Properties

Label 8624.2.a.bz.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
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] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1078)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{3} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} +5.00000 q^{9} +1.00000 q^{11} +4.24264 q^{13} -2.82843 q^{17} +4.24264 q^{19} -6.00000 q^{23} -5.00000 q^{25} -5.65685 q^{27} -4.00000 q^{29} +7.07107 q^{31} -2.82843 q^{33} +2.00000 q^{37} -12.0000 q^{39} +2.82843 q^{41} -10.0000 q^{43} +12.7279 q^{47} +8.00000 q^{51} +2.00000 q^{53} -12.0000 q^{57} -11.3137 q^{59} -9.89949 q^{61} -8.00000 q^{67} +16.9706 q^{69} -16.0000 q^{71} -8.48528 q^{73} +14.1421 q^{75} +8.00000 q^{79} +1.00000 q^{81} +12.7279 q^{83} +11.3137 q^{87} +7.07107 q^{89} -20.0000 q^{93} -7.07107 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{9} + 2 q^{11} - 12 q^{23} - 10 q^{25} - 8 q^{29} + 4 q^{37} - 24 q^{39} - 20 q^{43} + 16 q^{51} + 4 q^{53} - 24 q^{57} - 16 q^{67} - 32 q^{71} + 16 q^{79} + 2 q^{81} - 40 q^{93} + 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) 4.24264 0.973329 0.486664 0.873589i \(-0.338214\pi\)
0.486664 + 0.873589i \(0.338214\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.00000 −1.25109 −0.625543 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −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 \(-0.281000\pi\)
0.635001 + 0.772512i \(0.281000\pi\)
\(32\) 0 0
\(33\) −2.82843 −0.492366
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) −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 \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.7279 1.85656 0.928279 0.371884i \(-0.121288\pi\)
0.928279 + 0.371884i \(0.121288\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 8.00000 1.12022
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) −9.89949 −1.26750 −0.633750 0.773538i \(-0.718485\pi\)
−0.633750 + 0.773538i \(0.718485\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) 16.9706 2.04302
\(70\) 0 0
\(71\) −16.0000 −1.89885 −0.949425 0.313993i \(-0.898333\pi\)
−0.949425 + 0.313993i \(0.898333\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 0 0
\(75\) 14.1421 1.63299
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 12.7279 1.39707 0.698535 0.715575i \(-0.253835\pi\)
0.698535 + 0.715575i \(0.253835\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 11.3137 1.21296
\(88\) 0 0
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −20.0000 −2.07390
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −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
\(100\) 0 0
\(101\) −1.41421 −0.140720 −0.0703598 0.997522i \(-0.522415\pi\)
−0.0703598 + 0.997522i \(0.522415\pi\)
\(102\) 0 0
\(103\) 1.41421 0.139347 0.0696733 0.997570i \(-0.477804\pi\)
0.0696733 + 0.997570i \(0.477804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.0000 1.35343 0.676716 0.736245i \(-0.263403\pi\)
0.676716 + 0.736245i \(0.263403\pi\)
\(108\) 0 0
\(109\) −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 \(0.228811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 21.2132 1.96116
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −8.00000 −0.721336
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 28.2843 2.49029
\(130\) 0 0
\(131\) −7.07107 −0.617802 −0.308901 0.951094i \(-0.599961\pi\)
−0.308901 + 0.951094i \(0.599961\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 −0.512615 −0.256307 0.966595i \(-0.582506\pi\)
−0.256307 + 0.966595i \(0.582506\pi\)
\(138\) 0 0
\(139\) −18.3848 −1.55938 −0.779688 0.626168i \(-0.784622\pi\)
−0.779688 + 0.626168i \(0.784622\pi\)
\(140\) 0 0
\(141\) −36.0000 −3.03175
\(142\) 0 0
\(143\) 4.24264 0.354787
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.00000 0.655386 0.327693 0.944784i \(-0.393729\pi\)
0.327693 + 0.944784i \(0.393729\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) −14.1421 −1.14332
