Properties

Label 1386.2.a.o.1.1
Level $1386$
Weight $2$
Character 1386.1
Self dual yes
Analytic conductor $11.067$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1386,2,Mod(1,1386)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1386, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1386.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1386.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: \(11.0672657201\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) 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 \(-3.16228\) of defining polynomial
Character \(\chi\) \(=\) 1386.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.16228 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -3.16228 q^{5} -1.00000 q^{7} +1.00000 q^{8} -3.16228 q^{10} -1.00000 q^{11} +2.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.16228 q^{17} +5.16228 q^{19} -3.16228 q^{20} -1.00000 q^{22} -6.32456 q^{23} +5.00000 q^{25} +2.00000 q^{26} -1.00000 q^{28} +4.00000 q^{29} +9.16228 q^{31} +1.00000 q^{32} +7.16228 q^{34} +3.16228 q^{35} +8.32456 q^{37} +5.16228 q^{38} -3.16228 q^{40} -7.16228 q^{41} -6.32456 q^{43} -1.00000 q^{44} -6.32456 q^{46} +11.4868 q^{47} +1.00000 q^{49} +5.00000 q^{50} +2.00000 q^{52} +4.32456 q^{53} +3.16228 q^{55} -1.00000 q^{56} +4.00000 q^{58} -1.67544 q^{59} +10.0000 q^{61} +9.16228 q^{62} +1.00000 q^{64} -6.32456 q^{65} +4.32456 q^{67} +7.16228 q^{68} +3.16228 q^{70} -8.00000 q^{71} -5.48683 q^{73} +8.32456 q^{74} +5.16228 q^{76} +1.00000 q^{77} -4.32456 q^{79} -3.16228 q^{80} -7.16228 q^{82} -5.16228 q^{83} -22.6491 q^{85} -6.32456 q^{86} -1.00000 q^{88} +16.3246 q^{89} -2.00000 q^{91} -6.32456 q^{92} +11.4868 q^{94} -16.3246 q^{95} -10.6491 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{7} + 2 q^{8} - 2 q^{11} + 4 q^{13} - 2 q^{14} + 2 q^{16} + 8 q^{17} + 4 q^{19} - 2 q^{22} + 10 q^{25} + 4 q^{26} - 2 q^{28} + 8 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 4 q^{37} + 4 q^{38} - 8 q^{41} - 2 q^{44} + 4 q^{47} + 2 q^{49} + 10 q^{50} + 4 q^{52} - 4 q^{53} - 2 q^{56} + 8 q^{58} - 16 q^{59} + 20 q^{61} + 12 q^{62} + 2 q^{64} - 4 q^{67} + 8 q^{68} - 16 q^{71} + 8 q^{73} + 4 q^{74} + 4 q^{76} + 2 q^{77} + 4 q^{79} - 8 q^{82} - 4 q^{83} - 20 q^{85} - 2 q^{88} + 20 q^{89} - 4 q^{91} + 4 q^{94} - 20 q^{95} + 4 q^{97} + 2 q^{98}+O(q^{100}) \) 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
\(4\) 1.00000 0.500000
\(5\) −3.16228 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −3.16228 −1.00000
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 7.16228 1.73711 0.868554 0.495595i \(-0.165050\pi\)
0.868554 + 0.495595i \(0.165050\pi\)
\(18\) 0 0
\(19\) 5.16228 1.18431 0.592154 0.805825i \(-0.298278\pi\)
0.592154 + 0.805825i \(0.298278\pi\)
\(20\) −3.16228 −0.707107
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −6.32456 −1.31876 −0.659380 0.751809i \(-0.729181\pi\)
−0.659380 + 0.751809i \(0.729181\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −1.00000 −0.188982
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) 9.16228 1.64559 0.822797 0.568336i \(-0.192412\pi\)
0.822797 + 0.568336i \(0.192412\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 7.16228 1.22832
\(35\) 3.16228 0.534522
\(36\) 0 0
\(37\) 8.32456 1.36855 0.684274 0.729225i \(-0.260119\pi\)
0.684274 + 0.729225i \(0.260119\pi\)
\(38\) 5.16228 0.837432
\(39\) 0 0
\(40\) −3.16228 −0.500000
\(41\) −7.16228 −1.11856 −0.559280 0.828979i \(-0.688922\pi\)
−0.559280 + 0.828979i \(0.688922\pi\)
\(42\) 0 0
\(43\) −6.32456 −0.964486 −0.482243 0.876038i \(-0.660178\pi\)
−0.482243 + 0.876038i \(0.660178\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −6.32456 −0.932505
\(47\) 11.4868 1.67553 0.837763 0.546033i \(-0.183863\pi\)
0.837763 + 0.546033i \(0.183863\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.00000 0.707107
\(51\) 0 0
\(52\) 2.00000 0.277350
\(53\) 4.32456 0.594023 0.297012 0.954874i \(-0.404010\pi\)
0.297012 + 0.954874i \(0.404010\pi\)
\(54\) 0 0
\(55\) 3.16228 0.426401
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 4.00000 0.525226
\(59\) −1.67544 −0.218124 −0.109062 0.994035i \(-0.534785\pi\)
−0.109062 + 0.994035i \(0.534785\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 9.16228 1.16361
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.32456 −0.784465
\(66\) 0 0
\(67\) 4.32456 0.528329 0.264164 0.964478i \(-0.414904\pi\)
