Properties

Label 6384.2.a.bk.1.2
Level $6384$
Weight $2$
Character 6384.1
Self dual yes
Analytic conductor $50.976$
Analytic rank $1$
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] = [6384,2,Mod(1,6384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6384.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6384 = 2^{4} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6384.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: \(50.9764966504\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3192)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{13} -3.46410 q^{15} -3.46410 q^{17} +1.00000 q^{19} -1.00000 q^{21} -4.00000 q^{23} +7.00000 q^{25} -1.00000 q^{27} -0.535898 q^{29} -10.9282 q^{31} +3.46410 q^{35} -2.00000 q^{37} +2.00000 q^{39} +6.00000 q^{41} -6.92820 q^{43} +3.46410 q^{45} -9.46410 q^{47} +1.00000 q^{49} +3.46410 q^{51} -3.46410 q^{53} -1.00000 q^{57} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} -6.92820 q^{65} -14.9282 q^{67} +4.00000 q^{69} -13.4641 q^{71} -0.928203 q^{73} -7.00000 q^{75} +1.00000 q^{81} +5.46410 q^{83} -12.0000 q^{85} +0.535898 q^{87} -15.8564 q^{89} -2.00000 q^{91} +10.9282 q^{93} +3.46410 q^{95} +10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{7} + 2 q^{9} - 4 q^{13} + 2 q^{19} - 2 q^{21} - 8 q^{23} + 14 q^{25} - 2 q^{27} - 8 q^{29} - 8 q^{31} - 4 q^{37} + 4 q^{39} + 12 q^{41} - 12 q^{47} + 2 q^{49} - 2 q^{57} - 8 q^{59} + 12 q^{61} + 2 q^{63} - 16 q^{67} + 8 q^{69} - 20 q^{71} + 12 q^{73} - 14 q^{75} + 2 q^{81} + 4 q^{83} - 24 q^{85} + 8 q^{87} - 4 q^{89} - 4 q^{91} + 8 q^{93} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −3.46410 −0.894427
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −0.535898 −0.0995138 −0.0497569 0.998761i \(-0.515845\pi\)
−0.0497569 + 0.998761i \(0.515845\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0 0
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −6.92820 −1.05654 −0.528271 0.849076i \(-0.677159\pi\)
−0.528271 + 0.849076i \(0.677159\pi\)
\(44\) 0 0
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) −9.46410 −1.38048 −0.690241 0.723580i \(-0.742495\pi\)
−0.690241 + 0.723580i \(0.742495\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.46410 0.485071
\(52\) 0 0
\(53\) −3.46410 −0.475831 −0.237915 0.971286i \(-0.576464\pi\)
−0.237915 + 0.971286i \(0.576464\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −6.92820 −0.859338
\(66\) 0 0
\(67\) −14.9282 −1.82377 −0.911885 0.410445i \(-0.865373\pi\)
−0.911885 + 0.410445i \(0.865373\pi\)
\(68\) 0 0
\(69\) 4.00000 0.481543
\(70\) 0 0
\(71\) −13.4641 −1.59789 −0.798947 0.601401i \(-0.794609\pi\)
−0.798947 + 0.601401i \(0.794609\pi\)
\(72\) 0 0
\(73\) −0.928203 −0.108638 −0.0543190 0.998524i \(-0.517299\pi\)
−0.0543190 + 0.998524i \(0.517299\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.46410 0.599763 0.299882 0.953976i \(-0.403053\pi\)
0.299882 + 0.953976i \(0.403053\pi\)
\(84\) 0 0
\(85\) −12.0000 −1.30158
\(86\) 0 0
\(87\) 0.535898 0.0574543
\(88\) 0 0
\(89\) −15.8564 −1.68078 −0.840388 0.541985i \(-0.817673\pi\)
−0.840388 + 0.541985i \(0.817673\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 10.9282 1.13320
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.3923 −1.03407 −0.517036 0.855963i \(-0.672965\pi\)
−0.517036 + 0.855963i \(0.672965\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) −3.46410 −0.338062
\(106\) 0 0
\(107\) 12.3923 1.19801 0.599005 0.800746i \(-0.295563\pi\)
0.599005 + 0.800746i \(0.295563\pi\)
\(108\) 0 0
\(109\) −12.9282 −1.23830 −0.619149 0.785274i \(-0.712522\pi\)
