Properties

Label 9216.2.a.bi.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 $2$

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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: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 768)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.93185\) 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)
 
\(f(q)\) \(=\) \(q-3.86370 q^{5} +2.44949 q^{7} +O(q^{10})\) \(q-3.86370 q^{5} +2.44949 q^{7} -1.46410 q^{11} +4.24264 q^{13} +3.46410 q^{17} -7.46410 q^{19} -2.82843 q^{23} +9.92820 q^{25} -8.76268 q^{29} +7.34847 q^{31} -9.46410 q^{35} +0.656339 q^{37} -4.53590 q^{41} +3.46410 q^{43} +2.82843 q^{47} -1.00000 q^{49} +4.62158 q^{53} +5.65685 q^{55} +13.8564 q^{59} -0.656339 q^{61} -16.3923 q^{65} -14.9282 q^{67} -6.41473 q^{71} +4.00000 q^{73} -3.58630 q^{77} -2.44949 q^{79} +5.46410 q^{83} -13.3843 q^{85} +4.92820 q^{89} +10.3923 q^{91} +28.8391 q^{95} +1.07180 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{11} - 16 q^{19} + 12 q^{25} - 24 q^{35} - 32 q^{41} - 4 q^{49} - 24 q^{65} - 32 q^{67} + 16 q^{73} + 8 q^{83} - 8 q^{89} + 32 q^{97}+O(q^{100}) \) 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
\(5\) −3.86370 −1.72790 −0.863950 0.503577i \(-0.832017\pi\)
−0.863950 + 0.503577i \(0.832017\pi\)
\(6\) 0 0
\(7\) 2.44949 0.925820 0.462910 0.886405i \(-0.346805\pi\)
0.462910 + 0.886405i \(0.346805\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.46410 −0.441443 −0.220722 0.975337i \(-0.570841\pi\)
−0.220722 + 0.975337i \(0.570841\pi\)
\(12\) 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
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −7.46410 −1.71238 −0.856191 0.516659i \(-0.827175\pi\)
−0.856191 + 0.516659i \(0.827175\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 9.92820 1.98564
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.76268 −1.62719 −0.813595 0.581432i \(-0.802492\pi\)
−0.813595 + 0.581432i \(0.802492\pi\)
\(30\) 0 0
\(31\) 7.34847 1.31982 0.659912 0.751343i \(-0.270594\pi\)
0.659912 + 0.751343i \(0.270594\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −9.46410 −1.59973
\(36\) 0 0
\(37\) 0.656339 0.107901 0.0539507 0.998544i \(-0.482819\pi\)
0.0539507 + 0.998544i \(0.482819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.53590 −0.708388 −0.354194 0.935172i \(-0.615245\pi\)
−0.354194 + 0.935172i \(0.615245\pi\)
\(42\) 0 0
\(43\) 3.46410 0.528271 0.264135 0.964486i \(-0.414913\pi\)
0.264135 + 0.964486i \(0.414913\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.62158 0.634823 0.317411 0.948288i \(-0.397186\pi\)
0.317411 + 0.948288i \(0.397186\pi\)
\(54\) 0 0
\(55\) 5.65685 0.762770
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 13.8564 1.80395 0.901975 0.431788i \(-0.142117\pi\)
0.901975 + 0.431788i \(0.142117\pi\)
\(60\) 0 0
\(61\) −0.656339 −0.0840356 −0.0420178 0.999117i \(-0.513379\pi\)
−0.0420178 + 0.999117i \(0.513379\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −16.3923 −2.03322
\(66\) 0 0
\(67\) −14.9282 −1.82377 −0.911885 0.410445i \(-0.865373\pi\)
−0.911885 + 0.410445i \(0.865373\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.41473 −0.761288 −0.380644 0.924722i \(-0.624298\pi\)
−0.380644 + 0.924722i \(0.624298\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.58630 −0.408697
\(78\) 0 0
\(79\) −2.44949 −0.275589 −0.137795 0.990461i \(-0.544001\pi\)
−0.137795 + 0.990461i \(0.544001\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.46410 0.599763 0.299882 0.953976i \(-0.403053\pi\)
0.299882 + 0.953976i \(0.403053\pi\)
\(84\) 0 0
\(85\) −13.3843 −1.45173
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.92820 0.522388 0.261194 0.965286i \(-0.415884\pi\)
0.261194 + 0.965286i \(0.415884\pi\)
\(90\) 0 0
\(91\) 10.3923 1.08941
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 28.8391 2.95883
\(96\) 0 0
\(97\) 1.07180 0.108824 0.0544122 0.998519i \(-0.482671\pi\)
