Properties

Label 2842.2.a.j.1.1
Level $2842$
Weight $2$
Character 2842.1
Self dual yes
Analytic conductor $22.693$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2842,2,Mod(1,2842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2842, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2842 = 2 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2842.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: \(22.6934842544\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2842.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.41421 q^{5} +1.00000 q^{8} -3.00000 q^{9} -1.41421 q^{10} -2.00000 q^{11} +7.07107 q^{13} +1.00000 q^{16} -1.41421 q^{17} -3.00000 q^{18} -2.82843 q^{19} -1.41421 q^{20} -2.00000 q^{22} -6.00000 q^{23} -3.00000 q^{25} +7.07107 q^{26} -1.00000 q^{29} +1.41421 q^{31} +1.00000 q^{32} -1.41421 q^{34} -3.00000 q^{36} -4.00000 q^{37} -2.82843 q^{38} -1.41421 q^{40} -7.07107 q^{41} -4.00000 q^{43} -2.00000 q^{44} +4.24264 q^{45} -6.00000 q^{46} +9.89949 q^{47} -3.00000 q^{50} +7.07107 q^{52} -6.00000 q^{53} +2.82843 q^{55} -1.00000 q^{58} +1.41421 q^{59} +1.41421 q^{62} +1.00000 q^{64} -10.0000 q^{65} +4.00000 q^{67} -1.41421 q^{68} -2.00000 q^{71} -3.00000 q^{72} -12.7279 q^{73} -4.00000 q^{74} -2.82843 q^{76} -12.0000 q^{79} -1.41421 q^{80} +9.00000 q^{81} -7.07107 q^{82} -9.89949 q^{83} +2.00000 q^{85} -4.00000 q^{86} -2.00000 q^{88} -7.07107 q^{89} +4.24264 q^{90} -6.00000 q^{92} +9.89949 q^{94} +4.00000 q^{95} +4.24264 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} - 6 q^{9} - 4 q^{11} + 2 q^{16} - 6 q^{18} - 4 q^{22} - 12 q^{23} - 6 q^{25} - 2 q^{29} + 2 q^{32} - 6 q^{36} - 8 q^{37} - 8 q^{43} - 4 q^{44} - 12 q^{46} - 6 q^{50} - 12 q^{53} - 2 q^{58} + 2 q^{64} - 20 q^{65} + 8 q^{67} - 4 q^{71} - 6 q^{72} - 8 q^{74} - 24 q^{79} + 18 q^{81} + 4 q^{85} - 8 q^{86} - 4 q^{88} - 12 q^{92} + 8 q^{95} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) −3.00000 −1.00000
\(10\) −1.41421 −0.447214
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 7.07107 1.96116 0.980581 0.196116i \(-0.0628330\pi\)
0.980581 + 0.196116i \(0.0628330\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.41421 −0.342997 −0.171499 0.985184i \(-0.554861\pi\)
−0.171499 + 0.985184i \(0.554861\pi\)
\(18\) −3.00000 −0.707107
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) −1.41421 −0.316228
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 7.07107 1.38675
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.41421 −0.242536
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −2.82843 −0.458831
\(39\) 0 0
\(40\) −1.41421 −0.223607
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) −2.00000 −0.301511
\(45\) 4.24264 0.632456
\(46\) −6.00000 −0.884652
\(47\) 9.89949 1.44399 0.721995 0.691898i \(-0.243225\pi\)
0.721995 + 0.691898i \(0.243225\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) 7.07107 0.980581
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) −1.00000 −0.131306
\(59\) 1.41421 0.184115 0.0920575 0.995754i \(-0.470656\pi\)
0.0920575 + 0.995754i \(0.470656\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −10.0000 −1.24035
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) −1.41421 −0.171499
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) −3.00000 −0.353553
\(73\) −12.7279 −1.48969 −0.744845 0.667237i \(-0.767477\pi\)
−0.744845 + 0.667237i \(0.767477\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −2.82843 −0.324443
\(77\) 0 0
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) −1.41421 −0.158114
\(81\) 9.00000 1.00000
\(82\) −7.07107 −0.780869
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −2.00000 −0.213201
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 4.24264 0.447214
