Properties

Label 1078.2.a.v.1.2
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1078,2,Mod(1,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, 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("1078.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.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: \(8.60787333789\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.82843 q^{3} +1.00000 q^{4} +2.82843 q^{6} +1.00000 q^{8} +5.00000 q^{9} -1.00000 q^{11} +2.82843 q^{12} +4.24264 q^{13} +1.00000 q^{16} -2.82843 q^{17} +5.00000 q^{18} -4.24264 q^{19} -1.00000 q^{22} +6.00000 q^{23} +2.82843 q^{24} -5.00000 q^{25} +4.24264 q^{26} +5.65685 q^{27} -4.00000 q^{29} -7.07107 q^{31} +1.00000 q^{32} -2.82843 q^{33} -2.82843 q^{34} +5.00000 q^{36} +2.00000 q^{37} -4.24264 q^{38} +12.0000 q^{39} +2.82843 q^{41} +10.0000 q^{43} -1.00000 q^{44} +6.00000 q^{46} -12.7279 q^{47} +2.82843 q^{48} -5.00000 q^{50} -8.00000 q^{51} +4.24264 q^{52} +2.00000 q^{53} +5.65685 q^{54} -12.0000 q^{57} -4.00000 q^{58} +11.3137 q^{59} -9.89949 q^{61} -7.07107 q^{62} +1.00000 q^{64} -2.82843 q^{66} +8.00000 q^{67} -2.82843 q^{68} +16.9706 q^{69} +16.0000 q^{71} +5.00000 q^{72} -8.48528 q^{73} +2.00000 q^{74} -14.1421 q^{75} -4.24264 q^{76} +12.0000 q^{78} -8.00000 q^{79} +1.00000 q^{81} +2.82843 q^{82} -12.7279 q^{83} +10.0000 q^{86} -11.3137 q^{87} -1.00000 q^{88} +7.07107 q^{89} +6.00000 q^{92} -20.0000 q^{93} -12.7279 q^{94} +2.82843 q^{96} -7.07107 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{8} + 10 q^{9} - 2 q^{11} + 2 q^{16} + 10 q^{18} - 2 q^{22} + 12 q^{23} - 10 q^{25} - 8 q^{29} + 2 q^{32} + 10 q^{36} + 4 q^{37} + 24 q^{39} + 20 q^{43} - 2 q^{44} + 12 q^{46} - 10 q^{50} - 16 q^{51} + 4 q^{53} - 24 q^{57} - 8 q^{58} + 2 q^{64} + 16 q^{67} + 32 q^{71} + 10 q^{72} + 4 q^{74} + 24 q^{78} - 16 q^{79} + 2 q^{81} + 20 q^{86} - 2 q^{88} + 12 q^{92} - 40 q^{93} - 10 q^{99}+O(q^{100}) \) 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
\(3\) 2.82843 1.63299 0.816497 0.577350i \(-0.195913\pi\)
0.816497 + 0.577350i \(0.195913\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.82843 1.15470
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 2.82843 0.816497
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 5.00000 1.17851
\(19\) −4.24264 −0.973329 −0.486664 0.873589i \(-0.661786\pi\)
−0.486664 + 0.873589i \(0.661786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 2.82843 0.577350
\(25\) −5.00000 −1.00000
\(26\) 4.24264 0.832050
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) −4.00000 −0.742781 −0.371391 0.928477i \(-0.621119\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) 0 0
\(31\) −7.07107 −1.27000 −0.635001 0.772512i \(-0.719000\pi\)
−0.635001 + 0.772512i \(0.719000\pi\)
\(32\) 1.00000 0.176777
\(33\) −2.82843 −0.492366
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 5.00000 0.833333
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −4.24264 −0.688247
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) 2.82843 0.441726 0.220863 0.975305i \(-0.429113\pi\)
0.220863 + 0.975305i \(0.429113\pi\)
\(42\) 0 0
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −12.7279 −1.85656 −0.928279 0.371884i \(-0.878712\pi\)
−0.928279 + 0.371884i \(0.878712\pi\)
\(48\) 2.82843 0.408248
\(49\) 0 0
\(50\) −5.00000 −0.707107
\(51\) −8.00000 −1.12022
\(52\) 4.24264 0.588348
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 5.65685 0.769800
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) −4.00000 −0.525226
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) −9.89949 −1.26750 −0.633750 0.773538i \(-0.718485\pi\)
−0.633750 + 0.773538i \(0.718485\pi\)
\(62\) −7.07107 −0.898027
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −2.82843 −0.348155
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) −2.82843 −0.342997
\(69\) 16.9706 2.04302
\(70\) 0 0
\(71\) 16.0000 1.89885 0.949425 0.313993i \(-0.101667\pi\)
0.949425 + 0.313993i \(0.101667\pi\)
\(72\) 5.00000 0.589256