−0.619149 + 0.785274i \(0.712522\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 11.4641 1.07845 0.539226 0.842161i \(-0.318717\pi\)
0.539226 + 0.842161i \(0.318717\pi\)
\(114\) 0 0
\(115\) −13.8564 −1.29212
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −6.00000 −0.541002
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 18.9282 1.67961 0.839803 0.542891i \(-0.182670\pi\)
0.839803 + 0.542891i \(0.182670\pi\)
\(128\) 0 0
\(129\) 6.92820 0.609994
\(130\) 0 0
\(131\) 8.39230 0.733239 0.366620 0.930371i \(-0.380515\pi\)
0.366620 + 0.930371i \(0.380515\pi\)
\(132\) 0 0
\(133\) 1.00000 0.0867110
\(134\) 0 0
\(135\) −3.46410 −0.298142
\(136\) 0 0
\(137\) 2.00000 0.170872 0.0854358 0.996344i \(-0.472772\pi\)
0.0854358 + 0.996344i \(0.472772\pi\)
\(138\) 0 0
\(139\) 6.92820 0.587643 0.293821 0.955860i \(-0.405073\pi\)
0.293821 + 0.955860i \(0.405073\pi\)
\(140\) 0 0
\(141\) 9.46410 0.797021
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.85641 −0.154166
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −4.92820 −0.403734 −0.201867 0.979413i \(-0.564701\pi\)
−0.201867 + 0.979413i \(0.564701\pi\)
\(150\) 0 0
\(151\) 2.92820 0.238294 0.119147 0.992877i \(-0.461984\pi\)
0.119147 + 0.992877i \(0.461984\pi\)
\(152\) 0 0
\(153\) −3.46410 −0.280056
\(154\) 0 0
\(155\) −37.8564 −3.04070
\(156\) 0 0
\(157\) 11.8564 0.946244 0.473122 0.880997i \(-0.343127\pi\)
0.473122 + 0.880997i \(0.343127\pi\)
\(158\) 0 0
\(159\) 3.46410 0.274721
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) 14.9282 1.16927 0.584634 0.811297i \(-0.301238\pi\)
0.584634 + 0.811297i \(0.301238\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.92820 0.226591 0.113296 0.993561i \(-0.463859\pi\)
0.113296 + 0.993561i \(0.463859\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) 7.00000 0.529150
\(176\) 0 0
\(177\) 4.00000 0.300658
\(178\) 0 0
\(179\) −6.53590 −0.488516 −0.244258 0.969710i \(-0.578544\pi\)
−0.244258 + 0.969710i \(0.578544\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) −6.00000 −0.443533
\(184\) 0 0
\(185\) −6.92820 −0.509372
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 17.8564 1.29204 0.646022 0.763319i \(-0.276431\pi\)
0.646022 + 0.763319i \(0.276431\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) 6.92820 0.496139
\(196\) 0 0
\(197\) 19.8564 1.41471 0.707355 0.706858i \(-0.249888\pi\)
0.707355 + 0.706858i \(0.249888\pi\)
\(198\) 0 0
\(199\) 5.07180 0.359530 0.179765 0.983710i \(-0.442466\pi\)
0.179765 + 0.983710i \(0.442466\pi\)
\(200\) 0 0
\(201\) 14.9282 1.05295
\(202\) 0 0
\(203\) −0.535898 −0.0376127
\(204\) 0 0
\(205\) 20.7846 1.45166
\(206\) 0 0
\(207\) −4.00000 −0.278019
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −25.8564 −1.78003 −0.890014 0.455933i \(-0.849306\pi\)
−0.890014 + 0.455933i \(0.849306\pi\)
\(212\) 0 0
\(213\) 13.4641 0.922545
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) −10.9282 −0.741855
\(218\) 0 0
\(219\) 0.928203 0.0627222
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) 24.7846 1.65970 0.829850 0.557986i \(-0.188426\pi\)
0.829850 + 0.557986i \(0.188426\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 6.92820 0.459841 0.229920 0.973209i \(-0.426153\pi\)
0.229920 + 0.973209i \(0.426153\pi\)
\(228\) 0 0
\(229\) −18.7846 −1.24132 −0.620661 0.784079i \(-0.713136\pi\)
−0.620661 + 0.784079i \(0.713136\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.92820 0.322857 0.161429 0.986884i \(-0.448390\pi\)
0.161429 + 0.986884i \(0.448390\pi\)
\(234\) 0 0
\(235\) −32.7846 −2.13863
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.92820 0.448148 0.224074 0.974572i \(-0.428064\pi\)
0.224074 + 0.974572i \(0.428064\pi\)