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) 0 0
\(94\) 9.89949 1.02105
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 4.24264 0.430775 0.215387 0.976529i \(-0.430899\pi\)
0.215387 + 0.976529i \(0.430899\pi\)
\(98\) 0 0
\(99\) 6.00000 0.603023
\(100\) −3.00000 −0.300000
\(101\) −11.3137 −1.12576 −0.562878 0.826540i \(-0.690306\pi\)
−0.562878 + 0.826540i \(0.690306\pi\)
\(102\) 0 0
\(103\) 14.1421 1.39347 0.696733 0.717331i \(-0.254636\pi\)
0.696733 + 0.717331i \(0.254636\pi\)
\(104\) 7.07107 0.693375
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 2.82843 0.269680
\(111\) 0 0
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 8.48528 0.791257
\(116\) −1.00000 −0.0928477
\(117\) −21.2132 −1.96116
\(118\) 1.41421 0.130189
\(119\) 0 0
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 1.41421 0.127000
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −10.0000 −0.877058
\(131\) −19.7990 −1.72985 −0.864923 0.501905i \(-0.832633\pi\)
−0.864923 + 0.501905i \(0.832633\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) −1.41421 −0.121268
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −15.5563 −1.31947 −0.659736 0.751497i \(-0.729332\pi\)
−0.659736 + 0.751497i \(0.729332\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) −14.1421 −1.18262
\(144\) −3.00000 −0.250000
\(145\) 1.41421 0.117444
\(146\) −12.7279 −1.05337
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) −2.82843 −0.229416
\(153\) 4.24264 0.342997
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 14.1421 1.12867 0.564333 0.825547i \(-0.309134\pi\)
0.564333 + 0.825547i \(0.309134\pi\)
\(158\) −12.0000 −0.954669
\(159\) 0 0
\(160\) −1.41421 −0.111803
\(161\) 0 0
\(162\) 9.00000 0.707107
\(163\) −6.00000 −0.469956 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(164\) −7.07107 −0.552158
\(165\) 0 0
\(166\) −9.89949 −0.768350
\(167\) −5.65685 −0.437741 −0.218870 0.975754i \(-0.570237\pi\)
−0.218870 + 0.975754i \(0.570237\pi\)
\(168\) 0 0
\(169\) 37.0000 2.84615
\(170\) 2.00000 0.153393
\(171\) 8.48528 0.648886
\(172\) −4.00000 −0.304997
\(173\) 18.3848 1.39777 0.698884 0.715235i \(-0.253680\pi\)
0.698884 + 0.715235i \(0.253680\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.00000 −0.150756
\(177\) 0 0
\(178\) −7.07107 −0.529999
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 4.24264 0.316228
\(181\) −15.5563 −1.15629 −0.578147 0.815933i \(-0.696224\pi\)
−0.578147 + 0.815933i \(0.696224\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 5.65685 0.415900
\(186\) 0 0
\(187\) 2.82843 0.206835
\(188\) 9.89949 0.721995
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 4.24264 0.304604
\(195\) 0 0
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 6.00000 0.426401
\(199\) 25.4558 1.80452 0.902258 0.431196i \(-0.141908\pi\)
0.902258 + 0.431196i \(0.141908\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) −11.3137 −0.796030
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 14.1421 0.985329
\(207\) 18.0000 1.25109
\(208\) 7.07107 0.490290
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 5.65685 0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 0 0
\(220\) 2.82843 0.190693
\(221\) −10.0000 −0.672673
\(222\) 0 0
\(223\) 25.4558 1.70465 0.852325 0.523013i \(-0.175192\pi\)
0.852325 + 0.523013i \(0.175192\pi\)
\(224\) 0 0
\(225\) 9.00000 0.600000
\(226\) 2.00000 0.133038
\(227\) −7.07107 −0.469323 −0.234662 0.972077i \(-0.575398\pi\)
−0.234662 + 0.972077i \(0.575398\pi\)
\(228\) 0 0
\(229\) 25.4558 1.68217 0.841085 0.540903i \(-0.181918\pi\)
0.841085 + 0.540903i \(0.181918\pi\)
\(230\) 8.48528 0.559503
\(231\) 0 0
\(232\) −1.00000 −0.0656532