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.48528 0.677199 0.338600 0.940931i \(-0.390047\pi\)
0.338600 + 0.940931i \(0.390047\pi\)
\(158\) 0 0
\(159\) −5.65685 −0.448618
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.1421 1.09435 0.547176 0.837018i \(-0.315703\pi\)
0.547176 + 0.837018i \(0.315703\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 21.2132 1.62221
\(172\) 0 0
\(173\) 7.07107 0.537603 0.268802 0.963196i \(-0.413372\pi\)
0.268802 + 0.963196i \(0.413372\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 32.0000 2.40527
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 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
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.82843 −0.206835
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −10.0000 −0.723575 −0.361787 0.932261i \(-0.617833\pi\)
−0.361787 + 0.932261i \(0.617833\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −9.89949 −0.701757 −0.350878 0.936421i \(-0.614117\pi\)
−0.350878 + 0.936421i \(0.614117\pi\)
\(200\) 0 0
\(201\) 22.6274 1.59601
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −30.0000 −2.08514
\(208\) 0 0
\(209\) 4.24264 0.293470
\(210\) 0 0
\(211\) −26.0000 −1.78991 −0.894957 0.446153i \(-0.852794\pi\)
−0.894957 + 0.446153i \(0.852794\pi\)
\(212\) 0 0
\(213\) 45.2548 3.10081
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 24.0000 1.62177
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −21.2132 −1.42054 −0.710271 0.703929i \(-0.751427\pi\)
−0.710271 + 0.703929i \(0.751427\pi\)
\(224\) 0 0
\(225\) −25.0000 −1.66667
\(226\) 0 0
\(227\) 7.07107 0.469323 0.234662 0.972077i \(-0.424602\pi\)
0.234662 + 0.972077i \(0.424602\pi\)
\(228\) 0 0
\(229\) −22.6274 −1.49526 −0.747631 0.664114i \(-0.768809\pi\)
−0.747631 + 0.664114i \(0.768809\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 26.0000 1.70332 0.851658 0.524097i \(-0.175597\pi\)
0.851658 + 0.524097i \(0.175597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −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 \(0.588113\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 18.0000 1.14531
\(248\) 0 0
\(249\) −36.0000 −2.28141
\(250\) 0 0
\(251\) 5.65685 0.357057 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 7.07107 0.441081 0.220541 0.975378i \(-0.429218\pi\)
0.220541 + 0.975378i \(0.429218\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −20.0000 −1.23797
\(262\) 0 0
\(263\) −4.00000 −0.246651 −0.123325 0.992366i \(-0.539356\pi\)
−0.123325 + 0.992366i \(0.539356\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −20.0000 −1.22398
\(268\) 0 0
\(269\) 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 \(-0.582972\pi\)
−0.257722 + 0.966219i \(0.582972\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.00000 −0.301511
\(276\) 0 0
\(277\) −8.00000 −0.480673 −0.240337 0.970690i \(-0.577258\pi\)
−0.240337 + 0.970690i \(0.577258\pi\)
\(278\) 0 0
\(279\) 35.3553 2.11667
\(280\) 0 0
\(281\) 22.0000 1.31241 0.656205 0.754583i \(-0.272161\pi\)
0.656205 + 0.754583i \(0.272161\pi\)
\(282\) 0 0
\(283\) −7.07107 −0.420331 −0.210166 0.977666i \(-0.567400\pi\)
−0.210166 + 0.977666i \(0.567400\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) 0 0
\(293\) −21.2132 −1.23929 −0.619644 0.784883i \(-0.712723\pi\)
−0.619644 + 0.784883i \(0.712723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.65685 −0.328244
\(298\) 0 0
\(299\) −25.4558 −1.47215
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 4.00000 0.229794
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −21.2132 −1.21070 −0.605351 0.795959i \(-0.706967\pi\)
−0.605351 + 0.795959i \(0.706967\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 1.41421 0.0801927 0.0400963 0.999196i \(-0.487234\pi\)
0.0400963 + 0.999196i \(0.487234\pi\)
\(312\) 0 0