0.264164 + 0.964478i \(0.414904\pi\)
\(68\) 7.16228 0.868554
\(69\) 0 0
\(70\) 3.16228 0.377964
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −5.48683 −0.642185 −0.321093 0.947048i \(-0.604050\pi\)
−0.321093 + 0.947048i \(0.604050\pi\)
\(74\) 8.32456 0.967710
\(75\) 0 0
\(76\) 5.16228 0.592154
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −4.32456 −0.486550 −0.243275 0.969957i \(-0.578222\pi\)
−0.243275 + 0.969957i \(0.578222\pi\)
\(80\) −3.16228 −0.353553
\(81\) 0 0
\(82\) −7.16228 −0.790941
\(83\) −5.16228 −0.566634 −0.283317 0.959026i \(-0.591435\pi\)
−0.283317 + 0.959026i \(0.591435\pi\)
\(84\) 0 0
\(85\) −22.6491 −2.45664
\(86\) −6.32456 −0.681994
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 16.3246 1.73040 0.865200 0.501427i \(-0.167192\pi\)
0.865200 + 0.501427i \(0.167192\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −6.32456 −0.659380
\(93\) 0 0
\(94\) 11.4868 1.18478
\(95\) −16.3246 −1.67486
\(96\) 0 0
\(97\) −10.6491 −1.08125 −0.540627 0.841263i \(-0.681813\pi\)
−0.540627 + 0.841263i \(0.681813\pi\)
\(98\) 1.00000 0.101015
\(99\) 0 0
\(100\) 5.00000 0.500000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 13.8114 1.36088 0.680438 0.732805i \(-0.261789\pi\)
0.680438 + 0.732805i \(0.261789\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 4.32456 0.420038
\(107\) −14.3246 −1.38481 −0.692404 0.721510i \(-0.743448\pi\)
−0.692404 + 0.721510i \(0.743448\pi\)
\(108\) 0 0
\(109\) −8.00000 −0.766261 −0.383131 0.923694i \(-0.625154\pi\)
−0.383131 + 0.923694i \(0.625154\pi\)
\(110\) 3.16228 0.301511
\(111\) 0 0
\(112\) −1.00000 −0.0944911
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) 20.0000 1.86501
\(116\) 4.00000 0.371391
\(117\) 0 0
\(118\) −1.67544 −0.154237
\(119\) −7.16228 −0.656565
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 10.0000 0.905357
\(123\) 0 0
\(124\) 9.16228 0.822797
\(125\) 0 0
\(126\) 0 0
\(127\) 20.9737 1.86111 0.930556 0.366150i \(-0.119324\pi\)
0.930556 + 0.366150i \(0.119324\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −6.32456 −0.554700
\(131\) 5.16228 0.451030 0.225515 0.974240i \(-0.427593\pi\)
0.225515 + 0.974240i \(0.427593\pi\)
\(132\) 0 0
\(133\) −5.16228 −0.447626
\(134\) 4.32456 0.373585
\(135\) 0 0
\(136\) 7.16228 0.614160
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) −7.48683 −0.635025 −0.317512 0.948254i \(-0.602848\pi\)
−0.317512 + 0.948254i \(0.602848\pi\)
\(140\) 3.16228 0.267261
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −12.6491 −1.05045
\(146\) −5.48683 −0.454094
\(147\) 0 0
\(148\) 8.32456 0.684274
\(149\) 2.00000 0.163846 0.0819232 0.996639i \(-0.473894\pi\)
0.0819232 + 0.996639i \(0.473894\pi\)
\(150\) 0 0
\(151\) 20.6491 1.68040 0.840200 0.542276i \(-0.182437\pi\)
0.840200 + 0.542276i \(0.182437\pi\)
\(152\) 5.16228 0.418716
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) −28.9737 −2.32722
\(156\) 0 0
\(157\) −5.48683 −0.437897 −0.218948 0.975736i \(-0.570263\pi\)
−0.218948 + 0.975736i \(0.570263\pi\)
\(158\) −4.32456 −0.344043
\(159\) 0 0
\(160\) −3.16228 −0.250000
\(161\) 6.32456 0.498445
\(162\) 0 0
\(163\) −16.6491 −1.30406 −0.652029 0.758194i \(-0.726082\pi\)
−0.652029 + 0.758194i \(0.726082\pi\)
\(164\) −7.16228 −0.559280
\(165\) 0 0
\(166\) −5.16228 −0.400670
\(167\) −6.32456 −0.489409 −0.244704 0.969598i \(-0.578691\pi\)
−0.244704 + 0.969598i \(0.578691\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) −22.6491 −1.73711
\(171\) 0 0
\(172\) −6.32456 −0.482243
\(173\) −12.3246 −0.937019 −0.468509 0.883459i \(-0.655209\pi\)
−0.468509 + 0.883459i \(0.655209\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 16.3246 1.22358
\(179\) −3.67544 −0.274716 −0.137358 0.990521i \(-0.543861\pi\)
−0.137358 + 0.990521i \(0.543861\pi\)
\(180\) 0 0
\(181\) 17.4868 1.29979 0.649893 0.760026i \(-0.274814\pi\)
0.649893 + 0.760026i \(0.274814\pi\)
\(182\) −2.00000 −0.148250
\(183\) 0 0
\(184\) −6.32456 −0.466252
\(185\) −26.3246 −1.93542
\(186\) 0 0
\(187\) −7.16228 −0.523758
\(188\) 11.4868 0.837763
\(189\) 0 0
\(190\) −16.3246 −1.18431
\(191\) −5.67544 −0.410661 −0.205330 0.978693i \(-0.565827\pi\)
−0.205330 + 0.978693i \(0.565827\pi\)
\(192\) 0 0
\(193\) 10.6491 0.766540 0.383270 0.923636i \(-0.374798\pi\)
0.383270 + 0.923636i \(0.374798\pi\)
\(194\) −10.6491 −0.764562
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 24.0000 1.70993 0.854965 0.518686i \(-0.173579\pi\)