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −21.2132 −1.38675
\(235\) −14.0000 −0.913259
\(236\) 1.41421 0.0920575
\(237\) 0 0
\(238\) 0 0
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −11.3137 −0.728780 −0.364390 0.931246i \(-0.618722\pi\)
−0.364390 + 0.931246i \(0.618722\pi\)
\(242\) −7.00000 −0.449977
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.0000 −1.27257
\(248\) 1.41421 0.0898027
\(249\) 0 0
\(250\) 11.3137 0.715542
\(251\) 2.82843 0.178529 0.0892644 0.996008i \(-0.471548\pi\)
0.0892644 + 0.996008i \(0.471548\pi\)
\(252\) 0 0
\(253\) 12.0000 0.754434
\(254\) 16.0000 1.00393
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 14.1421 0.882162 0.441081 0.897467i \(-0.354595\pi\)
0.441081 + 0.897467i \(0.354595\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −10.0000 −0.620174
\(261\) 3.00000 0.185695
\(262\) −19.7990 −1.22319
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 8.48528 0.521247
\(266\) 0 0
\(267\) 0 0
\(268\) 4.00000 0.244339
\(269\) 11.3137 0.689809 0.344904 0.938638i \(-0.387911\pi\)
0.344904 + 0.938638i \(0.387911\pi\)
\(270\) 0 0
\(271\) −15.5563 −0.944981 −0.472490 0.881336i \(-0.656645\pi\)
−0.472490 + 0.881336i \(0.656645\pi\)
\(272\) −1.41421 −0.0857493
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 6.00000 0.361814
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) −15.5563 −0.933008
\(279\) −4.24264 −0.254000
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 12.7279 0.756596 0.378298 0.925684i \(-0.376509\pi\)
0.378298 + 0.925684i \(0.376509\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) −14.1421 −0.836242
\(287\) 0 0
\(288\) −3.00000 −0.176777
\(289\) −15.0000 −0.882353
\(290\) 1.41421 0.0830455
\(291\) 0 0
\(292\) −12.7279 −0.744845
\(293\) 2.82843 0.165238 0.0826192 0.996581i \(-0.473671\pi\)
0.0826192 + 0.996581i \(0.473671\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) −4.00000 −0.232495
\(297\) 0 0
\(298\) −10.0000 −0.579284
\(299\) −42.4264 −2.45358
\(300\) 0 0
\(301\) 0 0
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −2.82843 −0.162221
\(305\) 0 0
\(306\) 4.24264 0.242536
\(307\) −11.3137 −0.645707 −0.322854 0.946449i \(-0.604642\pi\)
−0.322854 + 0.946449i \(0.604642\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.00000 −0.113592
\(311\) −32.5269 −1.84443 −0.922216 0.386675i \(-0.873623\pi\)
−0.922216 + 0.386675i \(0.873623\pi\)
\(312\) 0 0
\(313\) 16.9706 0.959233 0.479616 0.877478i \(-0.340776\pi\)
0.479616 + 0.877478i \(0.340776\pi\)
\(314\) 14.1421 0.798087
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) −1.41421 −0.0790569
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 9.00000 0.500000
\(325\) −21.2132 −1.17670
\(326\) −6.00000 −0.332309
\(327\) 0 0
\(328\) −7.07107 −0.390434
\(329\) 0 0
\(330\) 0 0
\(331\) 12.0000 0.659580 0.329790 0.944054i \(-0.393022\pi\)
0.329790 + 0.944054i \(0.393022\pi\)
\(332\) −9.89949 −0.543305
\(333\) 12.0000 0.657596
\(334\) −5.65685 −0.309529
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 37.0000 2.01253
\(339\) 0 0
\(340\) 2.00000 0.108465
\(341\) −2.82843 −0.153168
\(342\) 8.48528 0.458831
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 18.3848 0.988372
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 18.3848 0.984115 0.492057 0.870563i \(-0.336245\pi\)
0.492057 + 0.870563i \(0.336245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.00000 −0.106600
\(353\) 28.2843 1.50542 0.752710 0.658352i \(-0.228746\pi\)
0.752710 + 0.658352i \(0.228746\pi\)
\(354\) 0 0
\(355\) 2.82843 0.150117
\(356\) −7.07107 −0.374766
\(357\) 0 0
\(358\) 4.00000 0.211407
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 4.24264 0.223607
\(361\) −11.0000 −0.578947
