Properties

Label 9216.2.a.z.1.1
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $4$
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] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.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: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.765367\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.02734 q^{5} -4.61313 q^{7} +O(q^{10})\) \(q-4.02734 q^{5} -4.61313 q^{7} +1.53073 q^{11} -5.57900 q^{13} +1.29769 q^{17} -4.35916 q^{19} +4.00000 q^{23} +11.2195 q^{25} +1.86256 q^{29} +2.77791 q^{31} +18.5786 q^{35} +3.97682 q^{37} +3.03188 q^{41} -3.03188 q^{43} +9.65685 q^{47} +14.2809 q^{49} +8.58995 q^{53} -6.16478 q^{55} +5.73418 q^{59} -7.64725 q^{61} +22.4685 q^{65} +8.79565 q^{67} -6.88311 q^{71} -3.50114 q^{73} -7.06147 q^{77} -8.18252 q^{79} +4.12612 q^{83} -5.22625 q^{85} -7.98642 q^{89} +25.7366 q^{91} +17.5558 q^{95} -1.40461 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 8 q^{13} + 16 q^{23} + 4 q^{25} - 8 q^{31} + 16 q^{35} - 8 q^{37} + 16 q^{47} + 4 q^{49} - 16 q^{55} + 16 q^{59} - 24 q^{61} + 8 q^{65} + 16 q^{67} + 16 q^{71} - 8 q^{73} - 16 q^{77} - 24 q^{79} + 8 q^{89} + 16 q^{91} + 32 q^{95} - 16 q^{97}+O(q^{100}) \) 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
\(3\) 0 0
\(4\) 0 0
\(5\) −4.02734 −1.80108 −0.900540 0.434772i \(-0.856829\pi\)
−0.900540 + 0.434772i \(0.856829\pi\)
\(6\) 0 0
\(7\) −4.61313 −1.74360 −0.871799 0.489864i \(-0.837046\pi\)
−0.871799 + 0.489864i \(0.837046\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.53073 0.461534 0.230767 0.973009i \(-0.425877\pi\)
0.230767 + 0.973009i \(0.425877\pi\)
\(12\) 0 0
\(13\) −5.57900 −1.54734 −0.773668 0.633591i \(-0.781580\pi\)
−0.773668 + 0.633591i \(0.781580\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.29769 0.314737 0.157368 0.987540i \(-0.449699\pi\)
0.157368 + 0.987540i \(0.449699\pi\)
\(18\) 0 0
\(19\) −4.35916 −1.00006 −0.500030 0.866008i \(-0.666678\pi\)
−0.500030 + 0.866008i \(0.666678\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 11.2195 2.24389
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.86256 0.345868 0.172934 0.984933i \(-0.444675\pi\)
0.172934 + 0.984933i \(0.444675\pi\)
\(30\) 0 0
\(31\) 2.77791 0.498927 0.249464 0.968384i \(-0.419746\pi\)
0.249464 + 0.968384i \(0.419746\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 18.5786 3.14036
\(36\) 0 0
\(37\) 3.97682 0.653786 0.326893 0.945061i \(-0.393998\pi\)
0.326893 + 0.945061i \(0.393998\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.03188 0.473499 0.236750 0.971571i \(-0.423918\pi\)
0.236750 + 0.971571i \(0.423918\pi\)
\(42\) 0 0
\(43\) −3.03188 −0.462357 −0.231178 0.972911i \(-0.574258\pi\)
−0.231178 + 0.972911i \(0.574258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.65685 1.40860 0.704298 0.709904i \(-0.251262\pi\)
0.704298 + 0.709904i \(0.251262\pi\)
\(48\) 0 0
\(49\) 14.2809 2.04013
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.58995 1.17992 0.589960 0.807432i \(-0.299143\pi\)
0.589960 + 0.807432i \(0.299143\pi\)
\(54\) 0 0
\(55\) −6.16478 −0.831259
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.73418 0.746527 0.373263 0.927725i \(-0.378239\pi\)
0.373263 + 0.927725i \(0.378239\pi\)
\(60\) 0 0
\(61\) −7.64725 −0.979131 −0.489565 0.871967i \(-0.662845\pi\)
−0.489565 + 0.871967i \(0.662845\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 22.4685 2.78688
\(66\) 0 0
\(67\) 8.79565 1.07456 0.537280 0.843404i \(-0.319452\pi\)
0.537280 + 0.843404i \(0.319452\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.88311 −0.816874 −0.408437 0.912786i \(-0.633926\pi\)
−0.408437 + 0.912786i \(0.633926\pi\)
\(72\) 0 0
\(73\) −3.50114 −0.409778 −0.204889 0.978785i \(-0.565683\pi\)
−0.204889 + 0.978785i \(0.565683\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.06147 −0.804729
\(78\) 0 0
\(79\) −8.18252 −0.920606 −0.460303 0.887762i \(-0.652259\pi\)
−0.460303 + 0.887762i \(0.652259\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.12612 0.452901 0.226450 0.974023i \(-0.427288\pi\)
0.226450 + 0.974023i \(0.427288\pi\)
\(84\) 0 0