\(313\) −29.6985 −1.67866 −0.839329 0.543624i \(-0.817052\pi\)
−0.839329 + 0.543624i \(0.817052\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −4.00000 −0.223957
\(320\) 0 0
\(321\) −39.5980 −2.21014
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 0 0
\(325\) −21.2132 −1.17670
\(326\) 0 0
\(327\) 39.5980 2.18977
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 10.0000 0.547997
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) −45.2548 −2.45791
\(340\) 0 0
\(341\) 7.07107 0.382920
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −26.0000 −1.39575 −0.697877 0.716218i \(-0.745872\pi\)
−0.697877 + 0.716218i \(0.745872\pi\)
\(348\) 0 0
\(349\) −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
\(352\) 0 0
\(353\) 15.5563 0.827981 0.413990 0.910281i \(-0.364135\pi\)
0.413990 + 0.910281i \(0.364135\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) −2.82843 −0.148454
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 21.2132 1.10732 0.553660 0.832743i \(-0.313231\pi\)
0.553660 + 0.832743i \(0.313231\pi\)
\(368\) 0 0
\(369\) 14.1421 0.736210
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.9706 −0.874028
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) −56.5685 −2.89809
\(382\) 0 0
\(383\) 15.5563 0.794892 0.397446 0.917625i \(-0.369897\pi\)
0.397446 + 0.917625i \(0.369897\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −50.0000 −2.54164
\(388\) 0 0
\(389\) −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
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 25.4558 1.27759 0.638796 0.769376i \(-0.279433\pi\)
0.638796 + 0.769376i \(0.279433\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 30.0000 1.49441
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) 0 0
\(409\) 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
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 52.0000 2.54645
\(418\) 0 0
\(419\) −14.1421 −0.690889 −0.345444 0.938439i \(-0.612272\pi\)
−0.345444 + 0.938439i \(0.612272\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 63.6396 3.09426
\(424\) 0 0
\(425\) 14.1421 0.685994
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −12.0000 −0.579365
\(430\) 0 0
\(431\) −8.00000 −0.385346 −0.192673 0.981263i \(-0.561716\pi\)
−0.192673 + 0.981263i \(0.561716\pi\)
\(432\) 0 0
\(433\) 29.6985 1.42722 0.713609 0.700544i \(-0.247059\pi\)
0.713609 + 0.700544i \(0.247059\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −25.4558 −1.21772
\(438\) 0 0
\(439\) −25.4558 −1.21494 −0.607471 0.794342i \(-0.707816\pi\)
−0.607471 + 0.794342i \(0.707816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −22.6274 −1.07024
\(448\) 0 0
\(449\) 16.0000 0.755087 0.377543 0.925992i \(-0.376769\pi\)
0.377543 + 0.925992i \(0.376769\pi\)
\(450\) 0 0
\(451\) 2.82843 0.133185
\(452\) 0 0
\(453\) −22.6274 −1.06313
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 0 0
\(459\) 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 \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.4558 −1.17796 −0.588978 0.808149i \(-0.700470\pi\)
−0.588978 + 0.808149i \(0.700470\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −24.0000 −1.10586
\(472\) 0 0
\(473\) −10.0000 −0.459800
\(474\) 0 0
\(475\) −21.2132 −0.973329
\(476\) 0 0
\(477\) 10.0000 0.457869
\(478\) 0 0
\(479\) 28.2843 1.29234 0.646171 0.763193i \(-0.276369\pi\)
0.646171 + 0.763193i \(0.276369\pi\)
\(480\) 0 0
\(481\) 8.48528 0.386896
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 11.3137 0.509544
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −40.0000 −1.78707
\(502\) 0 0
\(503\) −19.7990 −0.882793 −0.441397 0.897312i \(-0.645517\pi\)
−0.441397 + 0.897312i \(0.645517\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −14.1421 −0.628074
\(508\) 0 0
\(509\) 16.9706 0.752207 0.376103 0.926578i \(-0.377264\pi\)
0.376103 + 0.926578i \(0.377264\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −24.0000 −1.05963