\(362\) −15.5563 −0.817624
\(363\) 0 0
\(364\) 0 0
\(365\) 18.0000 0.942163
\(366\) 0 0
\(367\) −21.2132 −1.10732 −0.553660 0.832743i \(-0.686769\pi\)
−0.553660 + 0.832743i \(0.686769\pi\)
\(368\) −6.00000 −0.312772
\(369\) 21.2132 1.10432
\(370\) 5.65685 0.294086
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) 9.89949 0.510527
\(377\) −7.07107 −0.364179
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 4.00000 0.205196
\(381\) 0 0
\(382\) 12.0000 0.613973
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.0000 −0.712581
\(387\) 12.0000 0.609994
\(388\) 4.24264 0.215387
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 8.48528 0.429119
\(392\) 0 0
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) 16.9706 0.853882
\(396\) 6.00000 0.301511
\(397\) −12.7279 −0.638796 −0.319398 0.947621i \(-0.603481\pi\)
−0.319398 + 0.947621i \(0.603481\pi\)
\(398\) 25.4558 1.27599
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 10.0000 0.498135
\(404\) −11.3137 −0.562878
\(405\) −12.7279 −0.632456
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) −18.3848 −0.909069 −0.454534 0.890729i \(-0.650194\pi\)
−0.454534 + 0.890729i \(0.650194\pi\)
\(410\) 10.0000 0.493865
\(411\) 0 0
\(412\) 14.1421 0.696733
\(413\) 0 0
\(414\) 18.0000 0.884652
\(415\) 14.0000 0.687233
\(416\) 7.07107 0.346688
\(417\) 0 0
\(418\) 5.65685 0.276686
\(419\) 4.24264 0.207267 0.103633 0.994616i \(-0.466953\pi\)
0.103633 + 0.994616i \(0.466953\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) 12.0000 0.584151
\(423\) −29.6985 −1.44399
\(424\) −6.00000 −0.291386
\(425\) 4.24264 0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 5.65685 0.272798
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 0 0
\(433\) −29.6985 −1.42722 −0.713609 0.700544i \(-0.752941\pi\)
−0.713609 + 0.700544i \(0.752941\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 16.9706 0.811812
\(438\) 0 0
\(439\) 22.6274 1.07995 0.539974 0.841682i \(-0.318434\pi\)
0.539974 + 0.841682i \(0.318434\pi\)
\(440\) 2.82843 0.134840
\(441\) 0 0
\(442\) −10.0000 −0.475651
\(443\) 34.0000 1.61539 0.807694 0.589601i \(-0.200715\pi\)
0.807694 + 0.589601i \(0.200715\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 25.4558 1.20537
\(447\) 0 0
\(448\) 0 0
\(449\) −2.00000 −0.0943858 −0.0471929 0.998886i \(-0.515028\pi\)
−0.0471929 + 0.998886i \(0.515028\pi\)
\(450\) 9.00000 0.424264
\(451\) 14.1421 0.665927
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) −7.07107 −0.331862
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) 25.4558 1.18947
\(459\) 0 0
\(460\) 8.48528 0.395628
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 30.0000 1.39422 0.697109 0.716965i \(-0.254469\pi\)
0.697109 + 0.716965i \(0.254469\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 19.7990 0.916188 0.458094 0.888904i \(-0.348532\pi\)
0.458094 + 0.888904i \(0.348532\pi\)
\(468\) −21.2132 −0.980581
\(469\) 0 0
\(470\) −14.0000 −0.645772
\(471\) 0 0
\(472\) 1.41421 0.0650945
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) 18.0000 0.824163
\(478\) −24.0000 −1.09773
\(479\) −4.24264 −0.193851 −0.0969256 0.995292i \(-0.530901\pi\)
−0.0969256 + 0.995292i \(0.530901\pi\)
\(480\) 0 0
\(481\) −28.2843 −1.28965
\(482\) −11.3137 −0.515325
\(483\) 0 0
\(484\) −7.00000 −0.318182
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) 32.0000 1.45006 0.725029 0.688718i \(-0.241826\pi\)
0.725029 + 0.688718i \(0.241826\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 1.41421 0.0636930
\(494\) −20.0000 −0.899843
\(495\) −8.48528 −0.381385
\(496\) 1.41421 0.0635001
\(497\) 0 0
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 11.3137 0.505964
\(501\) 0 0
\(502\) 2.82843 0.126239