0.0544122 + 0.998519i \(0.482671\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.4195 1.43480 0.717399 0.696663i \(-0.245333\pi\)
0.717399 + 0.696663i \(0.245333\pi\)
\(102\) 0 0
\(103\) −3.20736 −0.316031 −0.158016 0.987437i \(-0.550510\pi\)
−0.158016 + 0.987437i \(0.550510\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.00000 −0.386695 −0.193347 0.981130i \(-0.561934\pi\)
−0.193347 + 0.981130i \(0.561934\pi\)
\(108\) 0 0
\(109\) 1.41421 0.135457 0.0677285 0.997704i \(-0.478425\pi\)
0.0677285 + 0.997704i \(0.478425\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 10.9282 1.01906
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.48528 0.777844
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −19.0411 −1.70309
\(126\) 0 0
\(127\) 14.5211 1.28854 0.644268 0.764799i \(-0.277162\pi\)
0.644268 + 0.764799i \(0.277162\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −14.9282 −1.30428 −0.652142 0.758097i \(-0.726129\pi\)
−0.652142 + 0.758097i \(0.726129\pi\)
\(132\) 0 0
\(133\) −18.2832 −1.58536
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.3923 −1.22962 −0.614809 0.788676i \(-0.710767\pi\)
−0.614809 + 0.788676i \(0.710767\pi\)
\(138\) 0 0
\(139\) −6.92820 −0.587643 −0.293821 0.955860i \(-0.594927\pi\)
−0.293821 + 0.955860i \(0.594927\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −6.21166 −0.519445
\(144\) 0 0
\(145\) 33.8564 2.81162
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 5.37945 0.440702 0.220351 0.975421i \(-0.429280\pi\)
0.220351 + 0.975421i \(0.429280\pi\)
\(150\) 0 0
\(151\) 6.59059 0.536335 0.268167 0.963372i \(-0.413582\pi\)
0.268167 + 0.963372i \(0.413582\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −28.3923 −2.28052
\(156\) 0 0
\(157\) 2.17209 0.173352 0.0866758 0.996237i \(-0.472376\pi\)
0.0866758 + 0.996237i \(0.472376\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.92820 −0.546019
\(162\) 0 0
\(163\) −4.53590 −0.355279 −0.177639 0.984096i \(-0.556846\pi\)
−0.177639 + 0.984096i \(0.556846\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.72741 0.597965 0.298982 0.954259i \(-0.403353\pi\)
0.298982 + 0.954259i \(0.403353\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.2480 1.31134 0.655669 0.755048i \(-0.272387\pi\)
0.655669 + 0.755048i \(0.272387\pi\)
\(174\) 0 0
\(175\) 24.3190 1.83835
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −18.9282 −1.41476 −0.707380 0.706833i \(-0.750123\pi\)
−0.707380 + 0.706833i \(0.750123\pi\)
\(180\) 0 0
\(181\) 7.07107 0.525588 0.262794 0.964852i \(-0.415356\pi\)
0.262794 + 0.964852i \(0.415356\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.53590 −0.186443
\(186\) 0 0
\(187\) −5.07180 −0.370887
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 7.72741 0.559136 0.279568 0.960126i \(-0.409809\pi\)
0.279568 + 0.960126i \(0.409809\pi\)
\(192\) 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
\(197\) 8.76268 0.624315 0.312158 0.950030i \(-0.398948\pi\)
0.312158 + 0.950030i \(0.398948\pi\)
\(198\) 0 0
\(199\) 18.6622 1.32293 0.661463 0.749977i \(-0.269936\pi\)
0.661463 + 0.749977i \(0.269936\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −21.4641 −1.50648
\(204\) 0 0
\(205\) 17.5254 1.22402
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.9282 0.755920
\(210\) 0 0
\(211\) −6.92820 −0.476957 −0.238479 0.971148i \(-0.576649\pi\)
−0.238479 + 0.971148i \(0.576649\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.3843 −0.912799
\(216\) 0 0
\(217\) 18.0000 1.22192
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.6969 0.988623
\(222\) 0 0
\(223\) −14.5211 −0.972403 −0.486201 0.873847i \(-0.661618\pi\)
−0.486201 + 0.873847i \(0.661618\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.3205 0.751369 0.375684 0.926748i \(-0.377408\pi\)
0.375684 + 0.926748i \(0.377408\pi\)
\(228\) 0 0
\(229\) −16.8690 −1.11474 −0.557368 0.830265i \(-0.688189\pi\)