\(240\) 0 0
\(241\) −27.8564 −1.79439 −0.897194 0.441636i \(-0.854398\pi\)
−0.897194 + 0.441636i \(0.854398\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.46410 0.221313
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) −5.46410 −0.346273
\(250\) 0 0
\(251\) 8.39230 0.529718 0.264859 0.964287i \(-0.414675\pi\)
0.264859 + 0.964287i \(0.414675\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 12.0000 0.751469
\(256\) 0 0
\(257\) 22.7846 1.42126 0.710632 0.703563i \(-0.248409\pi\)
0.710632 + 0.703563i \(0.248409\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) −0.535898 −0.0331713
\(262\) 0 0
\(263\) 9.07180 0.559391 0.279695 0.960089i \(-0.409766\pi\)
0.279695 + 0.960089i \(0.409766\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 15.8564 0.970396
\(268\) 0 0
\(269\) −3.07180 −0.187291 −0.0936454 0.995606i \(-0.529852\pi\)
−0.0936454 + 0.995606i \(0.529852\pi\)
\(270\) 0 0
\(271\) 29.8564 1.81365 0.906824 0.421510i \(-0.138500\pi\)
0.906824 + 0.421510i \(0.138500\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.85641 0.231709 0.115855 0.993266i \(-0.463039\pi\)
0.115855 + 0.993266i \(0.463039\pi\)
\(278\) 0 0
\(279\) −10.9282 −0.654254
\(280\) 0 0
\(281\) 17.3205 1.03325 0.516627 0.856210i \(-0.327187\pi\)
0.516627 + 0.856210i \(0.327187\pi\)
\(282\) 0 0
\(283\) 9.07180 0.539262 0.269631 0.962964i \(-0.413098\pi\)
0.269631 + 0.962964i \(0.413098\pi\)
\(284\) 0 0
\(285\) −3.46410 −0.205196
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) 0 0
\(293\) −22.0000 −1.28525 −0.642627 0.766179i \(-0.722155\pi\)
−0.642627 + 0.766179i \(0.722155\pi\)
\(294\) 0 0
\(295\) −13.8564 −0.806751
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) −6.92820 −0.399335
\(302\) 0 0
\(303\) 10.3923 0.597022
\(304\) 0 0
\(305\) 20.7846 1.19012
\(306\) 0 0
\(307\) −1.07180 −0.0611707 −0.0305853 0.999532i \(-0.509737\pi\)
−0.0305853 + 0.999532i \(0.509737\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −17.4641 −0.990298 −0.495149 0.868808i \(-0.664887\pi\)
−0.495149 + 0.868808i \(0.664887\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 3.46410 0.195180
\(316\) 0 0
\(317\) 7.46410 0.419226 0.209613 0.977784i \(-0.432780\pi\)
0.209613 + 0.977784i \(0.432780\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −12.3923 −0.691671
\(322\) 0 0
\(323\) −3.46410 −0.192748
\(324\) 0 0
\(325\) −14.0000 −0.776580
\(326\) 0 0
\(327\) 12.9282 0.714931
\(328\) 0 0
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) −14.9282 −0.820528 −0.410264 0.911967i \(-0.634563\pi\)
−0.410264 + 0.911967i \(0.634563\pi\)
\(332\) 0 0
\(333\) −2.00000 −0.109599
\(334\) 0 0
\(335\) −51.7128 −2.82537
\(336\) 0 0
\(337\) 20.9282 1.14003 0.570016 0.821634i \(-0.306937\pi\)
0.570016 + 0.821634i \(0.306937\pi\)
\(338\) 0 0
\(339\) −11.4641 −0.622645
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 13.8564 0.746004
\(346\) 0 0
\(347\) −32.7846 −1.75997 −0.879985 0.475001i \(-0.842448\pi\)
−0.879985 + 0.475001i \(0.842448\pi\)
\(348\) 0 0
\(349\) 11.8564 0.634659 0.317329 0.948315i \(-0.397214\pi\)
0.317329 + 0.948315i \(0.397214\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −19.4641 −1.03597 −0.517985 0.855390i \(-0.673318\pi\)
−0.517985 + 0.855390i \(0.673318\pi\)
\(354\) 0 0
\(355\) −46.6410 −2.47545
\(356\) 0 0
\(357\) 3.46410 0.183340
\(358\) 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
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −3.21539 −0.168301
\(366\) 0 0
\(367\) −2.92820 −0.152851 −0.0764255 0.997075i \(-0.524351\pi\)
−0.0764255 + 0.997075i \(0.524351\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −3.46410 −0.179847