0.854965 + 0.518686i \(0.173579\pi\)
\(198\) 0 0
\(199\) 19.4868 1.38138 0.690692 0.723149i \(-0.257306\pi\)
0.690692 + 0.723149i \(0.257306\pi\)
\(200\) 5.00000 0.353553
\(201\) 0 0
\(202\) −6.00000 −0.422159
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 22.6491 1.58188
\(206\) 13.8114 0.962285
\(207\) 0 0
\(208\) 2.00000 0.138675
\(209\) −5.16228 −0.357082
\(210\) 0 0
\(211\) −18.3246 −1.26151 −0.630757 0.775980i \(-0.717256\pi\)
−0.630757 + 0.775980i \(0.717256\pi\)
\(212\) 4.32456 0.297012
\(213\) 0 0
\(214\) −14.3246 −0.979206
\(215\) 20.0000 1.36399
\(216\) 0 0
\(217\) −9.16228 −0.621976
\(218\) −8.00000 −0.541828
\(219\) 0 0
\(220\) 3.16228 0.213201
\(221\) 14.3246 0.963574
\(222\) 0 0
\(223\) −11.4868 −0.769215 −0.384608 0.923080i \(-0.625663\pi\)
−0.384608 + 0.923080i \(0.625663\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 2.00000 0.133038
\(227\) 7.48683 0.496919 0.248459 0.968642i \(-0.420076\pi\)
0.248459 + 0.968642i \(0.420076\pi\)
\(228\) 0 0
\(229\) −25.4868 −1.68422 −0.842109 0.539308i \(-0.818686\pi\)
−0.842109 + 0.539308i \(0.818686\pi\)
\(230\) 20.0000 1.31876
\(231\) 0 0
\(232\) 4.00000 0.262613
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −36.3246 −2.36955
\(236\) −1.67544 −0.109062
\(237\) 0 0
\(238\) −7.16228 −0.464262
\(239\) 12.6491 0.818203 0.409101 0.912489i \(-0.365842\pi\)
0.409101 + 0.912489i \(0.365842\pi\)
\(240\) 0 0
\(241\) 0.837722 0.0539624 0.0269812 0.999636i \(-0.491411\pi\)
0.0269812 + 0.999636i \(0.491411\pi\)
\(242\) 1.00000 0.0642824
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) −3.16228 −0.202031
\(246\) 0 0
\(247\) 10.3246 0.656936
\(248\) 9.16228 0.581805
\(249\) 0 0
\(250\) 0 0
\(251\) −13.6754 −0.863186 −0.431593 0.902068i \(-0.642048\pi\)
−0.431593 + 0.902068i \(0.642048\pi\)
\(252\) 0 0
\(253\) 6.32456 0.397621
\(254\) 20.9737 1.31600
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.67544 0.478781 0.239391 0.970923i \(-0.423052\pi\)
0.239391 + 0.970923i \(0.423052\pi\)
\(258\) 0 0
\(259\) −8.32456 −0.517263
\(260\) −6.32456 −0.392232
\(261\) 0 0
\(262\) 5.16228 0.318927
\(263\) 24.9737 1.53994 0.769971 0.638079i \(-0.220271\pi\)
0.769971 + 0.638079i \(0.220271\pi\)
\(264\) 0 0
\(265\) −13.6754 −0.840076
\(266\) −5.16228 −0.316520
\(267\) 0 0
\(268\) 4.32456 0.264164
\(269\) 11.1623 0.680576 0.340288 0.940321i \(-0.389475\pi\)
0.340288 + 0.940321i \(0.389475\pi\)
\(270\) 0 0
\(271\) −6.32456 −0.384189 −0.192095 0.981376i \(-0.561528\pi\)
−0.192095 + 0.981376i \(0.561528\pi\)
\(272\) 7.16228 0.434277
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −24.0000 −1.44202 −0.721010 0.692925i \(-0.756322\pi\)
−0.721010 + 0.692925i \(0.756322\pi\)
\(278\) −7.48683 −0.449030
\(279\) 0 0
\(280\) 3.16228 0.188982
\(281\) 22.6491 1.35113 0.675566 0.737299i \(-0.263899\pi\)
0.675566 + 0.737299i \(0.263899\pi\)
\(282\) 0 0
\(283\) −15.4868 −0.920597 −0.460298 0.887764i \(-0.652258\pi\)
−0.460298 + 0.887764i \(0.652258\pi\)
\(284\) −8.00000 −0.474713
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 7.16228 0.422776
\(288\) 0 0
\(289\) 34.2982 2.01754
\(290\) −12.6491 −0.742781
\(291\) 0 0
\(292\) −5.48683 −0.321093
\(293\) 24.9737 1.45898 0.729489 0.683993i \(-0.239758\pi\)
0.729489 + 0.683993i \(0.239758\pi\)
\(294\) 0 0
\(295\) 5.29822 0.308474
\(296\) 8.32456 0.483855
\(297\) 0 0
\(298\) 2.00000 0.115857
\(299\) −12.6491 −0.731517
\(300\) 0 0
\(301\) 6.32456 0.364541
\(302\) 20.6491 1.18822
\(303\) 0 0
\(304\) 5.16228 0.296077
\(305\) −31.6228 −1.81071
\(306\) 0 0
\(307\) −21.1623 −1.20779 −0.603897 0.797062i \(-0.706386\pi\)
−0.603897 + 0.797062i \(0.706386\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −28.9737 −1.64559
\(311\) −29.8114 −1.69045 −0.845224 0.534412i \(-0.820533\pi\)
−0.845224 + 0.534412i \(0.820533\pi\)
\(312\) 0 0
\(313\) 20.3246 1.14881 0.574406 0.818571i \(-0.305233\pi\)
0.574406 + 0.818571i \(0.305233\pi\)
\(314\) −5.48683 −0.309640
\(315\) 0 0
\(316\) −4.32456 −0.243275
\(317\) 12.9737 0.728674 0.364337 0.931267i \(-0.381296\pi\)
0.364337 + 0.931267i \(0.381296\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) −3.16228 −0.176777
\(321\) 0 0
\(322\) 6.32456 0.352454
\(323\) 36.9737 2.05727
\(324\) 0 0
\(325\) 10.0000 0.554700
\(326\) −16.6491 −0.922109
\(327\) 0 0
\(328\) −7.16228 −0.395471
\(329\) −11.4868 −0.633290