−0.557368 + 0.830265i \(0.688189\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.9282 −0.846955 −0.423477 0.905907i \(-0.639191\pi\)
−0.423477 + 0.905907i \(0.639191\pi\)
\(234\) 0 0
\(235\) −10.9282 −0.712877
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.72741 −0.499844 −0.249922 0.968266i \(-0.580405\pi\)
−0.249922 + 0.968266i \(0.580405\pi\)
\(240\) 0 0
\(241\) −26.7846 −1.72535 −0.862674 0.505760i \(-0.831212\pi\)
−0.862674 + 0.505760i \(0.831212\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.86370 0.246843
\(246\) 0 0
\(247\) −31.6675 −2.01495
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −6.53590 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(252\) 0 0
\(253\) 4.14110 0.260349
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 22.7846 1.42126 0.710632 0.703563i \(-0.248409\pi\)
0.710632 + 0.703563i \(0.248409\pi\)
\(258\) 0 0
\(259\) 1.60770 0.0998973
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −22.6274 −1.39527 −0.697633 0.716455i \(-0.745763\pi\)
−0.697633 + 0.716455i \(0.745763\pi\)
\(264\) 0 0
\(265\) −17.8564 −1.09691
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −10.2784 −0.626687 −0.313344 0.949640i \(-0.601449\pi\)
−0.313344 + 0.949640i \(0.601449\pi\)
\(270\) 0 0
\(271\) −3.20736 −0.194834 −0.0974168 0.995244i \(-0.531058\pi\)
−0.0974168 + 0.995244i \(0.531058\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −14.5359 −0.876548
\(276\) 0 0
\(277\) 12.5249 0.752545 0.376273 0.926509i \(-0.377206\pi\)
0.376273 + 0.926509i \(0.377206\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.928203 −0.0553720 −0.0276860 0.999617i \(-0.508814\pi\)
−0.0276860 + 0.999617i \(0.508814\pi\)
\(282\) 0 0
\(283\) −9.85641 −0.585903 −0.292951 0.956127i \(-0.594637\pi\)
−0.292951 + 0.956127i \(0.594637\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.1106 −0.655840
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.8332 −0.632884 −0.316442 0.948612i \(-0.602488\pi\)
−0.316442 + 0.948612i \(0.602488\pi\)
\(294\) 0 0
\(295\) −53.5370 −3.11705
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 8.48528 0.489083
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.53590 0.145205
\(306\) 0 0
\(307\) −22.9282 −1.30858 −0.654291 0.756243i \(-0.727033\pi\)
−0.654291 + 0.756243i \(0.727033\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −32.4254 −1.83867 −0.919337 0.393471i \(-0.871274\pi\)
−0.919337 + 0.393471i \(0.871274\pi\)
\(312\) 0 0
\(313\) 3.85641 0.217977 0.108988 0.994043i \(-0.465239\pi\)
0.108988 + 0.994043i \(0.465239\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.69213 −0.375867 −0.187934 0.982182i \(-0.560179\pi\)
−0.187934 + 0.982182i \(0.560179\pi\)
\(318\) 0 0
\(319\) 12.8295 0.718312
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −25.8564 −1.43869
\(324\) 0 0
\(325\) 42.1218 2.33650
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.92820 0.381964
\(330\) 0 0
\(331\) −1.07180 −0.0589113 −0.0294556 0.999566i \(-0.509377\pi\)
−0.0294556 + 0.999566i \(0.509377\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 57.6781 3.15129
\(336\) 0 0
\(337\) 9.85641 0.536913 0.268456 0.963292i \(-0.413486\pi\)
0.268456 + 0.963292i \(0.413486\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.7589 −0.582627
\(342\) 0 0
\(343\) −19.5959 −1.05808
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.53590 0.136134 0.0680671 0.997681i \(-0.478317\pi\)
0.0680671 + 0.997681i \(0.478317\pi\)
\(348\) 0 0
\(349\) 27.4249 1.46802 0.734010 0.679139i \(-0.237647\pi\)
0.734010 + 0.679139i \(0.237647\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.928203 0.0494033 0.0247016 0.999695i \(-0.492136\pi\)
0.0247016 + 0.999695i \(0.492136\pi\)
\(354\) 0 0
\(355\) 24.7846 1.31543
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.8380 −0.994234 −0.497117 0.867684i \(-0.665608\pi\)
−0.497117 + 0.867684i \(0.665608\pi\)
\(360\) 0 0
\(361\) 36.7128 1.93225