\(503\) 12.7279 0.567510 0.283755 0.958897i \(-0.408420\pi\)
0.283755 + 0.958897i \(0.408420\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) 16.0000 0.709885
\(509\) −1.41421 −0.0626839 −0.0313420 0.999509i \(-0.509978\pi\)
−0.0313420 + 0.999509i \(0.509978\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 14.1421 0.623783
\(515\) −20.0000 −0.881305
\(516\) 0 0
\(517\) −19.7990 −0.870759
\(518\) 0 0
\(519\) 0 0
\(520\) −10.0000 −0.438529
\(521\) 19.7990 0.867409 0.433705 0.901055i \(-0.357206\pi\)
0.433705 + 0.901055i \(0.357206\pi\)
\(522\) 3.00000 0.131306
\(523\) −15.5563 −0.680232 −0.340116 0.940384i \(-0.610466\pi\)
−0.340116 + 0.940384i \(0.610466\pi\)
\(524\) −19.7990 −0.864923
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 8.48528 0.368577
\(531\) −4.24264 −0.184115
\(532\) 0 0
\(533\) −50.0000 −2.16574
\(534\) 0 0
\(535\) 16.9706 0.733701
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 11.3137 0.487769
\(539\) 0 0
\(540\) 0 0
\(541\) 26.0000 1.11783 0.558914 0.829226i \(-0.311218\pi\)
0.558914 + 0.829226i \(0.311218\pi\)
\(542\) −15.5563 −0.668202
\(543\) 0 0
\(544\) −1.41421 −0.0606339
\(545\) 14.1421 0.605783
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 6.00000 0.255841
\(551\) 2.82843 0.120495
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) −15.5563 −0.659736
\(557\) 34.0000 1.44063 0.720313 0.693649i \(-0.243998\pi\)
0.720313 + 0.693649i \(0.243998\pi\)
\(558\) −4.24264 −0.179605
\(559\) −28.2843 −1.19630
\(560\) 0 0
\(561\) 0 0
\(562\) −12.0000 −0.506189
\(563\) −14.1421 −0.596020 −0.298010 0.954563i \(-0.596323\pi\)
−0.298010 + 0.954563i \(0.596323\pi\)
\(564\) 0 0
\(565\) −2.82843 −0.118993
\(566\) 12.7279 0.534994
\(567\) 0 0
\(568\) −2.00000 −0.0839181
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −14.1421 −0.591312
\(573\) 0 0
\(574\) 0 0
\(575\) 18.0000 0.750652
\(576\) −3.00000 −0.125000
\(577\) 21.2132 0.883117 0.441559 0.897232i \(-0.354426\pi\)
0.441559 + 0.897232i \(0.354426\pi\)
\(578\) −15.0000 −0.623918
\(579\) 0 0
\(580\) 1.41421 0.0587220
\(581\) 0 0
\(582\) 0 0
\(583\) 12.0000 0.496989
\(584\) −12.7279 −0.526685
\(585\) 30.0000 1.24035
\(586\) 2.82843 0.116841
\(587\) −4.24264 −0.175113 −0.0875563 0.996160i \(-0.527906\pi\)
−0.0875563 + 0.996160i \(0.527906\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −2.00000 −0.0823387
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −39.5980 −1.62609 −0.813047 0.582198i \(-0.802193\pi\)
−0.813047 + 0.582198i \(0.802193\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −42.4264 −1.73494
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −9.89949 −0.403809 −0.201904 0.979405i \(-0.564713\pi\)
−0.201904 + 0.979405i \(0.564713\pi\)
\(602\) 0 0
\(603\) −12.0000 −0.488678
\(604\) 10.0000 0.406894
\(605\) 9.89949 0.402472
\(606\) 0 0
\(607\) −12.7279 −0.516610 −0.258305 0.966063i \(-0.583164\pi\)
−0.258305 + 0.966063i \(0.583164\pi\)
\(608\) −2.82843 −0.114708
\(609\) 0 0
\(610\) 0 0
\(611\) 70.0000 2.83190
\(612\) 4.24264 0.171499
\(613\) 10.0000 0.403896 0.201948 0.979396i \(-0.435273\pi\)
0.201948 + 0.979396i \(0.435273\pi\)
\(614\) −11.3137 −0.456584
\(615\) 0 0
\(616\) 0 0
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) −31.1127 −1.25052 −0.625262 0.780415i \(-0.715008\pi\)
−0.625262 + 0.780415i \(0.715008\pi\)
\(620\) −2.00000 −0.0803219
\(621\) 0 0
\(622\) −32.5269 −1.30421
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 16.9706 0.678280
\(627\) 0 0
\(628\) 14.1421 0.564333
\(629\) 5.65685 0.225554
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) −12.0000 −0.477334
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) −22.6274 −0.897942
\(636\) 0 0
\(637\) 0 0