\(330\) 0 0
\(331\) −8.97367 −0.493237 −0.246619 0.969113i \(-0.579320\pi\)
−0.246619 + 0.969113i \(0.579320\pi\)
\(332\) −5.16228 −0.283317
\(333\) 0 0
\(334\) −6.32456 −0.346064
\(335\) −13.6754 −0.747169
\(336\) 0 0
\(337\) −19.6754 −1.07179 −0.535895 0.844285i \(-0.680026\pi\)
−0.535895 + 0.844285i \(0.680026\pi\)
\(338\) −9.00000 −0.489535
\(339\) 0 0
\(340\) −22.6491 −1.22832
\(341\) −9.16228 −0.496165
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −6.32456 −0.340997
\(345\) 0 0
\(346\) −12.3246 −0.662572
\(347\) −14.3246 −0.768982 −0.384491 0.923129i \(-0.625623\pi\)
−0.384491 + 0.923129i \(0.625623\pi\)
\(348\) 0 0
\(349\) 8.97367 0.480349 0.240175 0.970730i \(-0.422795\pi\)
0.240175 + 0.970730i \(0.422795\pi\)
\(350\) −5.00000 −0.267261
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 7.67544 0.408523 0.204261 0.978916i \(-0.434521\pi\)
0.204261 + 0.978916i \(0.434521\pi\)
\(354\) 0 0
\(355\) 25.2982 1.34269
\(356\) 16.3246 0.865200
\(357\) 0 0
\(358\) −3.67544 −0.194253
\(359\) 3.67544 0.193983 0.0969913 0.995285i \(-0.469078\pi\)
0.0969913 + 0.995285i \(0.469078\pi\)
\(360\) 0 0
\(361\) 7.64911 0.402585
\(362\) 17.4868 0.919088
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 17.3509 0.908187
\(366\) 0 0
\(367\) −1.16228 −0.0606704 −0.0303352 0.999540i \(-0.509657\pi\)
−0.0303352 + 0.999540i \(0.509657\pi\)
\(368\) −6.32456 −0.329690
\(369\) 0 0
\(370\) −26.3246 −1.36855
\(371\) −4.32456 −0.224520
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) −7.16228 −0.370353
\(375\) 0 0
\(376\) 11.4868 0.592388
\(377\) 8.00000 0.412021
\(378\) 0 0
\(379\) −13.2982 −0.683084 −0.341542 0.939867i \(-0.610949\pi\)
−0.341542 + 0.939867i \(0.610949\pi\)
\(380\) −16.3246 −0.837432
\(381\) 0 0
\(382\) −5.67544 −0.290381
\(383\) −9.16228 −0.468171 −0.234085 0.972216i \(-0.575209\pi\)
−0.234085 + 0.972216i \(0.575209\pi\)
\(384\) 0 0
\(385\) −3.16228 −0.161165
\(386\) 10.6491 0.542025
\(387\) 0 0
\(388\) −10.6491 −0.540627
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −45.2982 −2.29083
\(392\) 1.00000 0.0505076
\(393\) 0 0
\(394\) 24.0000 1.20910
\(395\) 13.6754 0.688086
\(396\) 0 0
\(397\) 21.4868 1.07839 0.539197 0.842180i \(-0.318728\pi\)
0.539197 + 0.842180i \(0.318728\pi\)
\(398\) 19.4868 0.976787
\(399\) 0 0
\(400\) 5.00000 0.250000
\(401\) 23.2982 1.16346 0.581729 0.813383i \(-0.302376\pi\)
0.581729 + 0.813383i \(0.302376\pi\)
\(402\) 0 0
\(403\) 18.3246 0.912811
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) −8.32456 −0.412633
\(408\) 0 0
\(409\) 27.1623 1.34309 0.671544 0.740965i \(-0.265631\pi\)
0.671544 + 0.740965i \(0.265631\pi\)
\(410\) 22.6491 1.11856
\(411\) 0 0
\(412\) 13.8114 0.680438
\(413\) 1.67544 0.0824432
\(414\) 0 0
\(415\) 16.3246 0.801341
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −5.16228 −0.252495
\(419\) −16.6491 −0.813362 −0.406681 0.913570i \(-0.633314\pi\)
−0.406681 + 0.913570i \(0.633314\pi\)
\(420\) 0 0
\(421\) −39.9473 −1.94691 −0.973457 0.228870i \(-0.926497\pi\)
−0.973457 + 0.228870i \(0.926497\pi\)
\(422\) −18.3246 −0.892025
\(423\) 0 0
\(424\) 4.32456 0.210019
\(425\) 35.8114 1.73711
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −14.3246 −0.692404
\(429\) 0 0
\(430\) 20.0000 0.964486
\(431\) −8.97367 −0.432246 −0.216123 0.976366i \(-0.569341\pi\)
−0.216123 + 0.976366i \(0.569341\pi\)
\(432\) 0 0
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) −9.16228 −0.439803
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −32.6491 −1.56182
\(438\) 0 0
\(439\) −6.32456 −0.301855 −0.150927 0.988545i \(-0.548226\pi\)
−0.150927 + 0.988545i \(0.548226\pi\)
\(440\) 3.16228 0.150756
\(441\) 0 0
\(442\) 14.3246 0.681350
\(443\) 4.32456 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(444\) 0 0
\(445\) −51.6228 −2.44715
\(446\) −11.4868 −0.543917
\(447\) 0 0
\(448\) −1.00000 −0.0472456
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 7.16228 0.337258
\(452\) 2.00000 0.0940721
\(453\) 0 0
\(454\) 7.48683 0.351374
\(455\) 6.32456 0.296500
\(456\) 0 0
\(457\) 17.3509 0.811640 0.405820 0.913953i \(-0.366986\pi\)
0.405820 + 0.913953i \(0.366986\pi\)
\(458\) −25.4868 −1.19092
\(459\) 0 0
\(460\) 20.0000 0.932505
\(461\) −23.2982 −1.08511 −0.542553 0.840021i \(-0.682542\pi\)
−0.542553 + 0.840021i \(0.682542\pi\)
\(462\) 0 0
\(463\) −28.6491 −1.33144 −0.665719 0.746203i \(-0.731875\pi\)