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −15.4548 −0.808942
\(366\) 0 0
\(367\) −16.3886 −0.855476 −0.427738 0.903903i \(-0.640689\pi\)
−0.427738 + 0.903903i \(0.640689\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.3205 0.587731
\(372\) 0 0
\(373\) −13.2827 −0.687753 −0.343877 0.939015i \(-0.611740\pi\)
−0.343877 + 0.939015i \(0.611740\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −37.1769 −1.91471
\(378\) 0 0
\(379\) −13.3205 −0.684229 −0.342114 0.939658i \(-0.611143\pi\)
−0.342114 + 0.939658i \(0.611143\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −28.8391 −1.47361 −0.736804 0.676106i \(-0.763666\pi\)
−0.736804 + 0.676106i \(0.763666\pi\)
\(384\) 0 0
\(385\) 13.8564 0.706188
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.52056 −0.482711 −0.241356 0.970437i \(-0.577592\pi\)
−0.241356 + 0.970437i \(0.577592\pi\)
\(390\) 0 0
\(391\) −9.79796 −0.495504
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 9.46410 0.476191
\(396\) 0 0
\(397\) −23.0807 −1.15839 −0.579193 0.815190i \(-0.696632\pi\)
−0.579193 + 0.815190i \(0.696632\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 18.3923 0.918468 0.459234 0.888315i \(-0.348124\pi\)
0.459234 + 0.888315i \(0.348124\pi\)
\(402\) 0 0
\(403\) 31.1769 1.55303
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.960947 −0.0476324
\(408\) 0 0
\(409\) −9.07180 −0.448571 −0.224286 0.974523i \(-0.572005\pi\)
−0.224286 + 0.974523i \(0.572005\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 33.9411 1.67013
\(414\) 0 0
\(415\) −21.1117 −1.03633
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.392305 0.0191653 0.00958267 0.999954i \(-0.496950\pi\)
0.00958267 + 0.999954i \(0.496950\pi\)
\(420\) 0 0
\(421\) 19.6975 0.959995 0.479998 0.877270i \(-0.340638\pi\)
0.479998 + 0.877270i \(0.340638\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 34.3923 1.66827
\(426\) 0 0
\(427\) −1.60770 −0.0778018
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.34418 −0.209252 −0.104626 0.994512i \(-0.533364\pi\)
−0.104626 + 0.994512i \(0.533364\pi\)
\(432\) 0 0
\(433\) −29.8564 −1.43481 −0.717404 0.696658i \(-0.754670\pi\)
−0.717404 + 0.696658i \(0.754670\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 21.1117 1.00991
\(438\) 0 0
\(439\) 5.83272 0.278381 0.139190 0.990266i \(-0.455550\pi\)
0.139190 + 0.990266i \(0.455550\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −35.3205 −1.67813 −0.839064 0.544033i \(-0.816897\pi\)
−0.839064 + 0.544033i \(0.816897\pi\)
\(444\) 0 0
\(445\) −19.0411 −0.902635
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.39230 0.301672 0.150836 0.988559i \(-0.451804\pi\)
0.150836 + 0.988559i \(0.451804\pi\)
\(450\) 0 0
\(451\) 6.64102 0.312713
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −40.1528 −1.88239
\(456\) 0 0
\(457\) −20.7846 −0.972263 −0.486132 0.873886i \(-0.661592\pi\)
−0.486132 + 0.873886i \(0.661592\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −31.1870 −1.45252 −0.726262 0.687418i \(-0.758744\pi\)
−0.726262 + 0.687418i \(0.758744\pi\)
\(462\) 0 0
\(463\) −29.2180 −1.35788 −0.678938 0.734196i \(-0.737560\pi\)
−0.678938 + 0.734196i \(0.737560\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −30.2487 −1.39974 −0.699872 0.714269i \(-0.746760\pi\)
−0.699872 + 0.714269i \(0.746760\pi\)
\(468\) 0 0
\(469\) −36.5665 −1.68848
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.07180 −0.233201
\(474\) 0 0
\(475\) −74.1051 −3.40018
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 29.0421 1.32697 0.663485 0.748190i \(-0.269077\pi\)
0.663485 + 0.748190i \(0.269077\pi\)
\(480\) 0 0
\(481\) 2.78461 0.126967
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.14110 −0.188038
\(486\) 0 0
\(487\) 11.4896 0.520642 0.260321 0.965522i \(-0.416172\pi\)
0.260321 + 0.965522i \(0.416172\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.85641 0.264296 0.132148 0.991230i \(-0.457813\pi\)