\(638\) 2.00000 0.0791808
\(639\) 6.00000 0.237356
\(640\) −1.41421 −0.0559017
\(641\) −38.0000 −1.50091 −0.750455 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 0 0
\(643\) 24.0416 0.948109 0.474055 0.880495i \(-0.342790\pi\)
0.474055 + 0.880495i \(0.342790\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 4.00000 0.157378
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 9.00000 0.353553
\(649\) −2.82843 −0.111025
\(650\) −21.2132 −0.832050
\(651\) 0 0
\(652\) −6.00000 −0.234978
\(653\) −50.0000 −1.95665 −0.978326 0.207072i \(-0.933606\pi\)
−0.978326 + 0.207072i \(0.933606\pi\)
\(654\) 0 0
\(655\) 28.0000 1.09405
\(656\) −7.07107 −0.276079
\(657\) 38.1838 1.48969
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −41.0122 −1.59519 −0.797595 0.603194i \(-0.793895\pi\)
−0.797595 + 0.603194i \(0.793895\pi\)
\(662\) 12.0000 0.466393
\(663\) 0 0
\(664\) −9.89949 −0.384175
\(665\) 0 0
\(666\) 12.0000 0.464991
\(667\) 6.00000 0.232321
\(668\) −5.65685 −0.218870
\(669\) 0 0
\(670\) −5.65685 −0.218543
\(671\) 0 0
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 37.0000 1.42308
\(677\) 42.4264 1.63058 0.815290 0.579053i \(-0.196578\pi\)
0.815290 + 0.579053i \(0.196578\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.00000 0.0766965
\(681\) 0 0
\(682\) −2.82843 −0.108306
\(683\) 36.0000 1.37750 0.688751 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 8.48528 0.324443
\(685\) 2.82843 0.108069
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −42.4264 −1.61632
\(690\) 0 0
\(691\) −1.41421 −0.0537992 −0.0268996 0.999638i \(-0.508563\pi\)
−0.0268996 + 0.999638i \(0.508563\pi\)
\(692\) 18.3848 0.698884
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 22.0000 0.834508
\(696\) 0 0
\(697\) 10.0000 0.378777
\(698\) 18.3848 0.695874
\(699\) 0 0
\(700\) 0 0
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) 11.3137 0.426705
\(704\) −2.00000 −0.0753778
\(705\) 0 0
\(706\) 28.2843 1.06449
\(707\) 0 0
\(708\) 0 0
\(709\) −34.0000 −1.27690 −0.638448 0.769665i \(-0.720423\pi\)
−0.638448 + 0.769665i \(0.720423\pi\)
\(710\) 2.82843 0.106149
\(711\) 36.0000 1.35011
\(712\) −7.07107 −0.264999
\(713\) −8.48528 −0.317776
\(714\) 0 0
\(715\) 20.0000 0.747958
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) 32.0000 1.19423
\(719\) 14.1421 0.527413 0.263706 0.964603i \(-0.415055\pi\)
0.263706 + 0.964603i \(0.415055\pi\)
\(720\) 4.24264 0.158114
\(721\) 0 0
\(722\) −11.0000 −0.409378
\(723\) 0 0
\(724\) −15.5563 −0.578147
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) 38.1838 1.41616 0.708079 0.706133i \(-0.249562\pi\)
0.708079 + 0.706133i \(0.249562\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 18.0000 0.666210
\(731\) 5.65685 0.209226
\(732\) 0 0
\(733\) −22.6274 −0.835763 −0.417881 0.908502i \(-0.637227\pi\)
−0.417881 + 0.908502i \(0.637227\pi\)
\(734\) −21.2132 −0.782994
\(735\) 0 0
\(736\) −6.00000 −0.221163
\(737\) −8.00000 −0.294684
\(738\) 21.2132 0.780869
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 5.65685 0.207950
\(741\) 0 0
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 14.1421 0.518128
\(746\) −14.0000 −0.512576
\(747\) 29.6985 1.08661
\(748\) 2.82843 0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −48.0000 −1.75154 −0.875772 0.482724i \(-0.839647\pi\)
−0.875772 + 0.482724i \(0.839647\pi\)
\(752\) 9.89949 0.360997
\(753\) 0 0
\(754\) −7.07107 −0.257513
\(755\) −14.1421 −0.514685
\(756\) 0 0
\(757\) 36.0000 1.30844 0.654221 0.756303i \(-0.272997\pi\)
0.654221 + 0.756303i \(0.272997\pi\)
\(758\) 20.0000 0.726433
\(759\) 0 0
\(760\) 4.00000 0.145095
\(761\) −11.3137 −0.410122 −0.205061 0.978749i \(-0.565739\pi\)