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 2.00000 0.232495
\(75\) −14.1421 −1.63299
\(76\) −4.24264 −0.486664
\(77\) 0 0
\(78\) 12.0000 1.35873
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 2.82843 0.312348
\(83\) −12.7279 −1.39707 −0.698535 0.715575i \(-0.746165\pi\)
−0.698535 + 0.715575i \(0.746165\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.0000 1.07833
\(87\) −11.3137 −1.21296
\(88\) −1.00000 −0.106600
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −20.0000 −2.07390
\(94\) −12.7279 −1.31278
\(95\) 0 0
\(96\) 2.82843 0.288675
\(97\) −7.07107 −0.717958 −0.358979 0.933346i \(-0.616875\pi\)
−0.358979 + 0.933346i \(0.616875\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) −5.00000 −0.500000
\(101\) −1.41421 −0.140720 −0.0703598 0.997522i \(-0.522415\pi\)
−0.0703598 + 0.997522i \(0.522415\pi\)
\(102\) −8.00000 −0.792118
\(103\) −1.41421 −0.139347 −0.0696733 0.997570i \(-0.522196\pi\)
−0.0696733 + 0.997570i \(0.522196\pi\)
\(104\) 4.24264 0.416025
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −14.0000 −1.35343 −0.676716 0.736245i \(-0.736597\pi\)
−0.676716 + 0.736245i \(0.736597\pi\)
\(108\) 5.65685 0.544331
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 0 0
\(111\) 5.65685 0.536925
\(112\) 0 0
\(113\) 16.0000 1.50515 0.752577 0.658505i \(-0.228811\pi\)
0.752577 + 0.658505i \(0.228811\pi\)
\(114\) −12.0000 −1.12390
\(115\) 0 0
\(116\) −4.00000 −0.371391
\(117\) 21.2132 1.96116
\(118\) 11.3137 1.04151
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −9.89949 −0.896258
\(123\) 8.00000 0.721336
\(124\) −7.07107 −0.635001
\(125\) 0 0
\(126\) 0 0
\(127\) −20.0000 −1.77471 −0.887357 0.461084i \(-0.847461\pi\)
−0.887357 + 0.461084i \(0.847461\pi\)
\(128\) 1.00000 0.0883883
\(129\) 28.2843 2.49029
\(130\) 0 0
\(131\) 7.07107 0.617802 0.308901 0.951094i \(-0.400039\pi\)
0.308901 + 0.951094i \(0.400039\pi\)
\(132\) −2.82843 −0.246183
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −2.82843 −0.242536
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 16.9706 1.44463
\(139\) 18.3848 1.55938 0.779688 0.626168i \(-0.215378\pi\)
0.779688 + 0.626168i \(0.215378\pi\)
\(140\) 0 0
\(141\) −36.0000 −3.03175
\(142\) 16.0000 1.34269
\(143\) −4.24264 −0.354787
\(144\) 5.00000 0.416667
\(145\) 0 0
\(146\) −8.48528 −0.702247
\(147\) 0 0
\(148\) 2.00000 0.164399
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) −14.1421 −1.15470
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −4.24264 −0.344124
\(153\) −14.1421 −1.14332
\(154\) 0 0
\(155\) 0 0
\(156\) 12.0000 0.960769
\(157\) 8.48528 0.677199 0.338600 0.940931i \(-0.390047\pi\)
0.338600 + 0.940931i \(0.390047\pi\)
\(158\) −8.00000 −0.636446
\(159\) 5.65685 0.448618
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 2.82843 0.220863
\(165\) 0 0
\(166\) −12.7279 −0.987878
\(167\) −14.1421 −1.09435 −0.547176 0.837018i \(-0.684297\pi\)
−0.547176 + 0.837018i \(0.684297\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −21.2132 −1.62221
\(172\) 10.0000 0.762493
\(173\) 7.07107 0.537603 0.268802 0.963196i \(-0.413372\pi\)
0.268802 + 0.963196i \(0.413372\pi\)
\(174\) −11.3137 −0.857690
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 32.0000 2.40527
\(178\) 7.07107 0.529999
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 11.3137 0.840941 0.420471 0.907306i \(-0.361865\pi\)
0.420471 + 0.907306i \(0.361865\pi\)
\(182\) 0 0
\(183\) −28.0000 −2.06982
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −20.0000 −1.46647
\(187\) 2.82843 0.206835
\(188\) −12.7279 −0.928279
\(189\) 0 0
\(190\) 0 0
\(191\) 10.0000 0.723575 0.361787 0.932261i \(-0.382167\pi\)
0.361787 + 0.932261i \(0.382167\pi\)
\(192\) 2.82843 0.204124
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −7.07107 −0.507673
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −5.00000 −0.355335
\(199\) 9.89949 0.701757 0.350878 0.936421i \(-0.385883\pi\)
0.350878 + 0.936421i \(0.385883\pi\)
\(200\) −5.00000 −0.353553
\(201\) 22.6274 1.59601
\(202\) −1.41421 −0.0995037
\(203\) 0 0