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 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 \(0.509862\pi\)
\(522\) 0 0
\(523\) −4.24264 −0.185518 −0.0927589 0.995689i \(-0.529569\pi\)
−0.0927589 + 0.995689i \(0.529569\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20.0000 −0.871214
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −56.5685 −2.45487
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −33.9411 −1.46467
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −30.0000 −1.28980 −0.644900 0.764267i \(-0.723101\pi\)
−0.644900 + 0.764267i \(0.723101\pi\)
\(542\) 0 0
\(543\) −32.0000 −1.37325
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) 0 0
\(549\) −49.4975 −2.11250
\(550\) 0 0
\(551\) −16.9706 −0.722970
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.00000 −0.254228 −0.127114 0.991888i \(-0.540571\pi\)
−0.127114 + 0.991888i \(0.540571\pi\)
\(558\) 0 0
\(559\) −42.4264 −1.79445
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 0 0
\(563\) −41.0122 −1.72846 −0.864229 0.503099i \(-0.832193\pi\)
−0.864229 + 0.503099i \(0.832193\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 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 \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 28.2843 1.18159
\(574\) 0 0
\(575\) 30.0000 1.25109
\(576\) 0 0
\(577\) −9.89949 −0.412121 −0.206061 0.978539i \(-0.566064\pi\)
−0.206061 + 0.978539i \(0.566064\pi\)
\(578\) 0 0
\(579\) −16.9706 −0.705273
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 2.00000 0.0828315
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.65685 −0.233483 −0.116742 0.993162i \(-0.537245\pi\)
−0.116742 + 0.993162i \(0.537245\pi\)
\(588\) 0 0
\(589\) 30.0000 1.23613
\(590\) 0 0
\(591\) 5.65685 0.232692
\(592\) 0 0
\(593\) 8.48528 0.348449 0.174224 0.984706i \(-0.444258\pi\)
0.174224 + 0.984706i \(0.444258\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 28.0000 1.14596
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 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
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16.9706 0.688814 0.344407 0.938820i \(-0.388080\pi\)
0.344407 + 0.938820i \(0.388080\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 54.0000 2.18461
\(612\) 0 0
\(613\) −16.0000 −0.646234 −0.323117 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) 14.1421 0.568420 0.284210 0.958762i \(-0.408269\pi\)
0.284210 + 0.958762i \(0.408269\pi\)
\(620\) 0 0
\(621\) 33.9411 1.36201
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) −12.0000 −0.479234
\(628\) 0 0
\(629\) −5.65685 −0.225554
\(630\) 0 0
\(631\) 30.0000 1.19428 0.597141 0.802137i \(-0.296303\pi\)
0.597141 + 0.802137i \(0.296303\pi\)
\(632\) 0 0
\(633\) 73.5391 2.92292
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −80.0000 −3.16475
\(640\) 0 0
\(641\) 44.0000 1.73790 0.868948 0.494904i \(-0.164797\pi\)
0.868948 + 0.494904i \(0.164797\pi\)
\(642\) 0 0
\(643\) 33.9411 1.33851 0.669254 0.743034i \(-0.266614\pi\)
0.669254 + 0.743034i \(0.266614\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.5269 −1.27876 −0.639382 0.768889i \(-0.720810\pi\)
−0.639382 + 0.768889i \(0.720810\pi\)
\(648\) 0 0
\(649\) −11.3137 −0.444102
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −42.4264 −1.65521
\(658\) 0 0
\(659\) −30.0000 −1.16863 −0.584317 0.811525i \(-0.698638\pi\)
−0.584317 + 0.811525i \(0.698638\pi\)
\(660\) 0 0
\(661\) −16.9706 −0.660078 −0.330039 0.943967i \(-0.607062\pi\)
−0.330039 + 0.943967i \(0.607062\pi\)
\(662\) 0 0
\(663\) 33.9411 1.31816
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000 0.929284
\(668\) 0 0
\(669\) 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 \(0.846929\pi\)
\(674\) 0 0
\(675\) 28.2843 1.08866
\(676\) 0 0
\(677\) 9.89949 0.380468 0.190234 0.981739i \(-0.439075\pi\)