0.132148 + 0.991230i \(0.457813\pi\)
\(492\) 0 0
\(493\) −30.3548 −1.36711
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.7128 −0.704816
\(498\) 0 0
\(499\) −4.00000 −0.179065 −0.0895323 0.995984i \(-0.528537\pi\)
−0.0895323 + 0.995984i \(0.528537\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −23.3853 −1.04270 −0.521349 0.853343i \(-0.674571\pi\)
−0.521349 + 0.853343i \(0.674571\pi\)
\(504\) 0 0
\(505\) −55.7128 −2.47919
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.24693 0.321215 0.160607 0.987018i \(-0.448655\pi\)
0.160607 + 0.987018i \(0.448655\pi\)
\(510\) 0 0
\(511\) 9.79796 0.433436
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 12.3923 0.546070
\(516\) 0 0
\(517\) −4.14110 −0.182126
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 21.3205 0.934068 0.467034 0.884239i \(-0.345322\pi\)
0.467034 + 0.884239i \(0.345322\pi\)
\(522\) 0 0
\(523\) −31.4641 −1.37583 −0.687915 0.725792i \(-0.741474\pi\)
−0.687915 + 0.725792i \(0.741474\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 25.4558 1.10887
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −19.2442 −0.833558
\(534\) 0 0
\(535\) 15.4548 0.668170
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.46410 0.0630633
\(540\) 0 0
\(541\) −28.1827 −1.21167 −0.605835 0.795590i \(-0.707161\pi\)
−0.605835 + 0.795590i \(0.707161\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.46410 −0.234056
\(546\) 0 0
\(547\) −23.1769 −0.990973 −0.495487 0.868616i \(-0.665010\pi\)
−0.495487 + 0.868616i \(0.665010\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 65.4056 2.78637
\(552\) 0 0
\(553\) −6.00000 −0.255146
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 28.0069 1.18669 0.593345 0.804949i \(-0.297807\pi\)
0.593345 + 0.804949i \(0.297807\pi\)
\(558\) 0 0
\(559\) 14.6969 0.621614
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.46410 0.0617045 0.0308523 0.999524i \(-0.490178\pi\)
0.0308523 + 0.999524i \(0.490178\pi\)
\(564\) 0 0
\(565\) 69.5467 2.92585
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.46410 0.145223 0.0726113 0.997360i \(-0.476867\pi\)
0.0726113 + 0.997360i \(0.476867\pi\)
\(570\) 0 0
\(571\) 31.7128 1.32714 0.663570 0.748114i \(-0.269041\pi\)
0.663570 + 0.748114i \(0.269041\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −28.0812 −1.17107
\(576\) 0 0
\(577\) −21.8564 −0.909894 −0.454947 0.890518i \(-0.650342\pi\)
−0.454947 + 0.890518i \(0.650342\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 13.3843 0.555273
\(582\) 0 0
\(583\) −6.76646 −0.280238
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −42.9282 −1.77184 −0.885918 0.463841i \(-0.846471\pi\)
−0.885918 + 0.463841i \(0.846471\pi\)
\(588\) 0 0
\(589\) −54.8497 −2.26004
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −37.7128 −1.54868 −0.774340 0.632770i \(-0.781918\pi\)
−0.774340 + 0.632770i \(0.781918\pi\)
\(594\) 0 0
\(595\) −32.7846 −1.34404
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.96953 −0.284767 −0.142384 0.989812i \(-0.545477\pi\)
−0.142384 + 0.989812i \(0.545477\pi\)
\(600\) 0 0
\(601\) 19.7128 0.804102 0.402051 0.915617i \(-0.368297\pi\)
0.402051 + 0.915617i \(0.368297\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.2185 1.39118
\(606\) 0 0
\(607\) −31.8434 −1.29248 −0.646241 0.763133i \(-0.723660\pi\)
−0.646241 + 0.763133i \(0.723660\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 0 0
\(613\) −6.11012 −0.246785 −0.123393 0.992358i \(-0.539377\pi\)
−0.123393 + 0.992358i \(0.539377\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.8564 1.60456 0.802279 0.596949i \(-0.203621\pi\)
0.802279 + 0.596949i \(0.203621\pi\)
\(618\) 0 0
\(619\) 12.0000 0.482321 0.241160 0.970485i \(-0.422472\pi\)
0.241160 + 0.970485i \(0.422472\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.0716 0.483638
\(624\) 0 0
\(625\) 23.9282 0.957128
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.27362 0.0906553