\(85\) −5.22625 −0.566867
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.98642 −0.846559 −0.423280 0.905999i \(-0.639121\pi\)
−0.423280 + 0.905999i \(0.639121\pi\)
\(90\) 0 0
\(91\) 25.7366 2.69793
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 17.5558 1.80119
\(96\) 0 0
\(97\) −1.40461 −0.142617 −0.0713084 0.997454i \(-0.522717\pi\)
−0.0713084 + 0.997454i \(0.522717\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.457942 0.0455669 0.0227835 0.999740i \(-0.492747\pi\)
0.0227835 + 0.999740i \(0.492747\pi\)
\(102\) 0 0
\(103\) −7.77563 −0.766155 −0.383078 0.923716i \(-0.625136\pi\)
−0.383078 + 0.923716i \(0.625136\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.7047 −1.42156 −0.710781 0.703414i \(-0.751658\pi\)
−0.710781 + 0.703414i \(0.751658\pi\)
\(108\) 0 0
\(109\) −1.09372 −0.104759 −0.0523795 0.998627i \(-0.516681\pi\)
−0.0523795 + 0.998627i \(0.516681\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 7.65685 0.720296 0.360148 0.932895i \(-0.382726\pi\)
0.360148 + 0.932895i \(0.382726\pi\)
\(114\) 0 0
\(115\) −16.1094 −1.50221
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.98642 −0.548775
\(120\) 0 0
\(121\) −8.65685 −0.786987
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −25.0479 −2.24035
\(126\) 0 0
\(127\) 0.900845 0.0799371 0.0399685 0.999201i \(-0.487274\pi\)
0.0399685 + 0.999201i \(0.487274\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −15.7206 −1.37352 −0.686758 0.726886i \(-0.740967\pi\)
−0.686758 + 0.726886i \(0.740967\pi\)
\(132\) 0 0
\(133\) 20.1094 1.74370
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.76377 0.150689 0.0753447 0.997158i \(-0.475994\pi\)
0.0753447 + 0.997158i \(0.475994\pi\)
\(138\) 0 0
\(139\) 18.3752 1.55856 0.779281 0.626675i \(-0.215584\pi\)
0.779281 + 0.626675i \(0.215584\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.53996 −0.714147
\(144\) 0 0
\(145\) −7.50114 −0.622936
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.68419 0.137975 0.0689873 0.997618i \(-0.478023\pi\)
0.0689873 + 0.997618i \(0.478023\pi\)
\(150\) 0 0
\(151\) −10.3118 −0.839165 −0.419582 0.907717i \(-0.637823\pi\)
−0.419582 + 0.907717i \(0.637823\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −11.1876 −0.898609
\(156\) 0 0
\(157\) −13.6337 −1.08809 −0.544043 0.839057i \(-0.683107\pi\)
−0.544043 + 0.839057i \(0.683107\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −18.4525 −1.45426
\(162\) 0 0
\(163\) −3.17476 −0.248666 −0.124333 0.992241i \(-0.539679\pi\)
−0.124333 + 0.992241i \(0.539679\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.4170 1.42515 0.712576 0.701595i \(-0.247528\pi\)
0.712576 + 0.701595i \(0.247528\pi\)
\(168\) 0 0
\(169\) 18.1252 1.39425
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.0341 −0.838909 −0.419455 0.907776i \(-0.637779\pi\)
−0.419455 + 0.907776i \(0.637779\pi\)
\(174\) 0 0
\(175\) −51.7568 −3.91245
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −0.795649 −0.0594696 −0.0297348 0.999558i \(-0.509466\pi\)
−0.0297348 + 0.999558i \(0.509466\pi\)
\(180\) 0 0
\(181\) −5.24943 −0.390187 −0.195093 0.980785i \(-0.562501\pi\)
−0.195093 + 0.980785i \(0.562501\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −16.0160 −1.17752
\(186\) 0 0
\(187\) 1.98642 0.145262
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 25.0060 1.80937 0.904687 0.426077i \(-0.140105\pi\)
0.904687 + 0.426077i \(0.140105\pi\)
\(192\) 0 0
\(193\) 22.9378 1.65110 0.825549 0.564331i \(-0.190866\pi\)
0.825549 + 0.564331i \(0.190866\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −15.0533 −1.07251 −0.536253 0.844058i \(-0.680161\pi\)
−0.536253 + 0.844058i \(0.680161\pi\)
\(198\) 0 0
\(199\) 19.7485 1.39993 0.699966 0.714176i \(-0.253198\pi\)
0.699966 + 0.714176i \(0.253198\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.59220 −0.603054
\(204\) 0 0
\(205\) −12.2104 −0.852811
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.67271 −0.461561
\(210\) 0 0
\(211\) 0.938533 0.0646112 0.0323056 0.999478i \(-0.489715\pi\)
0.0323056 + 0.999478i \(0.489715\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 12.2104 0.832742