0.190234 + 0.981739i \(0.439075\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 0 0
\(683\) −32.0000 −1.22445 −0.612223 0.790685i \(-0.709725\pi\)
−0.612223 + 0.790685i \(0.709725\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 64.0000 2.44175
\(688\) 0 0
\(689\) 8.48528 0.323263
\(690\) 0 0
\(691\) −39.5980 −1.50638 −0.753189 0.657804i \(-0.771485\pi\)
−0.753189 + 0.657804i \(0.771485\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −8.00000 −0.303022
\(698\) 0 0
\(699\) −73.5391 −2.78150
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) 0 0
\(703\) 8.48528 0.320028
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −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
\(712\) 0 0
\(713\) −42.4264 −1.58888
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −32.5269 −1.21305 −0.606525 0.795065i \(-0.707437\pi\)
−0.606525 + 0.795065i \(0.707437\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 24.0000 0.892570
\(724\) 0 0
\(725\) 20.0000 0.742781
\(726\) 0 0
\(727\) 46.6690 1.73086 0.865430 0.501031i \(-0.167046\pi\)
0.865430 + 0.501031i \(0.167046\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) 28.2843 1.04613
\(732\) 0 0
\(733\) 7.07107 0.261176 0.130588 0.991437i \(-0.458314\pi\)
0.130588 + 0.991437i \(0.458314\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) 12.0000 0.441427 0.220714 0.975339i \(-0.429161\pi\)
0.220714 + 0.975339i \(0.429161\pi\)
\(740\) 0 0
\(741\) −50.9117 −1.87029
\(742\) 0 0
\(743\) 8.00000 0.293492 0.146746 0.989174i \(-0.453120\pi\)
0.146746 + 0.989174i \(0.453120\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 63.6396 2.32845
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.00000 0.0729810 0.0364905 0.999334i \(-0.488382\pi\)
0.0364905 + 0.999334i \(0.488382\pi\)
\(752\) 0 0
\(753\) −16.0000 −0.583072
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 46.0000 1.67190 0.835949 0.548807i \(-0.184918\pi\)
0.835949 + 0.548807i \(0.184918\pi\)
\(758\) 0 0
\(759\) 16.9706 0.615992
\(760\) 0 0
\(761\) −33.9411 −1.23036 −0.615182 0.788385i \(-0.710918\pi\)
−0.615182 + 0.788385i \(0.710918\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −48.0000 −1.73318
\(768\) 0 0
\(769\) 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
\(772\) 0 0
\(773\) −48.0833 −1.72943 −0.864717 0.502259i \(-0.832502\pi\)
−0.864717 + 0.502259i \(0.832502\pi\)
\(774\) 0 0
\(775\) −35.3553 −1.27000
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.0000 0.429945
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 0 0
\(783\) 22.6274 0.808638
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −9.89949 −0.352879 −0.176439 0.984311i \(-0.556458\pi\)
−0.176439 + 0.984311i \(0.556458\pi\)
\(788\) 0 0
\(789\) 11.3137 0.402779
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −42.0000 −1.49146
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −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
\(800\) 0 0
\(801\) 35.3553 1.24922
\(802\) 0 0
\(803\) −8.48528 −0.299439
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −32.0000 −1.12645
\(808\) 0 0
\(809\) −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 \(-0.411942\pi\)
0.273129 + 0.961978i \(0.411942\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −42.4264 −1.48431
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.00000 −0.279202 −0.139601 0.990208i \(-0.544582\pi\)
−0.139601 + 0.990208i \(0.544582\pi\)
\(822\) 0 0
\(823\) 2.00000 0.0697156 0.0348578 0.999392i \(-0.488902\pi\)
0.0348578 + 0.999392i \(0.488902\pi\)
\(824\) 0 0
\(825\) 14.1421 0.492366
\(826\) 0 0
\(827\) −28.0000 −0.973655 −0.486828 0.873498i \(-0.661846\pi\)
−0.486828 + 0.873498i \(0.661846\pi\)
\(828\) 0 0
\(829\) 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
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −40.0000 −1.38260
\(838\) 0 0
\(839\) 43.8406 1.51355 0.756773 0.653678i \(-0.226775\pi\)