\(204\) −8.00000 −0.560112
\(205\) 0 0
\(206\) −1.41421 −0.0985329
\(207\) 30.0000 2.08514
\(208\) 4.24264 0.294174
\(209\) 4.24264 0.293470
\(210\) 0 0
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) 2.00000 0.137361
\(213\) 45.2548 3.10081
\(214\) −14.0000 −0.957020
\(215\) 0 0
\(216\) 5.65685 0.384900
\(217\) 0 0
\(218\) −14.0000 −0.948200
\(219\) −24.0000 −1.62177
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 5.65685 0.379663
\(223\) 21.2132 1.42054 0.710271 0.703929i \(-0.248573\pi\)
0.710271 + 0.703929i \(0.248573\pi\)
\(224\) 0 0
\(225\) −25.0000 −1.66667
\(226\) 16.0000 1.06430
\(227\) −7.07107 −0.469323 −0.234662 0.972077i \(-0.575398\pi\)
−0.234662 + 0.972077i \(0.575398\pi\)
\(228\) −12.0000 −0.794719
\(229\) −22.6274 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −4.00000 −0.262613
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 21.2132 1.38675
\(235\) 0 0
\(236\) 11.3137 0.736460
\(237\) −22.6274 −1.46981
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −8.48528 −0.546585 −0.273293 0.961931i \(-0.588113\pi\)
−0.273293 + 0.961931i \(0.588113\pi\)
\(242\) 1.00000 0.0642824
\(243\) −14.1421 −0.907218
\(244\) −9.89949 −0.633750
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) −18.0000 −1.14531
\(248\) −7.07107 −0.449013
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) −5.65685 −0.357057 −0.178529 0.983935i \(-0.557134\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 7.07107 0.441081 0.220541 0.975378i \(-0.429218\pi\)
0.220541 + 0.975378i \(0.429218\pi\)
\(258\) 28.2843 1.76090
\(259\) 0 0
\(260\) 0 0
\(261\) −20.0000 −1.23797
\(262\) 7.07107 0.436852
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) −2.82843 −0.174078
\(265\) 0 0
\(266\) 0 0
\(267\) 20.0000 1.22398
\(268\) 8.00000 0.488678
\(269\) 11.3137 0.689809 0.344904 0.938638i \(-0.387911\pi\)
0.344904 + 0.938638i \(0.387911\pi\)
\(270\) 0 0
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) −2.82843 −0.171499
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 5.00000 0.301511
\(276\) 16.9706 1.02151
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 18.3848 1.10265
\(279\) −35.3553 −2.11667
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) −36.0000 −2.14377
\(283\) 7.07107 0.420331 0.210166 0.977666i \(-0.432600\pi\)
0.210166 + 0.977666i \(0.432600\pi\)
\(284\) 16.0000 0.949425
\(285\) 0 0
\(286\) −4.24264 −0.250873
\(287\) 0 0
\(288\) 5.00000 0.294628
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) −20.0000 −1.17242
\(292\) −8.48528 −0.496564
\(293\) −21.2132 −1.23929 −0.619644 0.784883i \(-0.712723\pi\)
−0.619644 + 0.784883i \(0.712723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) −5.65685 −0.328244
\(298\) 8.00000 0.463428
\(299\) 25.4558 1.47215
\(300\) −14.1421 −0.816497
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) −4.00000 −0.229794
\(304\) −4.24264 −0.243332
\(305\) 0 0
\(306\) −14.1421 −0.808452
\(307\) 21.2132 1.21070 0.605351 0.795959i \(-0.293033\pi\)
0.605351 + 0.795959i \(0.293033\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) −1.41421 −0.0801927 −0.0400963 0.999196i \(-0.512766\pi\)
−0.0400963 + 0.999196i \(0.512766\pi\)
\(312\) 12.0000 0.679366
\(313\) −29.6985 −1.67866 −0.839329 0.543624i \(-0.817052\pi\)
−0.839329 + 0.543624i \(0.817052\pi\)
\(314\) 8.48528 0.478852
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 5.65685 0.317221
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) −39.5980 −2.21014
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 1.00000 0.0555556
\(325\) −21.2132 −1.17670
\(326\) −4.00000 −0.221540
\(327\) −39.5980 −2.18977
\(328\) 2.82843 0.156174
\(329\) 0 0
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) −12.7279 −0.698535
\(333\) 10.0000 0.547997
\(334\) −14.1421 −0.773823
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 5.00000 0.271964
\(339\) 45.2548 2.45791
\(340\) 0 0
\(341\) 7.07107 0.382920
\(342\) −21.2132 −1.14708
\(343\) 0 0
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 7.07107 0.380143