\(216\) 0 0
\(217\) −12.8149 −0.869929
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −7.23983 −0.487004
\(222\) 0 0
\(223\) 26.2783 1.75973 0.879863 0.475228i \(-0.157634\pi\)
0.879863 + 0.475228i \(0.157634\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.9082 0.724002 0.362001 0.932178i \(-0.382094\pi\)
0.362001 + 0.932178i \(0.382094\pi\)
\(228\) 0 0
\(229\) −18.8599 −1.24630 −0.623150 0.782103i \(-0.714147\pi\)
−0.623150 + 0.782103i \(0.714147\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −19.7662 −1.29493 −0.647464 0.762096i \(-0.724170\pi\)
−0.647464 + 0.762096i \(0.724170\pi\)
\(234\) 0 0
\(235\) −38.8914 −2.53700
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −14.3515 −0.928319 −0.464160 0.885752i \(-0.653644\pi\)
−0.464160 + 0.885752i \(0.653644\pi\)
\(240\) 0 0
\(241\) 21.7534 1.40126 0.700629 0.713525i \(-0.252903\pi\)
0.700629 + 0.713525i \(0.252903\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −57.5142 −3.67444
\(246\) 0 0
\(247\) 24.3197 1.54743
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −9.53073 −0.601575 −0.300787 0.953691i \(-0.597249\pi\)
−0.300787 + 0.953691i \(0.597249\pi\)
\(252\) 0 0
\(253\) 6.12293 0.384946
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.20435 0.0751253 0.0375627 0.999294i \(-0.488041\pi\)
0.0375627 + 0.999294i \(0.488041\pi\)
\(258\) 0 0
\(259\) −18.3456 −1.13994
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 27.5777 1.70052 0.850258 0.526367i \(-0.176446\pi\)
0.850258 + 0.526367i \(0.176446\pi\)
\(264\) 0 0
\(265\) −34.5946 −2.12513
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.6646 −0.894115 −0.447057 0.894505i \(-0.647528\pi\)
−0.447057 + 0.894505i \(0.647528\pi\)
\(270\) 0 0
\(271\) 20.3574 1.23663 0.618313 0.785932i \(-0.287816\pi\)
0.618313 + 0.785932i \(0.287816\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 17.1740 1.03563
\(276\) 0 0
\(277\) 23.8113 1.43068 0.715341 0.698776i \(-0.246271\pi\)
0.715341 + 0.698776i \(0.246271\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 17.1116 1.02079 0.510397 0.859939i \(-0.329498\pi\)
0.510397 + 0.859939i \(0.329498\pi\)
\(282\) 0 0
\(283\) −2.26582 −0.134689 −0.0673444 0.997730i \(-0.521453\pi\)
−0.0673444 + 0.997730i \(0.521453\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −13.9864 −0.825592
\(288\) 0 0
\(289\) −15.3160 −0.900941
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.90366 0.228054 0.114027 0.993478i \(-0.463625\pi\)
0.114027 + 0.993478i \(0.463625\pi\)
\(294\) 0 0
\(295\) −23.0935 −1.34456
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −22.3160 −1.29057
\(300\) 0 0
\(301\) 13.9864 0.806164
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 30.7981 1.76349
\(306\) 0 0
\(307\) −26.2323 −1.49716 −0.748578 0.663047i \(-0.769263\pi\)
−0.748578 + 0.663047i \(0.769263\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.12522 0.517444 0.258722 0.965952i \(-0.416699\pi\)
0.258722 + 0.965952i \(0.416699\pi\)
\(312\) 0 0
\(313\) −9.28093 −0.524589 −0.262295 0.964988i \(-0.584479\pi\)
−0.262295 + 0.964988i \(0.584479\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.3652 1.81781 0.908906 0.417000i \(-0.136919\pi\)
0.908906 + 0.417000i \(0.136919\pi\)
\(318\) 0 0
\(319\) 2.85108 0.159630
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.65685 −0.314756
\(324\) 0 0
\(325\) −62.5934 −3.47206
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −44.5483 −2.45603
\(330\) 0 0
\(331\) −21.8435 −1.20063 −0.600315 0.799764i \(-0.704958\pi\)
−0.600315 + 0.799764i \(0.704958\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −35.4231 −1.93537
\(336\) 0 0
\(337\) −28.8722 −1.57277 −0.786385 0.617736i \(-0.788050\pi\)
−0.786385 + 0.617736i \(0.788050\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4.25224 0.230272
\(342\) 0 0
\(343\) −33.5879 −1.81357
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 21.9697 1.17939 0.589697 0.807625i \(-0.299247\pi\)
0.589697 + 0.807625i \(0.299247\pi\)
\(348\) 0 0
\(349\) 20.0862 1.07519 0.537594 0.843204i \(-0.319333\pi\)
0.537594 + 0.843204i \(0.319333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.9570 1.96702 0.983511 0.180849i \(-0.0578844\pi\)