−0.205061 + 0.978749i \(0.565739\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 12.0000 0.434145
\(765\) −6.00000 −0.216930
\(766\) 0 0
\(767\) 10.0000 0.361079
\(768\) 0 0
\(769\) 4.24264 0.152994 0.0764968 0.997070i \(-0.475627\pi\)
0.0764968 + 0.997070i \(0.475627\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −2.82843 −0.101731 −0.0508657 0.998706i \(-0.516198\pi\)
−0.0508657 + 0.998706i \(0.516198\pi\)
\(774\) 12.0000 0.431331
\(775\) −4.24264 −0.152400
\(776\) 4.24264 0.152302
\(777\) 0 0
\(778\) −8.00000 −0.286814
\(779\) 20.0000 0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 8.48528 0.303433
\(783\) 0 0
\(784\) 0 0
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 41.0122 1.46193 0.730963 0.682417i \(-0.239071\pi\)
0.730963 + 0.682417i \(0.239071\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) 16.9706 0.603786
\(791\) 0 0
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −12.7279 −0.451697
\(795\) 0 0
\(796\) 25.4558 0.902258
\(797\) −25.4558 −0.901692 −0.450846 0.892602i \(-0.648878\pi\)
−0.450846 + 0.892602i \(0.648878\pi\)
\(798\) 0 0
\(799\) −14.0000 −0.495284
\(800\) −3.00000 −0.106066
\(801\) 21.2132 0.749532
\(802\) 2.00000 0.0706225
\(803\) 25.4558 0.898317
\(804\) 0 0
\(805\) 0 0
\(806\) 10.0000 0.352235
\(807\) 0 0
\(808\) −11.3137 −0.398015
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) −12.7279 −0.447214
\(811\) 46.6690 1.63877 0.819386 0.573242i \(-0.194315\pi\)
0.819386 + 0.573242i \(0.194315\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 8.48528 0.297226
\(816\) 0 0
\(817\) 11.3137 0.395817
\(818\) −18.3848 −0.642809
\(819\) 0 0
\(820\) 10.0000 0.349215
\(821\) 6.00000 0.209401 0.104701 0.994504i \(-0.466612\pi\)
0.104701 + 0.994504i \(0.466612\pi\)
\(822\) 0 0
\(823\) 36.0000 1.25488 0.627441 0.778664i \(-0.284103\pi\)
0.627441 + 0.778664i \(0.284103\pi\)
\(824\) 14.1421 0.492665
\(825\) 0 0
\(826\) 0 0
\(827\) 2.00000 0.0695468 0.0347734 0.999395i \(-0.488929\pi\)
0.0347734 + 0.999395i \(0.488929\pi\)
\(828\) 18.0000 0.625543
\(829\) 5.65685 0.196471 0.0982353 0.995163i \(-0.468680\pi\)
0.0982353 + 0.995163i \(0.468680\pi\)
\(830\) 14.0000 0.485947
\(831\) 0 0
\(832\) 7.07107 0.245145
\(833\) 0 0
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 5.65685 0.195646
\(837\) 0 0
\(838\) 4.24264 0.146560
\(839\) −52.3259 −1.80649 −0.903245 0.429124i \(-0.858822\pi\)
−0.903245 + 0.429124i \(0.858822\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −20.0000 −0.689246
\(843\) 0 0
\(844\) 12.0000 0.413057
\(845\) −52.3259 −1.80007
\(846\) −29.6985 −1.02105
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 0 0
\(850\) 4.24264 0.145521
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −5.65685 −0.193687 −0.0968435 0.995300i \(-0.530875\pi\)
−0.0968435 + 0.995300i \(0.530875\pi\)
\(854\) 0 0
\(855\) −12.0000 −0.410391
\(856\) −12.0000 −0.410152
\(857\) −28.2843 −0.966172 −0.483086 0.875573i \(-0.660484\pi\)
−0.483086 + 0.875573i \(0.660484\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 5.65685 0.192897
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) −10.0000 −0.340404 −0.170202 0.985409i \(-0.554442\pi\)
−0.170202 + 0.985409i \(0.554442\pi\)
\(864\) 0 0
\(865\) −26.0000 −0.884027
\(866\) −29.6985 −1.00920
\(867\) 0 0
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 28.2843 0.958376
\(872\) −10.0000 −0.338643
\(873\) −12.7279 −0.430775
\(874\) 16.9706 0.574038
\(875\) 0 0
\(876\) 0 0
\(877\) 34.0000 1.14810 0.574049 0.818821i \(-0.305372\pi\)
0.574049 + 0.818821i \(0.305372\pi\)
\(878\) 22.6274 0.763638
\(879\) 0 0
\(880\) 2.82843 0.0953463
\(881\) 15.5563 0.524107 0.262053 0.965053i \(-0.415600\pi\)
0.262053 + 0.965053i \(0.415600\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) −10.0000 −0.336336