−0.665719 + 0.746203i \(0.731875\pi\)
\(464\) 4.00000 0.185695
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −17.2982 −0.800466 −0.400233 0.916413i \(-0.631071\pi\)
−0.400233 + 0.916413i \(0.631071\pi\)
\(468\) 0 0
\(469\) −4.32456 −0.199689
\(470\) −36.3246 −1.67553
\(471\) 0 0
\(472\) −1.67544 −0.0771186
\(473\) 6.32456 0.290803
\(474\) 0 0
\(475\) 25.8114 1.18431
\(476\) −7.16228 −0.328282
\(477\) 0 0
\(478\) 12.6491 0.578557
\(479\) −21.2982 −0.973141 −0.486570 0.873641i \(-0.661752\pi\)
−0.486570 + 0.873641i \(0.661752\pi\)
\(480\) 0 0
\(481\) 16.6491 0.759134
\(482\) 0.837722 0.0381572
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 33.6754 1.52912
\(486\) 0 0
\(487\) −14.3246 −0.649108 −0.324554 0.945867i \(-0.605214\pi\)
−0.324554 + 0.945867i \(0.605214\pi\)
\(488\) 10.0000 0.452679
\(489\) 0 0
\(490\) −3.16228 −0.142857
\(491\) 8.64911 0.390329 0.195164 0.980771i \(-0.437476\pi\)
0.195164 + 0.980771i \(0.437476\pi\)
\(492\) 0 0
\(493\) 28.6491 1.29029
\(494\) 10.3246 0.464524
\(495\) 0 0
\(496\) 9.16228 0.411398
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −16.9737 −0.759846 −0.379923 0.925018i \(-0.624049\pi\)
−0.379923 + 0.925018i \(0.624049\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −13.6754 −0.610365
\(503\) −5.02633 −0.224113 −0.112057 0.993702i \(-0.535744\pi\)
−0.112057 + 0.993702i \(0.535744\pi\)
\(504\) 0 0
\(505\) 18.9737 0.844317
\(506\) 6.32456 0.281161
\(507\) 0 0
\(508\) 20.9737 0.930556
\(509\) 39.8114 1.76461 0.882304 0.470679i \(-0.155991\pi\)
0.882304 + 0.470679i \(0.155991\pi\)
\(510\) 0 0
\(511\) 5.48683 0.242723
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 7.67544 0.338549
\(515\) −43.6754 −1.92457
\(516\) 0 0
\(517\) −11.4868 −0.505190
\(518\) −8.32456 −0.365760
\(519\) 0 0
\(520\) −6.32456 −0.277350
\(521\) 20.3246 0.890435 0.445217 0.895422i \(-0.353126\pi\)
0.445217 + 0.895422i \(0.353126\pi\)
\(522\) 0 0
\(523\) −4.13594 −0.180852 −0.0904261 0.995903i \(-0.528823\pi\)
−0.0904261 + 0.995903i \(0.528823\pi\)
\(524\) 5.16228 0.225515
\(525\) 0 0
\(526\) 24.9737 1.08890
\(527\) 65.6228 2.85857
\(528\) 0 0
\(529\) 17.0000 0.739130
\(530\) −13.6754 −0.594023
\(531\) 0 0
\(532\) −5.16228 −0.223813
\(533\) −14.3246 −0.620465
\(534\) 0 0
\(535\) 45.2982 1.95841
\(536\) 4.32456 0.186792
\(537\) 0 0
\(538\) 11.1623 0.481240
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) 31.2982 1.34562 0.672808 0.739817i \(-0.265088\pi\)
0.672808 + 0.739817i \(0.265088\pi\)
\(542\) −6.32456 −0.271663
\(543\) 0 0
\(544\) 7.16228 0.307080
\(545\) 25.2982 1.08366
\(546\) 0 0
\(547\) −41.2982 −1.76578 −0.882892 0.469576i \(-0.844407\pi\)
−0.882892 + 0.469576i \(0.844407\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) −5.00000 −0.213201
\(551\) 20.6491 0.879682
\(552\) 0 0
\(553\) 4.32456 0.183899
\(554\) −24.0000 −1.01966
\(555\) 0 0
\(556\) −7.48683 −0.317512
\(557\) −33.2982 −1.41089 −0.705445 0.708764i \(-0.749253\pi\)
−0.705445 + 0.708764i \(0.749253\pi\)
\(558\) 0 0
\(559\) −12.6491 −0.535000
\(560\) 3.16228 0.133631
\(561\) 0 0
\(562\) 22.6491 0.955395
\(563\) 20.1359 0.848629 0.424314 0.905515i \(-0.360515\pi\)
0.424314 + 0.905515i \(0.360515\pi\)
\(564\) 0 0
\(565\) −6.32456 −0.266076
\(566\) −15.4868 −0.650960
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) 26.6491 1.11719 0.558594 0.829441i \(-0.311341\pi\)
0.558594 + 0.829441i \(0.311341\pi\)
\(570\) 0 0
\(571\) 12.6491 0.529349 0.264674 0.964338i \(-0.414736\pi\)
0.264674 + 0.964338i \(0.414736\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 7.16228 0.298948
\(575\) −31.6228 −1.31876
\(576\) 0 0
\(577\) 15.6754 0.652577 0.326289 0.945270i \(-0.394202\pi\)
0.326289 + 0.945270i \(0.394202\pi\)
\(578\) 34.2982 1.42662
\(579\) 0 0
\(580\) −12.6491 −0.525226
\(581\) 5.16228 0.214167
\(582\) 0 0
\(583\) −4.32456 −0.179105
\(584\) −5.48683 −0.227047
\(585\) 0 0
\(586\) 24.9737 1.03165
\(587\) 6.97367 0.287834 0.143917 0.989590i \(-0.454030\pi\)
0.143917 + 0.989590i \(0.454030\pi\)
\(588\) 0 0
\(589\) 47.2982 1.94889
\(590\) 5.29822 0.218124
\(591\) 0 0
\(592\) 8.32456 0.342137
\(593\) −16.4605 −0.675952 −0.337976 0.941155i \(-0.609742\pi\)
−0.337976 + 0.941155i \(0.609742\pi\)
\(594\) 0 0
\(595\) 22.6491 0.928523
\(596\) 2.00000 0.0819232
\(597\) 0 0
\(598\) −12.6491 −0.517261
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) 19.1623 0.781646 0.390823 0.920466i \(-0.372191\pi\)