\(372\) 0 0
\(373\) −7.85641 −0.406789 −0.203395 0.979097i \(-0.565197\pi\)
−0.203395 + 0.979097i \(0.565197\pi\)
\(374\) 0 0
\(375\) −6.92820 −0.357771
\(376\) 0 0
\(377\) 1.07180 0.0552003
\(378\) 0 0
\(379\) 28.7846 1.47857 0.739283 0.673395i \(-0.235165\pi\)
0.739283 + 0.673395i \(0.235165\pi\)
\(380\) 0 0
\(381\) −18.9282 −0.969721
\(382\) 0 0
\(383\) −18.9282 −0.967186 −0.483593 0.875293i \(-0.660669\pi\)
−0.483593 + 0.875293i \(0.660669\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −6.92820 −0.352180
\(388\) 0 0
\(389\) −7.07180 −0.358554 −0.179277 0.983799i \(-0.557376\pi\)
−0.179277 + 0.983799i \(0.557376\pi\)
\(390\) 0 0
\(391\) 13.8564 0.700749
\(392\) 0 0
\(393\) −8.39230 −0.423336
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.143594 0.00720675 0.00360338 0.999994i \(-0.498853\pi\)
0.00360338 + 0.999994i \(0.498853\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) 6.39230 0.319216 0.159608 0.987180i \(-0.448977\pi\)
0.159608 + 0.987180i \(0.448977\pi\)
\(402\) 0 0
\(403\) 21.8564 1.08875
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) −2.00000 −0.0986527
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 18.9282 0.929149
\(416\) 0 0
\(417\) −6.92820 −0.339276
\(418\) 0 0
\(419\) −2.53590 −0.123887 −0.0619434 0.998080i \(-0.519730\pi\)
−0.0619434 + 0.998080i \(0.519730\pi\)
\(420\) 0 0
\(421\) −15.0718 −0.734554 −0.367277 0.930112i \(-0.619710\pi\)
−0.367277 + 0.930112i \(0.619710\pi\)
\(422\) 0 0
\(423\) −9.46410 −0.460160
\(424\) 0 0
\(425\) −24.2487 −1.17624
\(426\) 0 0
\(427\) 6.00000 0.290360
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −37.4641 −1.80458 −0.902291 0.431127i \(-0.858116\pi\)
−0.902291 + 0.431127i \(0.858116\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 1.85641 0.0890079
\(436\) 0 0
\(437\) −4.00000 −0.191346
\(438\) 0 0
\(439\) −0.784610 −0.0374474 −0.0187237 0.999825i \(-0.505960\pi\)
−0.0187237 + 0.999825i \(0.505960\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 16.7846 0.797461 0.398730 0.917068i \(-0.369451\pi\)
0.398730 + 0.917068i \(0.369451\pi\)
\(444\) 0 0
\(445\) −54.9282 −2.60385
\(446\) 0 0
\(447\) 4.92820 0.233096
\(448\) 0 0
\(449\) −7.46410 −0.352253 −0.176126 0.984368i \(-0.556357\pi\)
−0.176126 + 0.984368i \(0.556357\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.92820 −0.137579
\(454\) 0 0
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) −3.85641 −0.180395 −0.0901975 0.995924i \(-0.528750\pi\)
−0.0901975 + 0.995924i \(0.528750\pi\)
\(458\) 0 0
\(459\) 3.46410 0.161690
\(460\) 0 0
\(461\) −12.5359 −0.583855 −0.291927 0.956440i \(-0.594297\pi\)
−0.291927 + 0.956440i \(0.594297\pi\)
\(462\) 0 0
\(463\) 19.7128 0.916132 0.458066 0.888918i \(-0.348542\pi\)
0.458066 + 0.888918i \(0.348542\pi\)
\(464\) 0 0
\(465\) 37.8564 1.75555
\(466\) 0 0
\(467\) −37.4641 −1.73363 −0.866816 0.498628i \(-0.833837\pi\)
−0.866816 + 0.498628i \(0.833837\pi\)
\(468\) 0 0
\(469\) −14.9282 −0.689320
\(470\) 0 0
\(471\) −11.8564 −0.546314
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 7.00000 0.321182
\(476\) 0 0
\(477\) −3.46410 −0.158610
\(478\) 0 0
\(479\) −22.5359 −1.02969 −0.514846 0.857283i \(-0.672151\pi\)
−0.514846 + 0.857283i \(0.672151\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) 4.00000 0.182006
\(484\) 0 0
\(485\) 34.6410 1.57297
\(486\) 0 0
\(487\) 27.7128 1.25579 0.627894 0.778299i \(-0.283917\pi\)
0.627894 + 0.778299i \(0.283917\pi\)
\(488\) 0 0
\(489\) −14.9282 −0.675077
\(490\) 0 0
\(491\) −5.07180 −0.228887 −0.114443 0.993430i \(-0.536508\pi\)
−0.114443 + 0.993430i \(0.536508\pi\)
\(492\) 0 0