\(630\) 0 0
\(631\) −23.5612 −0.937955 −0.468977 0.883210i \(-0.655377\pi\)
−0.468977 + 0.883210i \(0.655377\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −56.1051 −2.22646
\(636\) 0 0
\(637\) −4.24264 −0.168100
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 25.6077 1.01144 0.505722 0.862697i \(-0.331226\pi\)
0.505722 + 0.862697i \(0.331226\pi\)
\(642\) 0 0
\(643\) 28.5359 1.12535 0.562673 0.826680i \(-0.309773\pi\)
0.562673 + 0.826680i \(0.309773\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −40.9107 −1.60836 −0.804182 0.594383i \(-0.797396\pi\)
−0.804182 + 0.594383i \(0.797396\pi\)
\(648\) 0 0
\(649\) −20.2872 −0.796342
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.9754 0.977362 0.488681 0.872463i \(-0.337478\pi\)
0.488681 + 0.872463i \(0.337478\pi\)
\(654\) 0 0
\(655\) 57.6781 2.25367
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) 6.51626 0.253453 0.126727 0.991938i \(-0.459553\pi\)
0.126727 + 0.991938i \(0.459553\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 70.6410 2.73934
\(666\) 0 0
\(667\) 24.7846 0.959664
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.960947 0.0370969
\(672\) 0 0
\(673\) 8.00000 0.308377 0.154189 0.988041i \(-0.450724\pi\)
0.154189 + 0.988041i \(0.450724\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −3.30890 −0.127171 −0.0635857 0.997976i \(-0.520254\pi\)
−0.0635857 + 0.997976i \(0.520254\pi\)
\(678\) 0 0
\(679\) 2.62536 0.100752
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.1769 0.963368 0.481684 0.876345i \(-0.340025\pi\)
0.481684 + 0.876345i \(0.340025\pi\)
\(684\) 0 0
\(685\) 55.6076 2.12466
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 19.6077 0.746994
\(690\) 0 0
\(691\) −26.3923 −1.00401 −0.502005 0.864865i \(-0.667404\pi\)
−0.502005 + 0.864865i \(0.667404\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 26.7685 1.01539
\(696\) 0 0
\(697\) −15.7128 −0.595165
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 14.9743 0.565573 0.282787 0.959183i \(-0.408741\pi\)
0.282787 + 0.959183i \(0.408741\pi\)
\(702\) 0 0
\(703\) −4.89898 −0.184769
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 35.3205 1.32836
\(708\) 0 0
\(709\) −49.0913 −1.84366 −0.921832 0.387590i \(-0.873308\pi\)
−0.921832 + 0.387590i \(0.873308\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −20.7846 −0.778390
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 2.27362 0.0847919 0.0423959 0.999101i \(-0.486501\pi\)
0.0423959 + 0.999101i \(0.486501\pi\)
\(720\) 0 0
\(721\) −7.85641 −0.292588
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −86.9977 −3.23101
\(726\) 0 0
\(727\) −8.10634 −0.300648 −0.150324 0.988637i \(-0.548032\pi\)
−0.150324 + 0.988637i \(0.548032\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 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
\(737\) 21.8564 0.805091
\(738\) 0 0
\(739\) −36.0000 −1.32428 −0.662141 0.749380i \(-0.730352\pi\)
−0.662141 + 0.749380i \(0.730352\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −27.3233 −1.00240 −0.501198 0.865333i \(-0.667107\pi\)
−0.501198 + 0.865333i \(0.667107\pi\)
\(744\) 0 0
\(745\) −20.7846 −0.761489
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −9.79796 −0.358010
\(750\) 0 0
\(751\) 46.1886 1.68545 0.842723 0.538348i \(-0.180951\pi\)
0.842723 + 0.538348i \(0.180951\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −25.4641 −0.926734
\(756\) 0 0
\(757\) −33.6365 −1.22254 −0.611270 0.791422i \(-0.709341\pi\)
−0.611270 + 0.791422i \(0.709341\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 52.2487 1.89401 0.947007 0.321212i \(-0.104090\pi\)
0.947007 + 0.321212i \(0.104090\pi\)
\(762\) 0 0
\(763\) 3.46410 0.125409
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 58.7878 2.12270
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −48.9155 −1.75937 −0.879684 0.475560i \(-0.842246\pi\)
−0.879684 + 0.475560i \(0.842246\pi\)
\(774\) 0 0