0.983511 + 0.180849i \(0.0578844\pi\)
\(354\) 0 0
\(355\) 27.7206 1.47126
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 22.5264 1.18890 0.594449 0.804134i \(-0.297370\pi\)
0.594449 + 0.804134i \(0.297370\pi\)
\(360\) 0 0
\(361\) 0.00228335 0.000120176 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.1003 0.738043
\(366\) 0 0
\(367\) −0.535798 −0.0279684 −0.0139842 0.999902i \(-0.504451\pi\)
−0.0139842 + 0.999902i \(0.504451\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −39.6265 −2.05731
\(372\) 0 0
\(373\) −8.83803 −0.457616 −0.228808 0.973472i \(-0.573483\pi\)
−0.228808 + 0.973472i \(0.573483\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −10.3912 −0.535174
\(378\) 0 0
\(379\) 24.6887 1.26817 0.634087 0.773261i \(-0.281376\pi\)
0.634087 + 0.773261i \(0.281376\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −31.1290 −1.59062 −0.795308 0.606205i \(-0.792691\pi\)
−0.795308 + 0.606205i \(0.792691\pi\)
\(384\) 0 0
\(385\) 28.4389 1.44938
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −20.3614 −1.03236 −0.516182 0.856479i \(-0.672647\pi\)
−0.516182 + 0.856479i \(0.672647\pi\)
\(390\) 0 0
\(391\) 5.19077 0.262509
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 32.9538 1.65809
\(396\) 0 0
\(397\) 3.60090 0.180724 0.0903620 0.995909i \(-0.471198\pi\)
0.0903620 + 0.995909i \(0.471198\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.0160 0.899677 0.449838 0.893110i \(-0.351482\pi\)
0.449838 + 0.893110i \(0.351482\pi\)
\(402\) 0 0
\(403\) −15.4980 −0.772008
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.08746 0.301744
\(408\) 0 0
\(409\) −25.1572 −1.24395 −0.621973 0.783039i \(-0.713669\pi\)
−0.621973 + 0.783039i \(0.713669\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −26.4525 −1.30164
\(414\) 0 0
\(415\) −16.6173 −0.815711
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6.52845 0.318936 0.159468 0.987203i \(-0.449022\pi\)
0.159468 + 0.987203i \(0.449022\pi\)
\(420\) 0 0
\(421\) −19.8758 −0.968687 −0.484343 0.874878i \(-0.660941\pi\)
−0.484343 + 0.874878i \(0.660941\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 14.5594 0.706236
\(426\) 0 0
\(427\) 35.2777 1.70721
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 34.7368 1.67321 0.836606 0.547805i \(-0.184537\pi\)
0.836606 + 0.547805i \(0.184537\pi\)
\(432\) 0 0
\(433\) −27.0935 −1.30203 −0.651015 0.759065i \(-0.725657\pi\)
−0.651015 + 0.759065i \(0.725657\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −17.4366 −0.834108
\(438\) 0 0
\(439\) 3.52976 0.168466 0.0842331 0.996446i \(-0.473156\pi\)
0.0842331 + 0.996446i \(0.473156\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.71742 −0.271643 −0.135821 0.990733i \(-0.543367\pi\)
−0.135821 + 0.990733i \(0.543367\pi\)
\(444\) 0 0
\(445\) 32.1640 1.52472
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.9346 −1.36551 −0.682754 0.730648i \(-0.739218\pi\)
−0.682754 + 0.730648i \(0.739218\pi\)
\(450\) 0 0
\(451\) 4.64099 0.218536
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −103.650 −4.85919
\(456\) 0 0
\(457\) −33.6233 −1.57283 −0.786416 0.617697i \(-0.788066\pi\)
−0.786416 + 0.617697i \(0.788066\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5.27730 −0.245788 −0.122894 0.992420i \(-0.539218\pi\)
−0.122894 + 0.992420i \(0.539218\pi\)
\(462\) 0 0
\(463\) −11.1430 −0.517857 −0.258929 0.965896i \(-0.583369\pi\)
−0.258929 + 0.965896i \(0.583369\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −26.5013 −1.22633 −0.613167 0.789953i \(-0.710105\pi\)
−0.613167 + 0.789953i \(0.710105\pi\)
\(468\) 0 0
\(469\) −40.5754 −1.87360
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.64099 −0.213393
\(474\) 0 0
\(475\) −48.9074 −2.24403
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −22.4170 −1.02426 −0.512130 0.858908i \(-0.671143\pi\)
−0.512130 + 0.858908i \(0.671143\pi\)
\(480\) 0 0
\(481\) −22.1867 −1.01163
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.65685 0.256865
\(486\) 0 0
\(487\) 7.42235 0.336339 0.168169 0.985758i \(-0.446214\pi\)