\(493\) 1.85641 0.0836083
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13.4641 −0.603947
\(498\) 0 0
\(499\) 12.0000 0.537194 0.268597 0.963253i \(-0.413440\pi\)
0.268597 + 0.963253i \(0.413440\pi\)
\(500\) 0 0
\(501\) −2.92820 −0.130822
\(502\) 0 0
\(503\) −37.1769 −1.65764 −0.828818 0.559518i \(-0.810986\pi\)
−0.828818 + 0.559518i \(0.810986\pi\)
\(504\) 0 0
\(505\) −36.0000 −1.60198
\(506\) 0 0
\(507\) 9.00000 0.399704
\(508\) 0 0
\(509\) 1.21539 0.0538712 0.0269356 0.999637i \(-0.491425\pi\)
0.0269356 + 0.999637i \(0.491425\pi\)
\(510\) 0 0
\(511\) −0.928203 −0.0410613
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −27.7128 −1.22117
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) −26.7846 −1.17346 −0.586728 0.809784i \(-0.699584\pi\)
−0.586728 + 0.809784i \(0.699584\pi\)
\(522\) 0 0
\(523\) −22.9282 −1.00258 −0.501290 0.865279i \(-0.667141\pi\)
−0.501290 + 0.865279i \(0.667141\pi\)
\(524\) 0 0
\(525\) −7.00000 −0.305505
\(526\) 0 0
\(527\) 37.8564 1.64905
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) 42.9282 1.85595
\(536\) 0 0
\(537\) 6.53590 0.282045
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 2.00000 0.0858282
\(544\) 0 0
\(545\) −44.7846 −1.91836
\(546\) 0 0
\(547\) −39.7128 −1.69800 −0.848999 0.528395i \(-0.822794\pi\)
−0.848999 + 0.528395i \(0.822794\pi\)
\(548\) 0 0
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −0.535898 −0.0228300
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.92820 0.294086
\(556\) 0 0
\(557\) −18.7846 −0.795929 −0.397965 0.917401i \(-0.630283\pi\)
−0.397965 + 0.917401i \(0.630283\pi\)
\(558\) 0 0
\(559\) 13.8564 0.586064
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.07180 0.0451708 0.0225854 0.999745i \(-0.492810\pi\)
0.0225854 + 0.999745i \(0.492810\pi\)
\(564\) 0 0
\(565\) 39.7128 1.67073
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 27.4641 1.15136 0.575678 0.817677i \(-0.304738\pi\)
0.575678 + 0.817677i \(0.304738\pi\)
\(570\) 0 0
\(571\) 39.7128 1.66193 0.830965 0.556325i \(-0.187789\pi\)
0.830965 + 0.556325i \(0.187789\pi\)
\(572\) 0 0
\(573\) −17.8564 −0.745962
\(574\) 0 0
\(575\) −28.0000 −1.16768
\(576\) 0 0
\(577\) 34.0000 1.41544 0.707719 0.706494i \(-0.249724\pi\)
0.707719 + 0.706494i \(0.249724\pi\)
\(578\) 0 0
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 5.46410 0.226689
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.92820 −0.286446
\(586\) 0 0
\(587\) 8.39230 0.346387 0.173194 0.984888i \(-0.444591\pi\)
0.173194 + 0.984888i \(0.444591\pi\)
\(588\) 0 0
\(589\) −10.9282 −0.450289
\(590\) 0 0
\(591\) −19.8564 −0.816783
\(592\) 0 0
\(593\) 15.4641 0.635035 0.317517 0.948252i \(-0.397151\pi\)
0.317517 + 0.948252i \(0.397151\pi\)
\(594\) 0 0
\(595\) −12.0000 −0.491952
\(596\) 0 0
\(597\) −5.07180 −0.207575
\(598\) 0 0
\(599\) 15.6077 0.637713 0.318857 0.947803i \(-0.396701\pi\)
0.318857 + 0.947803i \(0.396701\pi\)
\(600\) 0 0
\(601\) 37.7128 1.53834 0.769169 0.639046i \(-0.220670\pi\)
0.769169 + 0.639046i \(0.220670\pi\)
\(602\) 0 0
\(603\) −14.9282 −0.607923
\(604\) 0 0
\(605\) −38.1051 −1.54919
\(606\) 0 0
\(607\) 19.7128 0.800118 0.400059 0.916489i \(-0.368990\pi\)
0.400059 + 0.916489i \(0.368990\pi\)
\(608\) 0 0
\(609\) 0.535898 0.0217157
\(610\) 0 0
\(611\) 18.9282 0.765753
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) −20.7846 −0.838116
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 38.9282 1.56466 0.782328 0.622866i \(-0.214032\pi\)
0.782328 + 0.622866i \(0.214032\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) 0 0
\(623\) −15.8564 −0.635274
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.92820 0.276246