\(775\) 72.9571 2.62070
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 33.8564 1.21303
\(780\) 0 0
\(781\) 9.39182 0.336066
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8.39230 −0.299534
\(786\) 0 0
\(787\) 23.4641 0.836405 0.418202 0.908354i \(-0.362660\pi\)
0.418202 + 0.908354i \(0.362660\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −44.0908 −1.56769
\(792\) 0 0
\(793\) −2.78461 −0.0988844
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −21.9439 −0.777292 −0.388646 0.921387i \(-0.627057\pi\)
−0.388646 + 0.921387i \(0.627057\pi\)
\(798\) 0 0
\(799\) 9.79796 0.346627
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −5.85641 −0.206668
\(804\) 0 0
\(805\) 26.7685 0.943466
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 6.67949 0.234838 0.117419 0.993082i \(-0.462538\pi\)
0.117419 + 0.993082i \(0.462538\pi\)
\(810\) 0 0
\(811\) −54.3923 −1.90997 −0.954986 0.296651i \(-0.904130\pi\)
−0.954986 + 0.296651i \(0.904130\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 17.5254 0.613887
\(816\) 0 0
\(817\) −25.8564 −0.904601
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.1784 0.878734 0.439367 0.898308i \(-0.355203\pi\)
0.439367 + 0.898308i \(0.355203\pi\)
\(822\) 0 0
\(823\) 9.97382 0.347666 0.173833 0.984775i \(-0.444385\pi\)
0.173833 + 0.984775i \(0.444385\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.7846 1.55731 0.778657 0.627450i \(-0.215901\pi\)
0.778657 + 0.627450i \(0.215901\pi\)
\(828\) 0 0
\(829\) −1.61729 −0.0561706 −0.0280853 0.999606i \(-0.508941\pi\)
−0.0280853 + 0.999606i \(0.508941\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.46410 −0.120024
\(834\) 0 0
\(835\) −29.8564 −1.03322
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 26.0106 0.897987 0.448994 0.893535i \(-0.351783\pi\)
0.448994 + 0.893535i \(0.351783\pi\)
\(840\) 0 0
\(841\) 47.7846 1.64775
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −19.3185 −0.664577
\(846\) 0 0
\(847\) −21.6937 −0.745404
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.85641 −0.0636368
\(852\) 0 0
\(853\) 47.0208 1.60996 0.804980 0.593301i \(-0.202176\pi\)
0.804980 + 0.593301i \(0.202176\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43.1769 −1.47490 −0.737448 0.675404i \(-0.763969\pi\)
−0.737448 + 0.675404i \(0.763969\pi\)
\(858\) 0 0
\(859\) 32.2487 1.10031 0.550156 0.835062i \(-0.314568\pi\)
0.550156 + 0.835062i \(0.314568\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 37.1213 1.26362 0.631812 0.775122i \(-0.282312\pi\)
0.631812 + 0.775122i \(0.282312\pi\)
\(864\) 0 0
\(865\) −66.6410 −2.26586
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 3.58630 0.121657
\(870\) 0 0
\(871\) −63.3350 −2.14602
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −46.6410 −1.57675
\(876\) 0 0
\(877\) −35.7071 −1.20574 −0.602871 0.797839i \(-0.705977\pi\)
−0.602871 + 0.797839i \(0.705977\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 36.6410 1.23447 0.617234 0.786780i \(-0.288253\pi\)
0.617234 + 0.786780i \(0.288253\pi\)
\(882\) 0 0
\(883\) 5.60770 0.188714 0.0943570 0.995538i \(-0.469920\pi\)
0.0943570 + 0.995538i \(0.469920\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 36.5665 1.22778 0.613891 0.789391i \(-0.289603\pi\)
0.613891 + 0.789391i \(0.289603\pi\)
\(888\) 0 0
\(889\) 35.5692 1.19295
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −21.1117 −0.706475
\(894\) 0 0
\(895\) 73.1330 2.44457
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −64.3923 −2.14760
\(900\) 0 0
\(901\) 16.0096 0.533358
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −27.3205 −0.908164
\(906\) 0 0
\(907\) 42.3923 1.40761 0.703807 0.710392i \(-0.251482\pi\)
0.703807 + 0.710392i \(0.251482\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −22.0726 −0.731298 −0.365649 0.930753i \(-0.619153\pi\)
−0.365649 + 0.930753i \(0.619153\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −36.5665 −1.20753