\(885\) 0 0
\(886\) 34.0000 1.14225
\(887\) 49.4975 1.66196 0.830981 0.556300i \(-0.187780\pi\)
0.830981 + 0.556300i \(0.187780\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 10.0000 0.335201
\(891\) −18.0000 −0.603023
\(892\) 25.4558 0.852325
\(893\) −28.0000 −0.936984
\(894\) 0 0
\(895\) −5.65685 −0.189088
\(896\) 0 0
\(897\) 0 0
\(898\) −2.00000 −0.0667409
\(899\) −1.41421 −0.0471667
\(900\) 9.00000 0.300000
\(901\) 8.48528 0.282686
\(902\) 14.1421 0.470882
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) 22.0000 0.731305
\(906\) 0 0
\(907\) −34.0000 −1.12895 −0.564476 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) −7.07107 −0.234662
\(909\) 33.9411 1.12576
\(910\) 0 0
\(911\) −28.0000 −0.927681 −0.463841 0.885919i \(-0.653529\pi\)
−0.463841 + 0.885919i \(0.653529\pi\)
\(912\) 0 0
\(913\) 19.7990 0.655251
\(914\) −8.00000 −0.264616
\(915\) 0 0
\(916\) 25.4558 0.841085
\(917\) 0 0
\(918\) 0 0
\(919\) 14.0000 0.461817 0.230909 0.972975i \(-0.425830\pi\)
0.230909 + 0.972975i \(0.425830\pi\)
\(920\) 8.48528 0.279751
\(921\) 0 0
\(922\) 0 0
\(923\) −14.1421 −0.465494
\(924\) 0 0
\(925\) 12.0000 0.394558
\(926\) 30.0000 0.985861
\(927\) −42.4264 −1.39347
\(928\) −1.00000 −0.0328266
\(929\) 45.2548 1.48476 0.742381 0.669977i \(-0.233696\pi\)
0.742381 + 0.669977i \(0.233696\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −14.0000 −0.458585
\(933\) 0 0
\(934\) 19.7990 0.647843
\(935\) −4.00000 −0.130814
\(936\) −21.2132 −0.693375
\(937\) −50.9117 −1.66321 −0.831606 0.555366i \(-0.812578\pi\)
−0.831606 + 0.555366i \(0.812578\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −14.0000 −0.456630
\(941\) −15.5563 −0.507122 −0.253561 0.967319i \(-0.581602\pi\)
−0.253561 + 0.967319i \(0.581602\pi\)
\(942\) 0 0
\(943\) 42.4264 1.38159
\(944\) 1.41421 0.0460287
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 0 0
\(949\) −90.0000 −2.92152
\(950\) 8.48528 0.275299
\(951\) 0 0
\(952\) 0 0
\(953\) −56.0000 −1.81402 −0.907009 0.421111i \(-0.861640\pi\)
−0.907009 + 0.421111i \(0.861640\pi\)
\(954\) 18.0000 0.582772
\(955\) −16.9706 −0.549155
\(956\) −24.0000 −0.776215
\(957\) 0 0
\(958\) −4.24264 −0.137073
\(959\) 0 0
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) −28.2843 −0.911922
\(963\) 36.0000 1.16008
\(964\) −11.3137 −0.364390
\(965\) 19.7990 0.637352
\(966\) 0 0
\(967\) −4.00000 −0.128631 −0.0643157 0.997930i \(-0.520486\pi\)
−0.0643157 + 0.997930i \(0.520486\pi\)
\(968\) −7.00000 −0.224989
\(969\) 0 0
\(970\) −6.00000 −0.192648
\(971\) 59.3970 1.90614 0.953070 0.302751i \(-0.0979050\pi\)
0.953070 + 0.302751i \(0.0979050\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 32.0000 1.02535
\(975\) 0 0
\(976\) 0 0
\(977\) −60.0000 −1.91957 −0.959785 0.280736i \(-0.909421\pi\)
−0.959785 + 0.280736i \(0.909421\pi\)
\(978\) 0 0
\(979\) 14.1421 0.451985
\(980\) 0 0
\(981\) 30.0000 0.957826
\(982\) −36.0000 −1.14881
\(983\) −41.0122 −1.30809 −0.654043 0.756457i \(-0.726928\pi\)
−0.654043 + 0.756457i \(0.726928\pi\)
\(984\) 0 0
\(985\) −14.1421 −0.450606
\(986\) 1.41421 0.0450377
\(987\) 0 0
\(988\) −20.0000 −0.636285
\(989\) 24.0000 0.763156
\(990\) −8.48528 −0.269680
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 1.41421 0.0449013
\(993\) 0 0
\(994\) 0 0
\(995\) −36.0000 −1.14128
\(996\) 0 0
\(997\) −33.9411 −1.07493 −0.537463 0.843287i \(-0.680617\pi\)
−0.537463 + 0.843287i \(0.680617\pi\)
\(998\) 32.0000 1.01294
\(999\) 0 0
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 2842.2.a.j.1.1 2
7.6 odd 2 inner 2842.2.a.j.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2842.2.a.j.1.1 2 1.1 even 1 trivial
2842.2.a.j.1.2 yes 2 7.6 odd 2 inner