0.390823 + 0.920466i \(0.372191\pi\)
\(602\) 6.32456 0.257770
\(603\) 0 0
\(604\) 20.6491 0.840200
\(605\) −3.16228 −0.128565
\(606\) 0 0
\(607\) 4.64911 0.188702 0.0943508 0.995539i \(-0.469922\pi\)
0.0943508 + 0.995539i \(0.469922\pi\)
\(608\) 5.16228 0.209358
\(609\) 0 0
\(610\) −31.6228 −1.28037
\(611\) 22.9737 0.929415
\(612\) 0 0
\(613\) −31.2982 −1.26412 −0.632062 0.774918i \(-0.717791\pi\)
−0.632062 + 0.774918i \(0.717791\pi\)
\(614\) −21.1623 −0.854040
\(615\) 0 0
\(616\) 1.00000 0.0402911
\(617\) 20.6491 0.831302 0.415651 0.909524i \(-0.363554\pi\)
0.415651 + 0.909524i \(0.363554\pi\)
\(618\) 0 0
\(619\) 23.3509 0.938551 0.469276 0.883052i \(-0.344515\pi\)
0.469276 + 0.883052i \(0.344515\pi\)
\(620\) −28.9737 −1.16361
\(621\) 0 0
\(622\) −29.8114 −1.19533
\(623\) −16.3246 −0.654029
\(624\) 0 0
\(625\) −25.0000 −1.00000
\(626\) 20.3246 0.812333
\(627\) 0 0
\(628\) −5.48683 −0.218948
\(629\) 59.6228 2.37732
\(630\) 0 0
\(631\) −27.6228 −1.09965 −0.549823 0.835281i \(-0.685305\pi\)
−0.549823 + 0.835281i \(0.685305\pi\)
\(632\) −4.32456 −0.172022
\(633\) 0 0
\(634\) 12.9737 0.515250
\(635\) −66.3246 −2.63201
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) −4.00000 −0.158362
\(639\) 0 0
\(640\) −3.16228 −0.125000
\(641\) 37.9473 1.49883 0.749415 0.662101i \(-0.230335\pi\)
0.749415 + 0.662101i \(0.230335\pi\)
\(642\) 0 0
\(643\) 34.9737 1.37923 0.689613 0.724178i \(-0.257780\pi\)
0.689613 + 0.724178i \(0.257780\pi\)
\(644\) 6.32456 0.249222
\(645\) 0 0
\(646\) 36.9737 1.45471
\(647\) 3.48683 0.137082 0.0685408 0.997648i \(-0.478166\pi\)
0.0685408 + 0.997648i \(0.478166\pi\)
\(648\) 0 0
\(649\) 1.67544 0.0657670
\(650\) 10.0000 0.392232
\(651\) 0 0
\(652\) −16.6491 −0.652029
\(653\) −43.2982 −1.69439 −0.847195 0.531282i \(-0.821711\pi\)
−0.847195 + 0.531282i \(0.821711\pi\)
\(654\) 0 0
\(655\) −16.3246 −0.637853
\(656\) −7.16228 −0.279640
\(657\) 0 0
\(658\) −11.4868 −0.447803
\(659\) 22.9737 0.894927 0.447463 0.894302i \(-0.352327\pi\)
0.447463 + 0.894302i \(0.352327\pi\)
\(660\) 0 0
\(661\) −22.1359 −0.860988 −0.430494 0.902593i \(-0.641661\pi\)
−0.430494 + 0.902593i \(0.641661\pi\)
\(662\) −8.97367 −0.348771
\(663\) 0 0
\(664\) −5.16228 −0.200335
\(665\) 16.3246 0.633039
\(666\) 0 0
\(667\) −25.2982 −0.979551
\(668\) −6.32456 −0.244704
\(669\) 0 0
\(670\) −13.6754 −0.528329
\(671\) −10.0000 −0.386046
\(672\) 0 0
\(673\) 13.6228 0.525119 0.262560 0.964916i \(-0.415433\pi\)
0.262560 + 0.964916i \(0.415433\pi\)
\(674\) −19.6754 −0.757870
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) −8.32456 −0.319939 −0.159969 0.987122i \(-0.551140\pi\)
−0.159969 + 0.987122i \(0.551140\pi\)
\(678\) 0 0
\(679\) 10.6491 0.408675
\(680\) −22.6491 −0.868554
\(681\) 0 0
\(682\) −9.16228 −0.350842
\(683\) −24.9737 −0.955591 −0.477795 0.878471i \(-0.658564\pi\)
−0.477795 + 0.878471i \(0.658564\pi\)
\(684\) 0 0
\(685\) 56.9210 2.17484
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −6.32456 −0.241121
\(689\) 8.64911 0.329505
\(690\) 0 0
\(691\) −6.97367 −0.265291 −0.132645 0.991164i \(-0.542347\pi\)
−0.132645 + 0.991164i \(0.542347\pi\)
\(692\) −12.3246 −0.468509
\(693\) 0 0
\(694\) −14.3246 −0.543753
\(695\) 23.6754 0.898061
\(696\) 0 0
\(697\) −51.2982 −1.94306
\(698\) 8.97367 0.339658
\(699\) 0 0
\(700\) −5.00000 −0.188982
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 42.9737 1.62078
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 7.67544 0.288869
\(707\) 6.00000 0.225653
\(708\) 0 0
\(709\) 40.3246 1.51442 0.757210 0.653171i \(-0.226562\pi\)
0.757210 + 0.653171i \(0.226562\pi\)
\(710\) 25.2982 0.949425
\(711\) 0 0
\(712\) 16.3246 0.611789
\(713\) −57.9473 −2.17014
\(714\) 0 0
\(715\) 6.32456 0.236525
\(716\) −3.67544 −0.137358
\(717\) 0 0
\(718\) 3.67544 0.137166
\(719\) 22.8377 0.851703 0.425852 0.904793i \(-0.359975\pi\)
0.425852 + 0.904793i \(0.359975\pi\)
\(720\) 0 0
\(721\) −13.8114 −0.514363
\(722\) 7.64911 0.284670
\(723\) 0 0
\(724\) 17.4868 0.649893
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) −40.1359 −1.48856 −0.744280 0.667868i \(-0.767207\pi\)
−0.744280 + 0.667868i \(0.767207\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 0 0
\(730\) 17.3509 0.642185
\(731\) −45.2982 −1.67542
\(732\) 0 0
\(733\) 5.35089 0.197640 0.0988198 0.995105i \(-0.468493\pi\)
0.0988198 + 0.995105i \(0.468493\pi\)
\(734\) −1.16228 −0.0429005