0.168169 + 0.985758i \(0.446214\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.3711 −0.783946 −0.391973 0.919977i \(-0.628207\pi\)
−0.391973 + 0.919977i \(0.628207\pi\)
\(492\) 0 0
\(493\) 2.41703 0.108857
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 31.7526 1.42430
\(498\) 0 0
\(499\) 37.2938 1.66950 0.834749 0.550631i \(-0.185613\pi\)
0.834749 + 0.550631i \(0.185613\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −13.8672 −0.618310 −0.309155 0.951012i \(-0.600046\pi\)
−0.309155 + 0.951012i \(0.600046\pi\)
\(504\) 0 0
\(505\) −1.84429 −0.0820697
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −34.6311 −1.53499 −0.767497 0.641052i \(-0.778498\pi\)
−0.767497 + 0.641052i \(0.778498\pi\)
\(510\) 0 0
\(511\) 16.1512 0.714487
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 31.3151 1.37991
\(516\) 0 0
\(517\) 14.7821 0.650115
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −34.8437 −1.52653 −0.763265 0.646086i \(-0.776405\pi\)
−0.763265 + 0.646086i \(0.776405\pi\)
\(522\) 0 0
\(523\) −25.5499 −1.11722 −0.558610 0.829430i \(-0.688665\pi\)
−0.558610 + 0.829430i \(0.688665\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.60488 0.157031
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.9148 −0.732662
\(534\) 0 0
\(535\) 59.2210 2.56035
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 21.8603 0.941590
\(540\) 0 0
\(541\) −32.8735 −1.41334 −0.706671 0.707542i \(-0.749804\pi\)
−0.706671 + 0.707542i \(0.749804\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 4.40477 0.188680
\(546\) 0 0
\(547\) 32.4414 1.38709 0.693546 0.720412i \(-0.256047\pi\)
0.693546 + 0.720412i \(0.256047\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −8.11918 −0.345889
\(552\) 0 0
\(553\) 37.7470 1.60517
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.25025 −0.0953462 −0.0476731 0.998863i \(-0.515181\pi\)
−0.0476731 + 0.998863i \(0.515181\pi\)
\(558\) 0 0
\(559\) 16.9148 0.715421
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.37608 −0.0579948 −0.0289974 0.999579i \(-0.509231\pi\)
−0.0289974 + 0.999579i \(0.509231\pi\)
\(564\) 0 0
\(565\) −30.8368 −1.29731
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −15.5043 −0.649975 −0.324988 0.945718i \(-0.605360\pi\)
−0.324988 + 0.945718i \(0.605360\pi\)
\(570\) 0 0
\(571\) −7.45659 −0.312049 −0.156024 0.987753i \(-0.549868\pi\)
−0.156024 + 0.987753i \(0.549868\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 44.8779 1.87154
\(576\) 0 0
\(577\) −25.9355 −1.07971 −0.539855 0.841758i \(-0.681521\pi\)
−0.539855 + 0.841758i \(0.681521\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −19.0343 −0.789676
\(582\) 0 0
\(583\) 13.1489 0.544573
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 33.6233 1.38778 0.693892 0.720079i \(-0.255895\pi\)
0.693892 + 0.720079i \(0.255895\pi\)
\(588\) 0 0
\(589\) −12.1094 −0.498957
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −44.0302 −1.80810 −0.904052 0.427422i \(-0.859422\pi\)
−0.904052 + 0.427422i \(0.859422\pi\)
\(594\) 0 0
\(595\) 24.1094 0.988387
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.09578 0.249067 0.124533 0.992215i \(-0.460257\pi\)
0.124533 + 0.992215i \(0.460257\pi\)
\(600\) 0 0
\(601\) −11.8126 −0.481845 −0.240922 0.970544i \(-0.577450\pi\)
−0.240922 + 0.970544i \(0.577450\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.8641 1.41743
\(606\) 0 0
\(607\) −4.66330 −0.189277 −0.0946387 0.995512i \(-0.530170\pi\)
−0.0946387 + 0.995512i \(0.530170\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −53.8756 −2.17957
\(612\) 0 0
\(613\) 28.5715 1.15399 0.576995 0.816748i \(-0.304225\pi\)
0.576995 + 0.816748i \(0.304225\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21.2482 0.855418 0.427709 0.903916i \(-0.359321\pi\)
0.427709 + 0.903916i \(0.359321\pi\)
\(618\) 0 0
\(619\) 11.4412 0.459861 0.229931 0.973207i \(-0.426150\pi\)
0.229931 + 0.973207i \(0.426150\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 36.8424 1.47606
\(624\) 0 0
\(625\) 44.7790 1.79116
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.16070 0.205770
\(630\) 0 0
\(631\) 16.9427 0.674478 0.337239 0.941419i \(-0.390507\pi\)