\(347\) 26.0000 1.39575 0.697877 0.716218i \(-0.254128\pi\)
0.697877 + 0.716218i \(0.254128\pi\)
\(348\) −11.3137 −0.606478
\(349\) −26.8701 −1.43832 −0.719161 0.694844i \(-0.755473\pi\)
−0.719161 + 0.694844i \(0.755473\pi\)
\(350\) 0 0
\(351\) 24.0000 1.28103
\(352\) −1.00000 −0.0533002
\(353\) 15.5563 0.827981 0.413990 0.910281i \(-0.364135\pi\)
0.413990 + 0.910281i \(0.364135\pi\)
\(354\) 32.0000 1.70078
\(355\) 0 0
\(356\) 7.07107 0.374766
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 12.0000 0.633336 0.316668 0.948536i \(-0.397436\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 11.3137 0.594635
\(363\) 2.82843 0.148454
\(364\) 0 0
\(365\) 0 0
\(366\) −28.0000 −1.46358
\(367\) −21.2132 −1.10732 −0.553660 0.832743i \(-0.686769\pi\)
−0.553660 + 0.832743i \(0.686769\pi\)
\(368\) 6.00000 0.312772
\(369\) 14.1421 0.736210
\(370\) 0 0
\(371\) 0 0
\(372\) −20.0000 −1.03695
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) −12.7279 −0.656392
\(377\) −16.9706 −0.874028
\(378\) 0 0
\(379\) 24.0000 1.23280 0.616399 0.787434i \(-0.288591\pi\)
0.616399 + 0.787434i \(0.288591\pi\)
\(380\) 0 0
\(381\) −56.5685 −2.89809
\(382\) 10.0000 0.511645
\(383\) −15.5563 −0.794892 −0.397446 0.917625i \(-0.630103\pi\)
−0.397446 + 0.917625i \(0.630103\pi\)
\(384\) 2.82843 0.144338
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 50.0000 2.54164
\(388\) −7.07107 −0.358979
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) −16.9706 −0.858238
\(392\) 0 0
\(393\) 20.0000 1.00887
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −5.00000 −0.251259
\(397\) 25.4558 1.27759 0.638796 0.769376i \(-0.279433\pi\)
0.638796 + 0.769376i \(0.279433\pi\)
\(398\) 9.89949 0.496217
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 22.6274 1.12855
\(403\) −30.0000 −1.49441
\(404\) −1.41421 −0.0703598
\(405\) 0 0
\(406\) 0 0
\(407\) −2.00000 −0.0991363
\(408\) −8.00000 −0.396059
\(409\) 28.2843 1.39857 0.699284 0.714844i \(-0.253502\pi\)
0.699284 + 0.714844i \(0.253502\pi\)
\(410\) 0 0
\(411\) −16.9706 −0.837096
\(412\) −1.41421 −0.0696733
\(413\) 0 0
\(414\) 30.0000 1.47442
\(415\) 0 0
\(416\) 4.24264 0.208013
\(417\) 52.0000 2.54645
\(418\) 4.24264 0.207514
\(419\) 14.1421 0.690889 0.345444 0.938439i \(-0.387728\pi\)
0.345444 + 0.938439i \(0.387728\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 26.0000 1.26566
\(423\) −63.6396 −3.09426
\(424\) 2.00000 0.0971286
\(425\) 14.1421 0.685994
\(426\) 45.2548 2.19260
\(427\) 0 0
\(428\) −14.0000 −0.676716
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 5.65685 0.272166
\(433\) 29.6985 1.42722 0.713609 0.700544i \(-0.247059\pi\)
0.713609 + 0.700544i \(0.247059\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) −25.4558 −1.21772
\(438\) −24.0000 −1.14676
\(439\) 25.4558 1.21494 0.607471 0.794342i \(-0.292184\pi\)
0.607471 + 0.794342i \(0.292184\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 5.65685 0.268462
\(445\) 0 0
\(446\) 21.2132 1.00447
\(447\) 22.6274 1.07024
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) −25.0000 −1.17851
\(451\) −2.82843 −0.133185
\(452\) 16.0000 0.752577
\(453\) −22.6274 −1.06313
\(454\) −7.07107 −0.331862
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) −22.6274 −1.05731
\(459\) −16.0000 −0.746816
\(460\) 0 0
\(461\) 1.41421 0.0658665 0.0329332 0.999458i \(-0.489515\pi\)
0.0329332 + 0.999458i \(0.489515\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) −4.00000 −0.185695
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) 25.4558 1.17796 0.588978 0.808149i \(-0.299530\pi\)
0.588978 + 0.808149i \(0.299530\pi\)
\(468\) 21.2132 0.980581
\(469\) 0 0
\(470\) 0 0
\(471\) 24.0000 1.10586
\(472\) 11.3137 0.520756
\(473\) −10.0000 −0.459800
\(474\) −22.6274 −1.03931
\(475\) 21.2132 0.973329
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) −28.2843 −1.29234 −0.646171 0.763193i \(-0.723631\pi\)
−0.646171 + 0.763193i \(0.723631\pi\)
\(480\) 0 0
\(481\) 8.48528 0.386896
\(482\) −8.48528 −0.386494