0.756773 + 0.653678i \(0.226775\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) −62.2254 −2.14316
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) −15.5563 −0.532639 −0.266320 0.963885i \(-0.585808\pi\)
−0.266320 + 0.963885i \(0.585808\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 16.9706 0.579703 0.289852 0.957072i \(-0.406394\pi\)
0.289852 + 0.957072i \(0.406394\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 25.4558 0.864526
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) −33.9411 −1.15005
\(872\) 0 0
\(873\) −35.3553 −1.19660
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) 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 \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.6274 −0.759754 −0.379877 0.925037i \(-0.624034\pi\)
−0.379877 + 0.925037i \(0.624034\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 0 0
\(893\) 54.0000 1.80704
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 72.0000 2.40401
\(898\) 0 0
\(899\) −28.2843 −0.943333
\(900\) 0 0
\(901\) −5.65685 −0.188457
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −16.0000 −0.531271 −0.265636 0.964073i \(-0.585582\pi\)
−0.265636 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) −7.07107 −0.234533
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) 12.7279 0.421233
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 60.0000 1.97707
\(922\) 0 0
\(923\) −67.8823 −2.23437
\(924\) 0 0
\(925\) −10.0000 −0.328798
\(926\) 0 0
\(927\) 7.07107 0.232244
\(928\) 0 0
\(929\) −26.8701 −0.881578 −0.440789 0.897611i \(-0.645301\pi\)
−0.440789 + 0.897611i \(0.645301\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4.00000 −0.130954
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −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 \(0.322127\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.00000 −0.129983 −0.0649913 0.997886i \(-0.520702\pi\)
−0.0649913 + 0.997886i \(0.520702\pi\)
\(948\) 0 0
\(949\) −36.0000 −1.16861
\(950\) 0 0
\(951\) −50.9117 −1.65092
\(952\) 0 0
\(953\) −46.0000 −1.49009 −0.745043 0.667016i \(-0.767571\pi\)
−0.745043 + 0.667016i \(0.767571\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 11.3137 0.365720
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 70.0000 2.25572
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 52.0000 1.67221 0.836104 0.548572i \(-0.184828\pi\)
0.836104 + 0.548572i \(0.184828\pi\)
\(968\) 0 0
\(969\) 33.9411 1.09035
\(970\) 0 0
\(971\) 14.1421 0.453843 0.226921 0.973913i \(-0.427134\pi\)
0.226921 + 0.973913i \(0.427134\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 60.0000 1.92154
\(976\) 0 0
\(977\) −32.0000 −1.02377 −0.511885 0.859054i \(-0.671053\pi\)
−0.511885 + 0.859054i \(0.671053\pi\)
\(978\) 0 0
\(979\) 7.07107 0.225992
\(980\) 0 0
\(981\) −70.0000 −2.23493
\(982\) 0 0
\(983\) 26.8701 0.857022 0.428511 0.903537i \(-0.359038\pi\)
0.428511 + 0.903537i \(0.359038\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 22.6274 0.718059
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −21.2132 −0.671829 −0.335914 0.941893i \(-0.609045\pi\)
−0.335914 + 0.941893i \(0.609045\pi\)
\(998\) 0 0
\(999\) −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 8624.2.a.bz.1.1 2
4.3 odd 2 1078.2.a.v.1.2 yes 2
7.6 odd 2 inner 8624.2.a.bz.1.2 2
12.11 even 2 9702.2.a.co.1.2 2
28.3 even 6 1078.2.e.o.177.2 4
28.11 odd 6 1078.2.e.o.177.1 4
28.19 even 6 1078.2.e.o.67.2 4
28.23 odd 6 1078.2.e.o.67.1 4
28.27 even 2 1078.2.a.v.1.1 2
84.83 odd 2 9702.2.a.co.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.v.1.1 2 28.27 even 2
1078.2.a.v.1.2 yes 2 4.3 odd 2
1078.2.e.o.67.1 4 28.23 odd 6
1078.2.e.o.67.2 4 28.19 even 6
1078.2.e.o.177.1 4 28.11 odd 6
1078.2.e.o.177.2 4 28.3 even 6
8624.2.a.bz.1.1 2 1.1 even 1 trivial
8624.2.a.bz.1.2 2 7.6 odd 2 inner
9702.2.a.co.1.1 2 84.83 odd 2
9702.2.a.co.1.2 2 12.11 even 2