\(735\) 0 0
\(736\) −6.32456 −0.233126
\(737\) −4.32456 −0.159297
\(738\) 0 0
\(739\) 3.35089 0.123264 0.0616322 0.998099i \(-0.480369\pi\)
0.0616322 + 0.998099i \(0.480369\pi\)
\(740\) −26.3246 −0.967710
\(741\) 0 0
\(742\) −4.32456 −0.158759
\(743\) 3.35089 0.122932 0.0614661 0.998109i \(-0.480422\pi\)
0.0614661 + 0.998109i \(0.480422\pi\)
\(744\) 0 0
\(745\) −6.32456 −0.231714
\(746\) −10.0000 −0.366126
\(747\) 0 0
\(748\) −7.16228 −0.261879
\(749\) 14.3246 0.523408
\(750\) 0 0
\(751\) 5.67544 0.207100 0.103550 0.994624i \(-0.466980\pi\)
0.103550 + 0.994624i \(0.466980\pi\)
\(752\) 11.4868 0.418882
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) −65.2982 −2.37645
\(756\) 0 0
\(757\) 29.6228 1.07666 0.538329 0.842735i \(-0.319056\pi\)
0.538329 + 0.842735i \(0.319056\pi\)
\(758\) −13.2982 −0.483013
\(759\) 0 0
\(760\) −16.3246 −0.592154
\(761\) −9.48683 −0.343897 −0.171949 0.985106i \(-0.555006\pi\)
−0.171949 + 0.985106i \(0.555006\pi\)
\(762\) 0 0
\(763\) 8.00000 0.289619
\(764\) −5.67544 −0.205330
\(765\) 0 0
\(766\) −9.16228 −0.331047
\(767\) −3.35089 −0.120994
\(768\) 0 0
\(769\) 25.4868 0.919079 0.459539 0.888157i \(-0.348015\pi\)
0.459539 + 0.888157i \(0.348015\pi\)
\(770\) −3.16228 −0.113961
\(771\) 0 0
\(772\) 10.6491 0.383270
\(773\) −5.86406 −0.210915 −0.105458 0.994424i \(-0.533631\pi\)
−0.105458 + 0.994424i \(0.533631\pi\)
\(774\) 0 0
\(775\) 45.8114 1.64559
\(776\) −10.6491 −0.382281
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) −36.9737 −1.32472
\(780\) 0 0
\(781\) 8.00000 0.286263
\(782\) −45.2982 −1.61986
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 17.3509 0.619280
\(786\) 0 0
\(787\) 6.46050 0.230292 0.115146 0.993349i \(-0.463266\pi\)
0.115146 + 0.993349i \(0.463266\pi\)
\(788\) 24.0000 0.854965
\(789\) 0 0
\(790\) 13.6754 0.486550
\(791\) −2.00000 −0.0711118
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 21.4868 0.762539
\(795\) 0 0
\(796\) 19.4868 0.690692
\(797\) 8.83772 0.313048 0.156524 0.987674i \(-0.449971\pi\)
0.156524 + 0.987674i \(0.449971\pi\)
\(798\) 0 0
\(799\) 82.2719 2.91057
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 23.2982 0.822689
\(803\) 5.48683 0.193626
\(804\) 0 0
\(805\) −20.0000 −0.704907
\(806\) 18.3246 0.645455
\(807\) 0 0
\(808\) −6.00000 −0.211079
\(809\) 43.6754 1.53555 0.767773 0.640721i \(-0.221365\pi\)
0.767773 + 0.640721i \(0.221365\pi\)
\(810\) 0 0
\(811\) −7.48683 −0.262898 −0.131449 0.991323i \(-0.541963\pi\)
−0.131449 + 0.991323i \(0.541963\pi\)
\(812\) −4.00000 −0.140372
\(813\) 0 0
\(814\) −8.32456 −0.291776
\(815\) 52.6491 1.84422
\(816\) 0 0
\(817\) −32.6491 −1.14225
\(818\) 27.1623 0.949707
\(819\) 0 0
\(820\) 22.6491 0.790941
\(821\) −52.5964 −1.83563 −0.917814 0.397010i \(-0.870048\pi\)
−0.917814 + 0.397010i \(0.870048\pi\)
\(822\) 0 0
\(823\) −9.02633 −0.314638 −0.157319 0.987548i \(-0.550285\pi\)
−0.157319 + 0.987548i \(0.550285\pi\)
\(824\) 13.8114 0.481143
\(825\) 0 0
\(826\) 1.67544 0.0582962
\(827\) 33.9473 1.18046 0.590232 0.807234i \(-0.299036\pi\)
0.590232 + 0.807234i \(0.299036\pi\)
\(828\) 0 0
\(829\) −12.4605 −0.432771 −0.216386 0.976308i \(-0.569427\pi\)
−0.216386 + 0.976308i \(0.569427\pi\)
\(830\) 16.3246 0.566634
\(831\) 0 0
\(832\) 2.00000 0.0693375
\(833\) 7.16228 0.248158
\(834\) 0 0
\(835\) 20.0000 0.692129
\(836\) −5.16228 −0.178541
\(837\) 0 0
\(838\) −16.6491 −0.575134
\(839\) −34.4605 −1.18971 −0.594854 0.803834i \(-0.702790\pi\)
−0.594854 + 0.803834i \(0.702790\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −39.9473 −1.37668
\(843\) 0 0
\(844\) −18.3246 −0.630757
\(845\) 28.4605 0.979071
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 4.32456 0.148506
\(849\) 0 0
\(850\) 35.8114 1.22832
\(851\) −52.6491 −1.80479
\(852\) 0 0
\(853\) −6.64911 −0.227661 −0.113831 0.993500i \(-0.536312\pi\)
−0.113831 + 0.993500i \(0.536312\pi\)
\(854\) −10.0000 −0.342193
\(855\) 0 0
\(856\) −14.3246 −0.489603
\(857\) 24.1886 0.826267 0.413134 0.910670i \(-0.364434\pi\)
0.413134 + 0.910670i \(0.364434\pi\)
\(858\) 0 0
\(859\) −30.3246 −1.03466 −0.517330 0.855786i \(-0.673074\pi\)
−0.517330 + 0.855786i \(0.673074\pi\)
\(860\) 20.0000 0.681994
\(861\) 0 0
\(862\) −8.97367 −0.305644
\(863\) 17.6754 0.601679 0.300840 0.953675i \(-0.402733\pi\)
0.300840 + 0.953675i \(0.402733\pi\)
\(864\) 0 0
\(865\) 38.9737 1.32514
\(866\) −34.0000 −1.15537
\(867\) 0 0