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −14.1421 −0.641500
\(487\) 38.0000 1.72194 0.860972 0.508652i \(-0.169856\pi\)
0.860972 + 0.508652i \(0.169856\pi\)
\(488\) −9.89949 −0.448129
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) 8.00000 0.360668
\(493\) 11.3137 0.509544
\(494\) −18.0000 −0.809858
\(495\) 0 0
\(496\) −7.07107 −0.317500
\(497\) 0 0
\(498\) −36.0000 −1.61320
\(499\) 36.0000 1.61158 0.805791 0.592200i \(-0.201741\pi\)
0.805791 + 0.592200i \(0.201741\pi\)
\(500\) 0 0
\(501\) −40.0000 −1.78707
\(502\) −5.65685 −0.252478
\(503\) 19.7990 0.882793 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −6.00000 −0.266733
\(507\) 14.1421 0.628074
\(508\) −20.0000 −0.887357
\(509\) 16.9706 0.752207 0.376103 0.926578i \(-0.377264\pi\)
0.376103 + 0.926578i \(0.377264\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −24.0000 −1.05963
\(514\) 7.07107 0.311891
\(515\) 0 0
\(516\) 28.2843 1.24515
\(517\) 12.7279 0.559773
\(518\) 0 0
\(519\) 20.0000 0.877903
\(520\) 0 0
\(521\) −1.41421 −0.0619578 −0.0309789 0.999520i \(-0.509862\pi\)
−0.0309789 + 0.999520i \(0.509862\pi\)
\(522\) −20.0000 −0.875376
\(523\) 4.24264 0.185518 0.0927589 0.995689i \(-0.470431\pi\)
0.0927589 + 0.995689i \(0.470431\pi\)
\(524\) 7.07107 0.308901
\(525\) 0 0
\(526\) 4.00000 0.174408
\(527\) 20.0000 0.871214
\(528\) −2.82843 −0.123091
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 56.5685 2.45487
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 20.0000 0.865485
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) −33.9411 −1.46467
\(538\) 11.3137 0.487769
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 8.48528 0.364474
\(543\) 32.0000 1.37325
\(544\) −2.82843 −0.121268
\(545\) 0 0
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) −6.00000 −0.256307
\(549\) −49.4975 −2.11250
\(550\) 5.00000 0.213201
\(551\) 16.9706 0.722970
\(552\) 16.9706 0.722315
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) 18.3848 0.779688
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) −35.3553 −1.49671
\(559\) 42.4264 1.79445
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 22.0000 0.928014
\(563\) 41.0122 1.72846 0.864229 0.503099i \(-0.167807\pi\)
0.864229 + 0.503099i \(0.167807\pi\)
\(564\) −36.0000 −1.51587
\(565\) 0 0
\(566\) 7.07107 0.297219
\(567\) 0 0
\(568\) 16.0000 0.671345
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) −4.24264 −0.177394
\(573\) 28.2843 1.18159
\(574\) 0 0
\(575\) −30.0000 −1.25109
\(576\) 5.00000 0.208333
\(577\) −9.89949 −0.412121 −0.206061 0.978539i \(-0.566064\pi\)
−0.206061 + 0.978539i \(0.566064\pi\)
\(578\) −9.00000 −0.374351
\(579\) 16.9706 0.705273
\(580\) 0 0
\(581\) 0 0
\(582\) −20.0000 −0.829027
\(583\) −2.00000 −0.0828315
\(584\) −8.48528 −0.351123
\(585\) 0 0
\(586\) −21.2132 −0.876309
\(587\) 5.65685 0.233483 0.116742 0.993162i \(-0.462755\pi\)
0.116742 + 0.993162i \(0.462755\pi\)
\(588\) 0 0
\(589\) 30.0000 1.23613
\(590\) 0 0
\(591\) −5.65685 −0.232692
\(592\) 2.00000 0.0821995
\(593\) 8.48528 0.348449 0.174224 0.984706i \(-0.444258\pi\)
0.174224 + 0.984706i \(0.444258\pi\)
\(594\) −5.65685 −0.232104
\(595\) 0 0
\(596\) 8.00000 0.327693
\(597\) 28.0000 1.14596
\(598\) 25.4558 1.04097
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −14.1421 −0.577350
\(601\) 11.3137 0.461496 0.230748 0.973014i \(-0.425883\pi\)
0.230748 + 0.973014i \(0.425883\pi\)
\(602\) 0 0
\(603\) 40.0000 1.62893
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −4.00000 −0.162489
\(607\) −16.9706 −0.688814 −0.344407 0.938820i \(-0.611920\pi\)
−0.344407 + 0.938820i \(0.611920\pi\)
\(608\) −4.24264 −0.172062
\(609\) 0 0
\(610\) 0 0
\(611\) −54.0000 −2.18461
\(612\) −14.1421 −0.571662
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 21.2132 0.856095
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) −4.00000 −0.160904
\(619\) −14.1421 −0.568420 −0.284210 0.958762i \(-0.591731\pi\)
−0.284210 + 0.958762i \(0.591731\pi\)
\(620\) 0 0
\(621\) 33.9411 1.36201
\(622\) −1.41421 −0.0567048
