1.27120 0.635601 0.772018i \(-0.280752\pi\)
0.635601 + 0.772018i \(0.280752\pi\)
\(908\) 0 0
\(909\) 11.3803 0.377461
\(910\) 0 0
\(911\) 15.8161 0.524011 0.262005 0.965066i \(-0.415616\pi\)
0.262005 + 0.965066i \(0.415616\pi\)
\(912\) 0 0
\(913\) 4.58304 0.151677
\(914\) 0 0
\(915\) 1.90792 0.0630740
\(916\) 0 0
\(917\) −9.13414 −0.301636
\(918\) 0 0
\(919\) 12.0546 0.397644 0.198822 0.980036i \(-0.436288\pi\)
0.198822 + 0.980036i \(0.436288\pi\)
\(920\) 0 0
\(921\) −22.4912 −0.741111
\(922\) 0 0
\(923\) 5.82430 0.191709
\(924\) 0 0
\(925\) −37.0224 −1.21729
\(926\) 0 0
\(927\) 5.13414 0.168627
\(928\) 0 0
\(929\) −1.98940 −0.0652701 −0.0326350 0.999467i \(-0.510390\pi\)
−0.0326350 + 0.999467i \(0.510390\pi\)
\(930\) 0 0
\(931\) −7.20627 −0.236176
\(932\) 0 0
\(933\) 33.1300 1.08463
\(934\) 0 0
\(935\) −0.436922 −0.0142889
\(936\) 0 0
\(937\) 1.31919 0.0430961 0.0215480 0.999768i \(-0.493141\pi\)
0.0215480 + 0.999768i \(0.493141\pi\)
\(938\) 0 0
\(939\) −19.3371 −0.631043
\(940\) 0 0
\(941\) 12.9107 0.420877 0.210438 0.977607i \(-0.432511\pi\)
0.210438 + 0.977607i \(0.432511\pi\)
\(942\) 0 0
\(943\) 20.4656 0.666451
\(944\) 0 0
\(945\) −0.152457 −0.00495944
\(946\) 0 0
\(947\) 46.7081 1.51781 0.758905 0.651202i \(-0.225735\pi\)
0.758905 + 0.651202i \(0.225735\pi\)
\(948\) 0 0
\(949\) 3.09865 0.100586
\(950\) 0 0
\(951\) 16.0860 0.521625
\(952\) 0 0
\(953\) −58.0355 −1.87995 −0.939977 0.341238i \(-0.889154\pi\)
−0.939977 + 0.341238i \(0.889154\pi\)
\(954\) 0 0
\(955\) 3.05102 0.0987286
\(956\) 0 0
\(957\) 2.00569 0.0648347
\(958\) 0 0
\(959\) −22.6005 −0.729808
\(960\) 0 0
\(961\) −27.8770 −0.899257
\(962\) 0 0
\(963\) −18.8781 −0.608339
\(964\) 0 0
\(965\) 0.133874 0.00430956
\(966\) 0 0
\(967\) 50.0057 1.60807 0.804037 0.594579i \(-0.202681\pi\)
0.804037 + 0.594579i \(0.202681\pi\)
\(968\) 0 0
\(969\) 53.6078 1.72213
\(970\) 0 0
\(971\) 52.3595 1.68030 0.840148 0.542357i \(-0.182468\pi\)
0.840148 + 0.542357i \(0.182468\pi\)
\(972\) 0 0
\(973\) 19.7929 0.634530
\(974\) 0 0
\(975\) 4.97676 0.159384
\(976\) 0 0
\(977\) −17.3098 −0.553791 −0.276895 0.960900i \(-0.589306\pi\)
−0.276895 + 0.960900i \(0.589306\pi\)
\(978\) 0 0
\(979\) −2.15410 −0.0688455
\(980\) 0 0
\(981\) 19.7440 0.630376
\(982\) 0 0
\(983\) 47.5409 1.51632 0.758159 0.652070i \(-0.226099\pi\)
0.758159 + 0.652070i \(0.226099\pi\)
\(984\) 0 0
\(985\) 0.0512808 0.00163394
\(986\) 0 0
\(987\) −3.59151 −0.114319
\(988\) 0 0
\(989\) 8.41784 0.267672
\(990\) 0 0
\(991\) 11.5809 0.367880 0.183940 0.982937i \(-0.441115\pi\)
0.183940 + 0.982937i \(0.441115\pi\)
\(992\) 0 0
\(993\) 27.9535 0.887078
\(994\) 0 0
\(995\) −0.900205 −0.0285384
\(996\) 0 0
\(997\) 16.5103 0.522886 0.261443 0.965219i \(-0.415802\pi\)
0.261443 + 0.965219i \(0.415802\pi\)
\(998\) 0 0
\(999\) −7.43905 −0.235361
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.1.2 4
3.2 odd 2 6552.2.a.bs.1.3 4
4.3 odd 2 4368.2.a.bs.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2184.2.a.v.1.2 4 1.1 even 1 trivial
4368.2.a.bs.1.2 4 4.3 odd 2
6552.2.a.bs.1.3 4 3.2 odd 2