0.337239 + 0.941419i \(0.390507\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −3.62801 −0.143973
\(636\) 0 0
\(637\) −79.6733 −3.15677
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 27.8596 1.10039 0.550193 0.835037i \(-0.314554\pi\)
0.550193 + 0.835037i \(0.314554\pi\)
\(642\) 0 0
\(643\) 20.6114 0.812834 0.406417 0.913688i \(-0.366778\pi\)
0.406417 + 0.913688i \(0.366778\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.7142 0.460534 0.230267 0.973127i \(-0.426040\pi\)
0.230267 + 0.973127i \(0.426040\pi\)
\(648\) 0 0
\(649\) 8.77751 0.344547
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −51.0435 −1.99749 −0.998743 0.0501153i \(-0.984041\pi\)
−0.998743 + 0.0501153i \(0.984041\pi\)
\(654\) 0 0
\(655\) 63.3122 2.47381
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.68401 0.221418 0.110709 0.993853i \(-0.464688\pi\)
0.110709 + 0.993853i \(0.464688\pi\)
\(660\) 0 0
\(661\) 36.2611 1.41039 0.705197 0.709012i \(-0.250859\pi\)
0.705197 + 0.709012i \(0.250859\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −80.9872 −3.14055
\(666\) 0 0
\(667\) 7.45022 0.288474
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −11.7059 −0.451902
\(672\) 0 0
\(673\) 17.6569 0.680622 0.340311 0.940313i \(-0.389468\pi\)
0.340311 + 0.940313i \(0.389468\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0866 −0.695124 −0.347562 0.937657i \(-0.612990\pi\)
−0.347562 + 0.937657i \(0.612990\pi\)
\(678\) 0 0
\(679\) 6.47966 0.248666
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 20.3448 0.778474 0.389237 0.921138i \(-0.372739\pi\)
0.389237 + 0.921138i \(0.372739\pi\)
\(684\) 0 0
\(685\) −7.10332 −0.271404
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −47.9233 −1.82573
\(690\) 0 0
\(691\) 20.1118 0.765089 0.382544 0.923937i \(-0.375048\pi\)
0.382544 + 0.923937i \(0.375048\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −74.0031 −2.80710
\(696\) 0 0
\(697\) 3.93444 0.149028
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −19.9263 −0.752606 −0.376303 0.926497i \(-0.622805\pi\)
−0.376303 + 0.926497i \(0.622805\pi\)
\(702\) 0 0
\(703\) −17.3356 −0.653825
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.11254 −0.0794504
\(708\) 0 0
\(709\) −3.30141 −0.123987 −0.0619935 0.998077i \(-0.519746\pi\)
−0.0619935 + 0.998077i \(0.519746\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 11.1116 0.416134
\(714\) 0 0
\(715\) 34.3933 1.28624
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −40.2104 −1.49959 −0.749797 0.661668i \(-0.769849\pi\)
−0.749797 + 0.661668i \(0.769849\pi\)
\(720\) 0 0
\(721\) 35.8699 1.33587
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 20.8969 0.776090
\(726\) 0 0
\(727\) −39.4272 −1.46227 −0.731137 0.682230i \(-0.761010\pi\)
−0.731137 + 0.682230i \(0.761010\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3.93444 −0.145521
\(732\) 0 0
\(733\) 20.7370 0.765938 0.382969 0.923761i \(-0.374902\pi\)
0.382969 + 0.923761i \(0.374902\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13.4638 0.495945
\(738\) 0 0
\(739\) −13.9529 −0.513266 −0.256633 0.966509i \(-0.582613\pi\)
−0.256633 + 0.966509i \(0.582613\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5.02418 −0.184319 −0.0921597 0.995744i \(-0.529377\pi\)
−0.0921597 + 0.995744i \(0.529377\pi\)
\(744\) 0 0
\(745\) −6.78282 −0.248503
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 67.8348 2.47863
\(750\) 0 0
\(751\) 5.44425 0.198664 0.0993318 0.995054i \(-0.468329\pi\)
0.0993318 + 0.995054i \(0.468329\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 41.5292 1.51140
\(756\) 0 0
\(757\) −14.7698 −0.536816 −0.268408 0.963305i \(-0.586498\pi\)
−0.268408 + 0.963305i \(0.586498\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.96584 0.0712617 0.0356308 0.999365i \(-0.488656\pi\)
0.0356308 + 0.999365i \(0.488656\pi\)
\(762\) 0 0
\(763\) 5.04545 0.182658
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −31.9910 −1.15513
\(768\) 0 0
\(769\) −7.09244 −0.255760 −0.127880 0.991790i \(-0.540817\pi\)
−0.127880 + 0.991790i \(0.540817\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 27.5696 0.991609 0.495804 0.868434i \(-0.334873\pi\)