\(623\) 0 0
\(624\) 12.0000 0.480384
\(625\) 25.0000 1.00000
\(626\) −29.6985 −1.18699
\(627\) 12.0000 0.479234
\(628\) 8.48528 0.338600
\(629\) −5.65685 −0.225554
\(630\) 0 0
\(631\) −30.0000 −1.19428 −0.597141 0.802137i \(-0.703697\pi\)
−0.597141 + 0.802137i \(0.703697\pi\)
\(632\) −8.00000 −0.318223
\(633\) 73.5391 2.92292
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) 5.65685 0.224309
\(637\) 0 0
\(638\) 4.00000 0.158362
\(639\) 80.0000 3.16475
\(640\) 0 0
\(641\) 44.0000 1.73790 0.868948 0.494904i \(-0.164797\pi\)
0.868948 + 0.494904i \(0.164797\pi\)
\(642\) −39.5980 −1.56281
\(643\) −33.9411 −1.33851 −0.669254 0.743034i \(-0.733386\pi\)
−0.669254 + 0.743034i \(0.733386\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 12.0000 0.472134
\(647\) 32.5269 1.27876 0.639382 0.768889i \(-0.279190\pi\)
0.639382 + 0.768889i \(0.279190\pi\)
\(648\) 1.00000 0.0392837
\(649\) −11.3137 −0.444102
\(650\) −21.2132 −0.832050
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) −39.5980 −1.54840
\(655\) 0 0
\(656\) 2.82843 0.110432
\(657\) −42.4264 −1.65521
\(658\) 0 0
\(659\) 30.0000 1.16863 0.584317 0.811525i \(-0.301362\pi\)
0.584317 + 0.811525i \(0.301362\pi\)
\(660\) 0 0
\(661\) −16.9706 −0.660078 −0.330039 0.943967i \(-0.607062\pi\)
−0.330039 + 0.943967i \(0.607062\pi\)
\(662\) 8.00000 0.310929
\(663\) −33.9411 −1.31816
\(664\) −12.7279 −0.493939
\(665\) 0 0
\(666\) 10.0000 0.387492
\(667\) −24.0000 −0.929284
\(668\) −14.1421 −0.547176
\(669\) 60.0000 2.31973
\(670\) 0 0
\(671\) 9.89949 0.382166
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) −22.0000 −0.847408
\(675\) −28.2843 −1.08866
\(676\) 5.00000 0.192308
\(677\) 9.89949 0.380468 0.190234 0.981739i \(-0.439075\pi\)
0.190234 + 0.981739i \(0.439075\pi\)
\(678\) 45.2548 1.73800
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 7.07107 0.270765
\(683\) 32.0000 1.22445 0.612223 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) −21.2132 −0.811107
\(685\) 0 0
\(686\) 0 0
\(687\) −64.0000 −2.44175
\(688\) 10.0000 0.381246
\(689\) 8.48528 0.323263
\(690\) 0 0
\(691\) 39.5980 1.50638 0.753189 0.657804i \(-0.228515\pi\)
0.753189 + 0.657804i \(0.228515\pi\)
\(692\) 7.07107 0.268802
\(693\) 0 0
\(694\) 26.0000 0.986947
\(695\) 0 0
\(696\) −11.3137 −0.428845
\(697\) −8.00000 −0.303022
\(698\) −26.8701 −1.01705
\(699\) 73.5391 2.78150
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 24.0000 0.905822
\(703\) −8.48528 −0.320028
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) 15.5563 0.585471
\(707\) 0 0
\(708\) 32.0000 1.20263
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) −40.0000 −1.50012
\(712\) 7.07107 0.264999
\(713\) −42.4264 −1.58888
\(714\) 0 0
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) 32.5269 1.21305 0.606525 0.795065i \(-0.292563\pi\)
0.606525 + 0.795065i \(0.292563\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.00000 −0.0372161
\(723\) −24.0000 −0.892570
\(724\) 11.3137 0.420471
\(725\) 20.0000 0.742781
\(726\) 2.82843 0.104973
\(727\) −46.6690 −1.73086 −0.865430 0.501031i \(-0.832954\pi\)
−0.865430 + 0.501031i \(0.832954\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) −28.2843 −1.04613
\(732\) −28.0000 −1.03491
\(733\) 7.07107 0.261176 0.130588 0.991437i \(-0.458314\pi\)
0.130588 + 0.991437i \(0.458314\pi\)
\(734\) −21.2132 −0.782994
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) −8.00000 −0.294684
\(738\) 14.1421 0.520579
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) −50.9117 −1.87029
\(742\) 0 0
\(743\) −8.00000 −0.293492 −0.146746 0.989174i \(-0.546880\pi\)
−0.146746 + 0.989174i \(0.546880\pi\)
\(744\) −20.0000 −0.733236
\(745\) 0 0
\(746\) −10.0000 −0.366126
\(747\) −63.6396 −2.32845
\(748\) 2.82843 0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −2.00000 −0.0729810 −0.0364905 0.999334i \(-0.511618\pi\)
−0.0364905 + 0.999334i \(0.511618\pi\)
\(752\) −12.7279 −0.464140
\(753\) −16.0000 −0.583072
\(754\) −16.9706 −0.618031
\(755\) 0 0
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 24.0000 0.871719