\(868\) −9.16228 −0.310988
\(869\) 4.32456 0.146700
\(870\) 0 0
\(871\) 8.64911 0.293064
\(872\) −8.00000 −0.270914
\(873\) 0 0
\(874\) −32.6491 −1.10437
\(875\) 0 0
\(876\) 0 0
\(877\) −23.3509 −0.788504 −0.394252 0.919002i \(-0.628996\pi\)
−0.394252 + 0.919002i \(0.628996\pi\)
\(878\) −6.32456 −0.213443
\(879\) 0 0
\(880\) 3.16228 0.106600
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) −13.2982 −0.447521 −0.223760 0.974644i \(-0.571833\pi\)
−0.223760 + 0.974644i \(0.571833\pi\)
\(884\) 14.3246 0.481787
\(885\) 0 0
\(886\) 4.32456 0.145286
\(887\) 16.0000 0.537227 0.268614 0.963248i \(-0.413434\pi\)
0.268614 + 0.963248i \(0.413434\pi\)
\(888\) 0 0
\(889\) −20.9737 −0.703434
\(890\) −51.6228 −1.73040
\(891\) 0 0
\(892\) −11.4868 −0.384608
\(893\) 59.2982 1.98434
\(894\) 0 0
\(895\) 11.6228 0.388507
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 36.6491 1.22232
\(900\) 0 0
\(901\) 30.9737 1.03188
\(902\) 7.16228 0.238478
\(903\) 0 0
\(904\) 2.00000 0.0665190
\(905\) −55.2982 −1.83818
\(906\) 0 0
\(907\) 53.6228 1.78052 0.890258 0.455457i \(-0.150524\pi\)
0.890258 + 0.455457i \(0.150524\pi\)
\(908\) 7.48683 0.248459
\(909\) 0 0
\(910\) 6.32456 0.209657
\(911\) −22.3246 −0.739646 −0.369823 0.929102i \(-0.620582\pi\)
−0.369823 + 0.929102i \(0.620582\pi\)
\(912\) 0 0
\(913\) 5.16228 0.170846
\(914\) 17.3509 0.573916
\(915\) 0 0
\(916\) −25.4868 −0.842109
\(917\) −5.16228 −0.170473
\(918\) 0 0
\(919\) −17.2982 −0.570616 −0.285308 0.958436i \(-0.592096\pi\)
−0.285308 + 0.958436i \(0.592096\pi\)
\(920\) 20.0000 0.659380
\(921\) 0 0
\(922\) −23.2982 −0.767286
\(923\) −16.0000 −0.526646
\(924\) 0 0
\(925\) 41.6228 1.36855
\(926\) −28.6491 −0.941468
\(927\) 0 0
\(928\) 4.00000 0.131306
\(929\) 33.6228 1.10313 0.551564 0.834133i \(-0.314031\pi\)
0.551564 + 0.834133i \(0.314031\pi\)
\(930\) 0 0
\(931\) 5.16228 0.169187
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −17.2982 −0.566015
\(935\) 22.6491 0.740705
\(936\) 0 0
\(937\) −42.1359 −1.37652 −0.688261 0.725464i \(-0.741625\pi\)
−0.688261 + 0.725464i \(0.741625\pi\)
\(938\) −4.32456 −0.141202
\(939\) 0 0
\(940\) −36.3246 −1.18478
\(941\) −27.6754 −0.902194 −0.451097 0.892475i \(-0.648967\pi\)
−0.451097 + 0.892475i \(0.648967\pi\)
\(942\) 0 0
\(943\) 45.2982 1.47511
\(944\) −1.67544 −0.0545311
\(945\) 0 0
\(946\) 6.32456 0.205629
\(947\) 2.70178 0.0877960 0.0438980 0.999036i \(-0.486022\pi\)
0.0438980 + 0.999036i \(0.486022\pi\)
\(948\) 0 0
\(949\) −10.9737 −0.356220
\(950\) 25.8114 0.837432
\(951\) 0 0
\(952\) −7.16228 −0.232131
\(953\) −4.97367 −0.161113 −0.0805564 0.996750i \(-0.525670\pi\)
−0.0805564 + 0.996750i \(0.525670\pi\)
\(954\) 0 0
\(955\) 17.9473 0.580762
\(956\) 12.6491 0.409101
\(957\) 0 0
\(958\) −21.2982 −0.688114
\(959\) 18.0000 0.581250
\(960\) 0 0
\(961\) 52.9473 1.70798
\(962\) 16.6491 0.536789
\(963\) 0 0
\(964\) 0.837722 0.0269812
\(965\) −33.6754 −1.08405
\(966\) 0 0
\(967\) −57.2982 −1.84259 −0.921293 0.388868i \(-0.872866\pi\)
−0.921293 + 0.388868i \(0.872866\pi\)
\(968\) 1.00000 0.0321412
\(969\) 0 0
\(970\) 33.6754 1.08125
\(971\) −7.35089 −0.235901 −0.117951 0.993019i \(-0.537632\pi\)
−0.117951 + 0.993019i \(0.537632\pi\)
\(972\) 0 0
\(973\) 7.48683 0.240017
\(974\) −14.3246 −0.458988
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −24.6491 −0.788595 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(978\) 0 0
\(979\) −16.3246 −0.521735
\(980\) −3.16228 −0.101015
\(981\) 0 0
\(982\) 8.64911 0.276004
\(983\) 48.1359 1.53530 0.767649 0.640870i \(-0.221426\pi\)
0.767649 + 0.640870i \(0.221426\pi\)
\(984\) 0 0
\(985\) −75.8947 −2.41821
\(986\) 28.6491 0.912374
\(987\) 0 0
\(988\) 10.3246 0.328468
\(989\) 40.0000 1.27193
\(990\) 0 0
\(991\) −61.2982 −1.94720 −0.973601 0.228256i \(-0.926698\pi\)
−0.973601 + 0.228256i \(0.926698\pi\)
\(992\) 9.16228 0.290903
\(993\) 0 0
\(994\) 8.00000 0.253745
\(995\) −61.6228 −1.95357
\(996\) 0 0
\(997\) 48.9737 1.55101 0.775506 0.631340i \(-0.217495\pi\)
0.775506 + 0.631340i \(0.217495\pi\)
\(998\) −16.9737 −0.537292
\(999\) 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 1386.2.a.o.1.1 yes 2
3.2 odd 2 1386.2.a.n.1.2 2
7.6 odd 2 9702.2.a.dc.1.2 2
21.20 even 2 9702.2.a.cn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1386.2.a.n.1.2 2 3.2 odd 2
1386.2.a.o.1.1 yes 2 1.1 even 1 trivial
9702.2.a.cn.1.1 2 21.20 even 2
9702.2.a.dc.1.2 2 7.6 odd 2