0.495804 + 0.868434i \(0.334873\pi\)
\(774\) 0 0
\(775\) 31.1667 1.11954
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.2164 −0.473528
\(780\) 0 0
\(781\) −10.5362 −0.377015
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 54.9074 1.95973
\(786\) 0 0
\(787\) 29.6529 1.05701 0.528506 0.848929i \(-0.322752\pi\)
0.528506 + 0.848929i \(0.322752\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −35.3220 −1.25591
\(792\) 0 0
\(793\) 42.6640 1.51504
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.43832 −0.192635 −0.0963177 0.995351i \(-0.530706\pi\)
−0.0963177 + 0.995351i \(0.530706\pi\)
\(798\) 0 0
\(799\) 12.5316 0.443337
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.35932 −0.189126
\(804\) 0 0
\(805\) 74.3145 2.61924
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −17.1548 −0.603131 −0.301566 0.953445i \(-0.597509\pi\)
−0.301566 + 0.953445i \(0.597509\pi\)
\(810\) 0 0
\(811\) −9.46624 −0.332404 −0.166202 0.986092i \(-0.553150\pi\)
−0.166202 + 0.986092i \(0.553150\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 12.7858 0.447868
\(816\) 0 0
\(817\) 13.2164 0.462384
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 19.1198 0.667285 0.333642 0.942700i \(-0.391722\pi\)
0.333642 + 0.942700i \(0.391722\pi\)
\(822\) 0 0
\(823\) −10.5257 −0.366902 −0.183451 0.983029i \(-0.558727\pi\)
−0.183451 + 0.983029i \(0.558727\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.9841 0.938330 0.469165 0.883110i \(-0.344555\pi\)
0.469165 + 0.883110i \(0.344555\pi\)
\(828\) 0 0
\(829\) 12.4704 0.433116 0.216558 0.976270i \(-0.430517\pi\)
0.216558 + 0.976270i \(0.430517\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 18.5323 0.642105
\(834\) 0 0
\(835\) −74.1716 −2.56681
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 29.4631 1.01718 0.508590 0.861009i \(-0.330167\pi\)
0.508590 + 0.861009i \(0.330167\pi\)
\(840\) 0 0
\(841\) −25.5309 −0.880375
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −72.9964 −2.51115
\(846\) 0 0
\(847\) 39.9352 1.37219
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.9073 0.545295
\(852\) 0 0
\(853\) −16.1955 −0.554525 −0.277262 0.960794i \(-0.589427\pi\)
−0.277262 + 0.960794i \(0.589427\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 28.3592 0.968730 0.484365 0.874866i \(-0.339051\pi\)
0.484365 + 0.874866i \(0.339051\pi\)
\(858\) 0 0
\(859\) 2.51562 0.0858319 0.0429159 0.999079i \(-0.486335\pi\)
0.0429159 + 0.999079i \(0.486335\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 13.8944 0.472971 0.236485 0.971635i \(-0.424004\pi\)
0.236485 + 0.971635i \(0.424004\pi\)
\(864\) 0 0
\(865\) 44.4382 1.51094
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −12.5253 −0.424891
\(870\) 0 0
\(871\) −49.0709 −1.66270
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 115.549 3.90627
\(876\) 0 0
\(877\) 26.7216 0.902323 0.451161 0.892442i \(-0.351010\pi\)
0.451161 + 0.892442i \(0.351010\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −11.5460 −0.388995 −0.194497 0.980903i \(-0.562308\pi\)
−0.194497 + 0.980903i \(0.562308\pi\)
\(882\) 0 0
\(883\) 53.1868 1.78988 0.894940 0.446187i \(-0.147218\pi\)
0.894940 + 0.446187i \(0.147218\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 44.8123 1.50465 0.752325 0.658792i \(-0.228932\pi\)
0.752325 + 0.658792i \(0.228932\pi\)
\(888\) 0 0
\(889\) −4.15571 −0.139378
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −42.0958 −1.40868
\(894\) 0 0
\(895\) 3.20435 0.107110
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.17401 0.172563
\(900\) 0 0
\(901\) 11.1471 0.371364
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21.1412 0.702758
\(906\) 0 0
\(907\) −29.3297 −0.973877 −0.486939 0.873436i \(-0.661887\pi\)
−0.486939 + 0.873436i \(0.661887\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.91483 0.162835 0.0814177 0.996680i \(-0.474055\pi\)
0.0814177 + 0.996680i \(0.474055\pi\)
\(912\) 0 0
\(913\) 6.31599 0.209029
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 72.5211 2.39486
\(918\) 0 0
\(919\) −18.9465 −0.624986 −0.312493 0.949920i \(-0.601164\pi\)