\(759\) −16.9706 −0.615992
\(760\) 0 0
\(761\) −33.9411 −1.23036 −0.615182 0.788385i \(-0.710918\pi\)
−0.615182 + 0.788385i \(0.710918\pi\)
\(762\) −56.5685 −2.04926
\(763\) 0 0
\(764\) 10.0000 0.361787
\(765\) 0 0
\(766\) −15.5563 −0.562074
\(767\) 48.0000 1.73318
\(768\) 2.82843 0.102062
\(769\) 5.65685 0.203991 0.101996 0.994785i \(-0.467477\pi\)
0.101996 + 0.994785i \(0.467477\pi\)
\(770\) 0 0
\(771\) 20.0000 0.720282
\(772\) 6.00000 0.215945
\(773\) −48.0833 −1.72943 −0.864717 0.502259i \(-0.832502\pi\)
−0.864717 + 0.502259i \(0.832502\pi\)
\(774\) 50.0000 1.79721
\(775\) 35.3553 1.27000
\(776\) −7.07107 −0.253837
\(777\) 0 0
\(778\) −30.0000 −1.07555
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) −16.9706 −0.606866
\(783\) −22.6274 −0.808638
\(784\) 0 0
\(785\) 0 0
\(786\) 20.0000 0.713376
\(787\) 9.89949 0.352879 0.176439 0.984311i \(-0.443542\pi\)
0.176439 + 0.984311i \(0.443542\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 11.3137 0.402779
\(790\) 0 0
\(791\) 0 0
\(792\) −5.00000 −0.177667
\(793\) −42.0000 −1.49146
\(794\) 25.4558 0.903394
\(795\) 0 0
\(796\) 9.89949 0.350878
\(797\) −2.82843 −0.100188 −0.0500940 0.998745i \(-0.515952\pi\)
−0.0500940 + 0.998745i \(0.515952\pi\)
\(798\) 0 0
\(799\) 36.0000 1.27359
\(800\) −5.00000 −0.176777
\(801\) 35.3553 1.24922
\(802\) −30.0000 −1.05934
\(803\) 8.48528 0.299439
\(804\) 22.6274 0.798007
\(805\) 0 0
\(806\) −30.0000 −1.05670
\(807\) 32.0000 1.12645
\(808\) −1.41421 −0.0497519
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −15.5563 −0.546257 −0.273129 0.961978i \(-0.588058\pi\)
−0.273129 + 0.961978i \(0.588058\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) −2.00000 −0.0701000
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) −42.4264 −1.48431
\(818\) 28.2843 0.988936
\(819\) 0 0
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) −16.9706 −0.591916
\(823\) −2.00000 −0.0697156 −0.0348578 0.999392i \(-0.511098\pi\)
−0.0348578 + 0.999392i \(0.511098\pi\)
\(824\) −1.41421 −0.0492665
\(825\) 14.1421 0.492366
\(826\) 0 0
\(827\) 28.0000 0.973655 0.486828 0.873498i \(-0.338154\pi\)
0.486828 + 0.873498i \(0.338154\pi\)
\(828\) 30.0000 1.04257
\(829\) 33.9411 1.17882 0.589412 0.807833i \(-0.299359\pi\)
0.589412 + 0.807833i \(0.299359\pi\)
\(830\) 0 0
\(831\) −22.6274 −0.784936
\(832\) 4.24264 0.147087
\(833\) 0 0
\(834\) 52.0000 1.80061
\(835\) 0 0
\(836\) 4.24264 0.146735
\(837\) −40.0000 −1.38260
\(838\) 14.1421 0.488532
\(839\) −43.8406 −1.51355 −0.756773 0.653678i \(-0.773225\pi\)
−0.756773 + 0.653678i \(0.773225\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) −2.00000 −0.0689246
\(843\) 62.2254 2.14316
\(844\) 26.0000 0.894957
\(845\) 0 0
\(846\) −63.6396 −2.18797
\(847\) 0 0
\(848\) 2.00000 0.0686803
\(849\) 20.0000 0.686398
\(850\) 14.1421 0.485071
\(851\) 12.0000 0.411355
\(852\) 45.2548 1.55041
\(853\) −15.5563 −0.532639 −0.266320 0.963885i \(-0.585808\pi\)
−0.266320 + 0.963885i \(0.585808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −14.0000 −0.478510
\(857\) 16.9706 0.579703 0.289852 0.957072i \(-0.406394\pi\)
0.289852 + 0.957072i \(0.406394\pi\)
\(858\) −12.0000 −0.409673
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000 0.272481
\(863\) 6.00000 0.204242 0.102121 0.994772i \(-0.467437\pi\)
0.102121 + 0.994772i \(0.467437\pi\)
\(864\) 5.65685 0.192450
\(865\) 0 0
\(866\) 29.6985 1.00920
\(867\) −25.4558 −0.864526
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 33.9411 1.15005
\(872\) −14.0000 −0.474100
\(873\) −35.3553 −1.19660
\(874\) −25.4558 −0.861057
\(875\) 0 0
\(876\) −24.0000 −0.810885
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 25.4558 0.859093
\(879\) −60.0000 −2.02375
\(880\) 0 0
\(881\) −12.7279 −0.428815 −0.214407 0.976744i \(-0.568782\pi\)
−0.214407 + 0.976744i \(0.568782\pi\)
\(882\) 0 0
\(883\) −12.0000 −0.403832 −0.201916 0.979403i \(-0.564717\pi\)
−0.201916 + 0.979403i \(0.564717\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 12.0000 0.403148