\(630\) 0 0
\(631\) 0.784610 0.0312348 0.0156174 0.999878i \(-0.495029\pi\)
0.0156174 + 0.999878i \(0.495029\pi\)
\(632\) 0 0
\(633\) 25.8564 1.02770
\(634\) 0 0
\(635\) 65.5692 2.60204
\(636\) 0 0
\(637\) −2.00000 −0.0792429
\(638\) 0 0
\(639\) −13.4641 −0.532632
\(640\) 0 0
\(641\) −43.1769 −1.70539 −0.852693 0.522413i \(-0.825032\pi\)
−0.852693 + 0.522413i \(0.825032\pi\)
\(642\) 0 0
\(643\) −3.21539 −0.126803 −0.0634013 0.997988i \(-0.520195\pi\)
−0.0634013 + 0.997988i \(0.520195\pi\)
\(644\) 0 0
\(645\) 24.0000 0.944999
\(646\) 0 0
\(647\) 42.2487 1.66097 0.830484 0.557042i \(-0.188064\pi\)
0.830484 + 0.557042i \(0.188064\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 10.9282 0.428310
\(652\) 0 0
\(653\) −45.7128 −1.78888 −0.894440 0.447187i \(-0.852426\pi\)
−0.894440 + 0.447187i \(0.852426\pi\)
\(654\) 0 0
\(655\) 29.0718 1.13593
\(656\) 0 0
\(657\) −0.928203 −0.0362127
\(658\) 0 0
\(659\) −32.1051 −1.25064 −0.625319 0.780369i \(-0.715031\pi\)
−0.625319 + 0.780369i \(0.715031\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 0 0
\(663\) −6.92820 −0.269069
\(664\) 0 0
\(665\) 3.46410 0.134332
\(666\) 0 0
\(667\) 2.14359 0.0830003
\(668\) 0 0
\(669\) −24.7846 −0.958228
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) −7.00000 −0.269430
\(676\) 0 0
\(677\) 1.21539 0.0467112 0.0233556 0.999727i \(-0.492565\pi\)
0.0233556 + 0.999727i \(0.492565\pi\)
\(678\) 0 0
\(679\) 10.0000 0.383765
\(680\) 0 0
\(681\) −6.92820 −0.265489
\(682\) 0 0
\(683\) −36.3923 −1.39251 −0.696256 0.717793i \(-0.745152\pi\)
−0.696256 + 0.717793i \(0.745152\pi\)
\(684\) 0 0
\(685\) 6.92820 0.264713
\(686\) 0 0
\(687\) 18.7846 0.716678
\(688\) 0 0
\(689\) 6.92820 0.263944
\(690\) 0 0
\(691\) −9.85641 −0.374955 −0.187478 0.982269i \(-0.560031\pi\)
−0.187478 + 0.982269i \(0.560031\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) −20.7846 −0.787273
\(698\) 0 0
\(699\) −4.92820 −0.186402
\(700\) 0 0
\(701\) −7.85641 −0.296732 −0.148366 0.988932i \(-0.547401\pi\)
−0.148366 + 0.988932i \(0.547401\pi\)
\(702\) 0 0
\(703\) −2.00000 −0.0754314
\(704\) 0 0
\(705\) 32.7846 1.23474
\(706\) 0 0
\(707\) −10.3923 −0.390843
\(708\) 0 0
\(709\) −2.00000 −0.0751116 −0.0375558 0.999295i \(-0.511957\pi\)
−0.0375558 + 0.999295i \(0.511957\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 43.7128 1.63706
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.92820 −0.258738
\(718\) 0 0
\(719\) −26.2487 −0.978912 −0.489456 0.872028i \(-0.662805\pi\)
−0.489456 + 0.872028i \(0.662805\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 27.8564 1.03599
\(724\) 0 0
\(725\) −3.75129 −0.139319
\(726\) 0 0
\(727\) −5.85641 −0.217202 −0.108601 0.994085i \(-0.534637\pi\)
−0.108601 + 0.994085i \(0.534637\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 0 0
\(733\) 22.7846 0.841569 0.420784 0.907161i \(-0.361755\pi\)
0.420784 + 0.907161i \(0.361755\pi\)
\(734\) 0 0
\(735\) −3.46410 −0.127775
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 52.7846 1.94171 0.970857 0.239661i \(-0.0770363\pi\)
0.970857 + 0.239661i \(0.0770363\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −7.60770 −0.279099 −0.139550 0.990215i \(-0.544565\pi\)
−0.139550 + 0.990215i \(0.544565\pi\)
\(744\) 0 0
\(745\) −17.0718 −0.625462
\(746\) 0 0
\(747\) 5.46410 0.199921
\(748\) 0 0
\(749\) 12.3923 0.452805
\(750\) 0 0
\(751\) −5.07180 −0.185072 −0.0925362 0.995709i \(-0.529497\pi\)
−0.0925362 + 0.995709i \(0.529497\pi\)
\(752\) 0 0
\(753\) −8.39230 −0.305833
\(754\) 0 0
\(755\) 10.1436 0.369163
\(756\) 0 0
\(757\) −43.5692 −1.58355 −0.791775 0.610813i \(-0.790843\pi\)