−0.312493 + 0.949920i \(0.601164\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 38.4008 1.26398
\(924\) 0 0
\(925\) 44.6178 1.46702
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.40296 0.275692 0.137846 0.990454i \(-0.455982\pi\)
0.137846 + 0.990454i \(0.455982\pi\)
\(930\) 0 0
\(931\) −62.2529 −2.04026
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −43.8100 −1.43121 −0.715605 0.698505i \(-0.753849\pi\)
−0.715605 + 0.698505i \(0.753849\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 16.5363 0.539069 0.269534 0.962991i \(-0.413130\pi\)
0.269534 + 0.962991i \(0.413130\pi\)
\(942\) 0 0
\(943\) 12.1275 0.394926
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 46.7639 1.51962 0.759812 0.650143i \(-0.225291\pi\)
0.759812 + 0.650143i \(0.225291\pi\)
\(948\) 0 0
\(949\) 19.5329 0.634064
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 43.4436 1.40728 0.703639 0.710558i \(-0.251557\pi\)
0.703639 + 0.710558i \(0.251557\pi\)
\(954\) 0 0
\(955\) −100.708 −3.25883
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −8.13651 −0.262742
\(960\) 0 0
\(961\) −23.2832 −0.751071
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −92.3782 −2.97376
\(966\) 0 0
\(967\) 41.3718 1.33043 0.665214 0.746653i \(-0.268340\pi\)
0.665214 + 0.746653i \(0.268340\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −55.1541 −1.76998 −0.884989 0.465612i \(-0.845834\pi\)
−0.884989 + 0.465612i \(0.845834\pi\)
\(972\) 0 0
\(973\) −84.7670 −2.71751
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.9120 1.50085 0.750424 0.660957i \(-0.229849\pi\)
0.750424 + 0.660957i \(0.229849\pi\)
\(978\) 0 0
\(979\) −12.2251 −0.390715
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.71195 −0.150288 −0.0751439 0.997173i \(-0.523942\pi\)
−0.0751439 + 0.997173i \(0.523942\pi\)
\(984\) 0 0
\(985\) 60.6249 1.93167
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.1275 −0.385632
\(990\) 0 0
\(991\) 13.3733 0.424817 0.212408 0.977181i \(-0.431869\pi\)
0.212408 + 0.977181i \(0.431869\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −79.5338 −2.52139
\(996\) 0 0
\(997\) −24.8445 −0.786833 −0.393416 0.919360i \(-0.628707\pi\)
−0.393416 + 0.919360i \(0.628707\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9216.2.a.z.1.1 4
3.2 odd 2 3072.2.a.p.1.4 4
4.3 odd 2 9216.2.a.bl.1.1 4
8.3 odd 2 9216.2.a.bm.1.4 4
8.5 even 2 9216.2.a.ba.1.4 4
12.11 even 2 3072.2.a.m.1.4 4
24.5 odd 2 3072.2.a.j.1.1 4
24.11 even 2 3072.2.a.s.1.1 4
32.3 odd 8 4608.2.k.bh.1153.4 8
32.5 even 8 4608.2.k.bc.3457.1 8
32.11 odd 8 4608.2.k.bh.3457.4 8
32.13 even 8 4608.2.k.bc.1153.1 8
32.19 odd 8 4608.2.k.be.1153.1 8
32.21 even 8 4608.2.k.bj.3457.4 8
32.27 odd 8 4608.2.k.be.3457.1 8
32.29 even 8 4608.2.k.bj.1153.4 8
48.5 odd 4 3072.2.d.j.1537.4 8
48.11 even 4 3072.2.d.e.1537.8 8
48.29 odd 4 3072.2.d.j.1537.5 8
48.35 even 4 3072.2.d.e.1537.1 8
96.5 odd 8 1536.2.j.i.385.2 yes 8
96.11 even 8 1536.2.j.e.385.1 8
96.29 odd 8 1536.2.j.f.1153.3 yes 8
96.35 even 8 1536.2.j.e.1153.1 yes 8
96.53 odd 8 1536.2.j.f.385.3 yes 8
96.59 even 8 1536.2.j.j.385.4 yes 8
96.77 odd 8 1536.2.j.i.1153.2 yes 8
96.83 even 8 1536.2.j.j.1153.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.j.e.385.1 8 96.11 even 8
1536.2.j.e.1153.1 yes 8 96.35 even 8
1536.2.j.f.385.3 yes 8 96.53 odd 8
1536.2.j.f.1153.3 yes 8 96.29 odd 8
1536.2.j.i.385.2 yes 8 96.5 odd 8
1536.2.j.i.1153.2 yes 8 96.77 odd 8
1536.2.j.j.385.4 yes 8 96.59 even 8
1536.2.j.j.1153.4 yes 8 96.83 even 8
3072.2.a.j.1.1 4 24.5 odd 2
3072.2.a.m.1.4 4 12.11 even 2
3072.2.a.p.1.4 4 3.2 odd 2
3072.2.a.s.1.1 4 24.11 even 2
3072.2.d.e.1537.1 8 48.35 even 4
3072.2.d.e.1537.8 8 48.11 even 4
3072.2.d.j.1537.4 8 48.5 odd 4
3072.2.d.j.1537.5 8 48.29 odd 4
4608.2.k.bc.1153.1 8 32.13 even 8
4608.2.k.bc.3457.1 8 32.5 even 8
4608.2.k.be.1153.1 8 32.19 odd 8
4608.2.k.be.3457.1 8 32.27 odd 8
4608.2.k.bh.1153.4 8 32.3 odd 8
4608.2.k.bh.3457.4 8 32.11 odd 8
4608.2.k.bj.1153.4 8 32.29 even 8
4608.2.k.bj.3457.4 8 32.21 even 8
9216.2.a.z.1.1 4 1.1 even 1 trivial
9216.2.a.ba.1.4 4 8.5 even 2
9216.2.a.bl.1.1 4 4.3 odd 2
9216.2.a.bm.1.4 4 8.3 odd 2