\(887\) 22.6274 0.759754 0.379877 0.925037i \(-0.375966\pi\)
0.379877 + 0.925037i \(0.375966\pi\)
\(888\) 5.65685 0.189832
\(889\) 0 0
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 21.2132 0.710271
\(893\) 54.0000 1.80704
\(894\) 22.6274 0.756774
\(895\) 0 0
\(896\) 0 0
\(897\) 72.0000 2.40401
\(898\) 16.0000 0.533927
\(899\) 28.2843 0.943333
\(900\) −25.0000 −0.833333
\(901\) −5.65685 −0.188457
\(902\) −2.82843 −0.0941763
\(903\) 0 0
\(904\) 16.0000 0.532152
\(905\) 0 0
\(906\) −22.6274 −0.751746
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) −7.07107 −0.234662
\(909\) −7.07107 −0.234533
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) −12.0000 −0.397360
\(913\) 12.7279 0.421233
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) −22.6274 −0.747631
\(917\) 0 0
\(918\) −16.0000 −0.528079
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 60.0000 1.97707
\(922\) 1.41421 0.0465746
\(923\) 67.8823 2.23437
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) −16.0000 −0.525793
\(927\) −7.07107 −0.232244
\(928\) −4.00000 −0.131306
\(929\) −26.8701 −0.881578 −0.440789 0.897611i \(-0.645301\pi\)
−0.440789 + 0.897611i \(0.645301\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 26.0000 0.851658
\(933\) −4.00000 −0.130954
\(934\) 25.4558 0.832941
\(935\) 0 0
\(936\) 21.2132 0.693375
\(937\) −8.48528 −0.277202 −0.138601 0.990348i \(-0.544261\pi\)
−0.138601 + 0.990348i \(0.544261\pi\)
\(938\) 0 0
\(939\) −84.0000 −2.74124
\(940\) 0 0
\(941\) 32.5269 1.06035 0.530174 0.847889i \(-0.322127\pi\)
0.530174 + 0.847889i \(0.322127\pi\)
\(942\) 24.0000 0.781962
\(943\) 16.9706 0.552638
\(944\) 11.3137 0.368230
\(945\) 0 0
\(946\) −10.0000 −0.325128
\(947\) 4.00000 0.129983 0.0649913 0.997886i \(-0.479298\pi\)
0.0649913 + 0.997886i \(0.479298\pi\)
\(948\) −22.6274 −0.734904
\(949\) −36.0000 −1.16861
\(950\) 21.2132 0.688247
\(951\) 50.9117 1.65092
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 10.0000 0.323762
\(955\) 0 0
\(956\) 0 0
\(957\) 11.3137 0.365720
\(958\) −28.2843 −0.913823
\(959\) 0 0
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 8.48528 0.273576
\(963\) −70.0000 −2.25572
\(964\) −8.48528 −0.273293
\(965\) 0 0
\(966\) 0 0
\(967\) −52.0000 −1.67221 −0.836104 0.548572i \(-0.815172\pi\)
−0.836104 + 0.548572i \(0.815172\pi\)
\(968\) 1.00000 0.0321412
\(969\) 33.9411 1.09035
\(970\) 0 0
\(971\) −14.1421 −0.453843 −0.226921 0.973913i \(-0.572866\pi\)
−0.226921 + 0.973913i \(0.572866\pi\)
\(972\) −14.1421 −0.453609
\(973\) 0 0
\(974\) 38.0000 1.21760
\(975\) −60.0000 −1.92154
\(976\) −9.89949 −0.316875
\(977\) −32.0000 −1.02377 −0.511885 0.859054i \(-0.671053\pi\)
−0.511885 + 0.859054i \(0.671053\pi\)
\(978\) −11.3137 −0.361773
\(979\) −7.07107 −0.225992
\(980\) 0 0
\(981\) −70.0000 −2.23493
\(982\) 12.0000 0.382935
\(983\) −26.8701 −0.857022 −0.428511 0.903537i \(-0.640962\pi\)
−0.428511 + 0.903537i \(0.640962\pi\)
\(984\) 8.00000 0.255031
\(985\) 0 0
\(986\) 11.3137 0.360302
\(987\) 0 0
\(988\) −18.0000 −0.572656
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −7.07107 −0.224507
\(993\) 22.6274 0.718059
\(994\) 0 0
\(995\) 0 0
\(996\) −36.0000 −1.14070
\(997\) −21.2132 −0.671829 −0.335914 0.941893i \(-0.609045\pi\)
−0.335914 + 0.941893i \(0.609045\pi\)
\(998\) 36.0000 1.13956
\(999\) 11.3137 0.357950
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 1078.2.a.v.1.2 yes 2
3.2 odd 2 9702.2.a.co.1.2 2
4.3 odd 2 8624.2.a.bz.1.1 2
7.2 even 3 1078.2.e.o.67.1 4
7.3 odd 6 1078.2.e.o.177.2 4
7.4 even 3 1078.2.e.o.177.1 4
7.5 odd 6 1078.2.e.o.67.2 4
7.6 odd 2 inner 1078.2.a.v.1.1 2
21.20 even 2 9702.2.a.co.1.1 2
28.27 even 2 8624.2.a.bz.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.v.1.1 2 7.6 odd 2 inner
1078.2.a.v.1.2 yes 2 1.1 even 1 trivial
1078.2.e.o.67.1 4 7.2 even 3
1078.2.e.o.67.2 4 7.5 odd 6
1078.2.e.o.177.1 4 7.4 even 3
1078.2.e.o.177.2 4 7.3 odd 6
8624.2.a.bz.1.1 2 4.3 odd 2
8624.2.a.bz.1.2 2 28.27 even 2
9702.2.a.co.1.1 2 21.20 even 2
9702.2.a.co.1.2 2 3.2 odd 2