−0.791775 + 0.610813i \(0.790843\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11.1769 0.405163 0.202581 0.979265i \(-0.435067\pi\)
0.202581 + 0.979265i \(0.435067\pi\)
\(762\) 0 0
\(763\) −12.9282 −0.468032
\(764\) 0 0
\(765\) −12.0000 −0.433861
\(766\) 0 0
\(767\) 8.00000 0.288863
\(768\) 0 0
\(769\) 51.5692 1.85963 0.929817 0.368023i \(-0.119965\pi\)
0.929817 + 0.368023i \(0.119965\pi\)
\(770\) 0 0
\(771\) −22.7846 −0.820568
\(772\) 0 0
\(773\) −11.8564 −0.426445 −0.213223 0.977004i \(-0.568396\pi\)
−0.213223 + 0.977004i \(0.568396\pi\)
\(774\) 0 0
\(775\) −76.4974 −2.74787
\(776\) 0 0
\(777\) 2.00000 0.0717496
\(778\) 0 0
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.535898 0.0191514
\(784\) 0 0
\(785\) 41.0718 1.46592
\(786\) 0 0
\(787\) −36.7846 −1.31123 −0.655615 0.755095i \(-0.727590\pi\)
−0.655615 + 0.755095i \(0.727590\pi\)
\(788\) 0 0
\(789\) −9.07180 −0.322965
\(790\) 0 0
\(791\) 11.4641 0.407617
\(792\) 0 0
\(793\) −12.0000 −0.426132
\(794\) 0 0
\(795\) 12.0000 0.425596
\(796\) 0 0
\(797\) 13.7128 0.485733 0.242866 0.970060i \(-0.421912\pi\)
0.242866 + 0.970060i \(0.421912\pi\)
\(798\) 0 0
\(799\) 32.7846 1.15984
\(800\) 0 0
\(801\) −15.8564 −0.560259
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −13.8564 −0.488374
\(806\) 0 0
\(807\) 3.07180 0.108132
\(808\) 0 0
\(809\) −11.8564 −0.416849 −0.208425 0.978038i \(-0.566834\pi\)
−0.208425 + 0.978038i \(0.566834\pi\)
\(810\) 0 0
\(811\) 4.00000 0.140459 0.0702295 0.997531i \(-0.477627\pi\)
0.0702295 + 0.997531i \(0.477627\pi\)
\(812\) 0 0
\(813\) −29.8564 −1.04711
\(814\) 0 0
\(815\) 51.7128 1.81142
\(816\) 0 0
\(817\) −6.92820 −0.242387
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −31.0718 −1.08441 −0.542207 0.840245i \(-0.682411\pi\)
−0.542207 + 0.840245i \(0.682411\pi\)
\(822\) 0 0
\(823\) −10.9282 −0.380933 −0.190467 0.981694i \(-0.561000\pi\)
−0.190467 + 0.981694i \(0.561000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15.3205 −0.532746 −0.266373 0.963870i \(-0.585825\pi\)
−0.266373 + 0.963870i \(0.585825\pi\)
\(828\) 0 0
\(829\) −20.1436 −0.699616 −0.349808 0.936821i \(-0.613753\pi\)
−0.349808 + 0.936821i \(0.613753\pi\)
\(830\) 0 0
\(831\) −3.85641 −0.133777
\(832\) 0 0
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) 10.1436 0.351034
\(836\) 0 0
\(837\) 10.9282 0.377734
\(838\) 0 0
\(839\) 10.1436 0.350196 0.175098 0.984551i \(-0.443976\pi\)
0.175098 + 0.984551i \(0.443976\pi\)
\(840\) 0 0
\(841\) −28.7128 −0.990097
\(842\) 0 0
\(843\) −17.3205 −0.596550
\(844\) 0 0
\(845\) −31.1769 −1.07252
\(846\) 0 0
\(847\) −11.0000 −0.377964
\(848\) 0 0
\(849\) −9.07180 −0.311343
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) 0 0
\(853\) −35.5692 −1.21787 −0.608933 0.793221i \(-0.708402\pi\)
−0.608933 + 0.793221i \(0.708402\pi\)
\(854\) 0 0
\(855\) 3.46410 0.118470
\(856\) 0 0
\(857\) −31.8564 −1.08819 −0.544097 0.839022i \(-0.683128\pi\)
−0.544097 + 0.839022i \(0.683128\pi\)
\(858\) 0 0
\(859\) 23.7128 0.809071 0.404535 0.914522i \(-0.367433\pi\)
0.404535 + 0.914522i \(0.367433\pi\)
\(860\) 0 0
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 27.3205 0.930001 0.465000 0.885310i \(-0.346054\pi\)
0.465000 + 0.885310i \(0.346054\pi\)
\(864\) 0 0
\(865\) 6.92820 0.235566
\(866\) 0 0
\(867\) 5.00000 0.169809
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 29.8564 1.01165
\(872\) 0 0
\(873\) 10.0000 0.338449
\(874\) 0 0
\(875\) 6.92820 0.234216
\(876\) 0 0
\(877\) 38.0000 1.28317 0.641584 0.767052i \(-0.278277\pi\)
0.641584 + 0.767052i \(0.278277\pi\)
\(878\) 0 0
\(879\) 22.0000 0.742042
\(880\) 0 0
\(881\) 6.67949 0.225038 0.112519 0.993650i \(-0.464108\pi\)
0.112519 + 0.993650i \(0.464108\pi\)
\(882\) 0 0
\(883\) 6.14359 0.206748 0.103374 0.994643i \(-0.467036\pi\)
0.103374 + 0.994643i \(0.467036\pi\)
\(884\) 0 0
\(885\) 13.8564 0.465778
\(886\) 0 0
\(887\) −46.6410 −1.56605 −0.783026 0.621989i \(-0.786325\pi\)
−0.783026 + 0.621989i \(0.786325\pi\)
\(888\) 0 0
\(889\) 18.9282 0.634832
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −9.46410 −0.316704
\(894\) 0 0
\(895\) −22.6410 −0.756806
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 0 0
\(899\) 5.85641 0.195322
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) 0 0
\(903\) 6.92820 0.230556
\(904\) 0 0
\(905\) −6.92820 −0.230301
\(906\) 0 0
\(907\) 20.0000 0.664089 0.332045 0.943264i \(-0.392262\pi\)
0.332045 + 0.943264i \(0.392262\pi\)
\(908\) 0 0
\(909\) −10.3923 −0.344691
\(910\) 0 0
\(911\) −50.5359 −1.67433 −0.837165 0.546951i \(-0.815788\pi\)
−0.837165 + 0.546951i \(0.815788\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) −20.7846 −0.687118
\(916\) 0 0
\(917\) 8.39230 0.277138
\(918\) 0 0
\(919\) −3.71281 −0.122474 −0.0612372 0.998123i \(-0.519505\pi\)
−0.0612372 + 0.998123i \(0.519505\pi\)
\(920\) 0 0
\(921\) 1.07180 0.0353169
\(922\) 0 0
\(923\) 26.9282 0.886353
\(924\) 0 0
\(925\) −14.0000 −0.460317
\(926\) 0 0
\(927\) −8.00000 −0.262754
\(928\) 0 0
\(929\) 20.5359 0.673761 0.336880 0.941547i \(-0.390628\pi\)
0.336880 + 0.941547i \(0.390628\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 17.4641 0.571749
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 53.7128 1.75472 0.877361 0.479832i \(-0.159302\pi\)
0.877361 + 0.479832i \(0.159302\pi\)
\(938\) 0 0
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 50.7846 1.65553 0.827765 0.561074i \(-0.189612\pi\)
0.827765 + 0.561074i \(0.189612\pi\)
\(942\) 0 0
\(943\) −24.0000 −0.781548
\(944\) 0 0
\(945\) −3.46410 −0.112687
\(946\) 0 0
\(947\) −2.14359 −0.0696574 −0.0348287 0.999393i \(-0.511089\pi\)
−0.0348287 + 0.999393i \(0.511089\pi\)
\(948\) 0 0
\(949\) 1.85641 0.0602615
\(950\) 0 0
\(951\) −7.46410 −0.242040
\(952\) 0 0
\(953\) 22.3923 0.725358 0.362679 0.931914i \(-0.381862\pi\)
0.362679 + 0.931914i \(0.381862\pi\)
\(954\) 0 0
\(955\) 61.8564 2.00163
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 0 0
\(963\) 12.3923 0.399336
\(964\) 0 0
\(965\) −48.4974 −1.56119
\(966\) 0 0
\(967\) 10.9282 0.351427 0.175714 0.984441i \(-0.443777\pi\)
0.175714 + 0.984441i \(0.443777\pi\)
\(968\) 0 0
\(969\) 3.46410 0.111283
\(970\) 0 0
\(971\) −6.14359 −0.197157 −0.0985786 0.995129i \(-0.531430\pi\)
−0.0985786 + 0.995129i \(0.531430\pi\)
\(972\) 0 0
\(973\) 6.92820 0.222108
\(974\) 0 0
\(975\) 14.0000 0.448359
\(976\) 0 0
\(977\) −43.1769 −1.38135 −0.690676 0.723164i \(-0.742687\pi\)
−0.690676 + 0.723164i \(0.742687\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −12.9282 −0.412766
\(982\) 0 0
\(983\) 40.0000 1.27580 0.637901 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 0 0
\(985\) 68.7846 2.19166
\(986\) 0 0
\(987\) 9.46410 0.301246
\(988\) 0 0
\(989\) 27.7128 0.881216
\(990\) 0 0
\(991\) 57.5692 1.82875 0.914373 0.404872i \(-0.132684\pi\)
0.914373 + 0.404872i \(0.132684\pi\)
\(992\) 0 0
\(993\) 14.9282 0.473732
\(994\) 0 0
\(995\) 17.5692 0.556982
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 0 0
\(999\) 2.00000 0.0632772
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 6384.2.a.bk.1.2 2
4.3 odd 2 3192.2.a.t.1.2 2
12.11 even 2 9576.2.a.bk.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3192.2.a.t.1.2 2 4.3 odd 2
6384.2.a.bk.1.2 2 1.1 even 1 trivial
9576.2.a.bk.1.1 2 12.11 even 2