\(918\) 0 0
\(919\) 0.582009 0.0191987 0.00959936 0.999954i \(-0.496944\pi\)
0.00959936 + 0.999954i \(0.496944\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −27.2154 −0.895805
\(924\) 0 0
\(925\) 6.51626 0.214253
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.3205 1.48692 0.743459 0.668782i \(-0.233184\pi\)
0.743459 + 0.668782i \(0.233184\pi\)
\(930\) 0 0
\(931\) 7.46410 0.244626
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.5959 0.640855
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 36.2891 1.18299 0.591495 0.806309i \(-0.298538\pi\)
0.591495 + 0.806309i \(0.298538\pi\)
\(942\) 0 0
\(943\) 12.8295 0.417785
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.0718 0.944706 0.472353 0.881409i \(-0.343405\pi\)
0.472353 + 0.881409i \(0.343405\pi\)
\(948\) 0 0
\(949\) 16.9706 0.550888
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −46.3923 −1.50279 −0.751397 0.659850i \(-0.770620\pi\)
−0.751397 + 0.659850i \(0.770620\pi\)
\(954\) 0 0
\(955\) −29.8564 −0.966131
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −35.2538 −1.13840
\(960\) 0 0
\(961\) 23.0000 0.741935
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −23.1822 −0.746262
\(966\) 0 0
\(967\) −10.3800 −0.333797 −0.166899 0.985974i \(-0.553375\pi\)
−0.166899 + 0.985974i \(0.553375\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.2487 0.713995 0.356998 0.934105i \(-0.383800\pi\)
0.356998 + 0.934105i \(0.383800\pi\)
\(972\) 0 0
\(973\) −16.9706 −0.544051
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 35.1769 1.12541 0.562705 0.826658i \(-0.309761\pi\)
0.562705 + 0.826658i \(0.309761\pi\)
\(978\) 0 0
\(979\) −7.21539 −0.230605
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 23.7370 0.757093 0.378547 0.925582i \(-0.376424\pi\)
0.378547 + 0.925582i \(0.376424\pi\)
\(984\) 0 0
\(985\) −33.8564 −1.07875
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −9.79796 −0.311557
\(990\) 0 0
\(991\) 12.2474 0.389053 0.194527 0.980897i \(-0.437683\pi\)
0.194527 + 0.980897i \(0.437683\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −72.1051 −2.28589
\(996\) 0 0
\(997\) 17.6269 0.558250 0.279125 0.960255i \(-0.409956\pi\)
0.279125 + 0.960255i \(0.409956\pi\)
\(998\) 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.bi.1.1 4
3.2 odd 2 3072.2.a.l.1.4 4
4.3 odd 2 9216.2.a.bc.1.1 4
8.3 odd 2 inner 9216.2.a.bi.1.4 4
8.5 even 2 9216.2.a.bc.1.4 4
12.11 even 2 3072.2.a.r.1.4 4
24.5 odd 2 3072.2.a.r.1.1 4
24.11 even 2 3072.2.a.l.1.1 4
32.3 odd 8 2304.2.k.l.577.4 8
32.5 even 8 2304.2.k.e.1729.2 8
32.11 odd 8 2304.2.k.l.1729.3 8
32.13 even 8 2304.2.k.e.577.1 8
32.19 odd 8 2304.2.k.e.577.2 8
32.21 even 8 2304.2.k.l.1729.4 8
32.27 odd 8 2304.2.k.e.1729.1 8
32.29 even 8 2304.2.k.l.577.3 8
48.5 odd 4 3072.2.d.h.1537.8 8
48.11 even 4 3072.2.d.h.1537.4 8
48.29 odd 4 3072.2.d.h.1537.1 8
48.35 even 4 3072.2.d.h.1537.5 8
96.5 odd 8 768.2.j.f.193.4 yes 8
96.11 even 8 768.2.j.e.193.3 yes 8
96.29 odd 8 768.2.j.e.577.1 yes 8
96.35 even 8 768.2.j.e.577.3 yes 8
96.53 odd 8 768.2.j.e.193.1 8
96.59 even 8 768.2.j.f.193.2 yes 8
96.77 odd 8 768.2.j.f.577.4 yes 8
96.83 even 8 768.2.j.f.577.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
768.2.j.e.193.1 8 96.53 odd 8
768.2.j.e.193.3 yes 8 96.11 even 8
768.2.j.e.577.1 yes 8 96.29 odd 8
768.2.j.e.577.3 yes 8 96.35 even 8
768.2.j.f.193.2 yes 8 96.59 even 8
768.2.j.f.193.4 yes 8 96.5 odd 8
768.2.j.f.577.2 yes 8 96.83 even 8
768.2.j.f.577.4 yes 8 96.77 odd 8
2304.2.k.e.577.1 8 32.13 even 8
2304.2.k.e.577.2 8 32.19 odd 8
2304.2.k.e.1729.1 8 32.27 odd 8
2304.2.k.e.1729.2 8 32.5 even 8
2304.2.k.l.577.3 8 32.29 even 8
2304.2.k.l.577.4 8 32.3 odd 8
2304.2.k.l.1729.3 8 32.11 odd 8
2304.2.k.l.1729.4 8 32.21 even 8
3072.2.a.l.1.1 4 24.11 even 2
3072.2.a.l.1.4 4 3.2 odd 2
3072.2.a.r.1.1 4 24.5 odd 2
3072.2.a.r.1.4 4 12.11 even 2
3072.2.d.h.1537.1 8 48.29 odd 4
3072.2.d.h.1537.4 8 48.11 even 4
3072.2.d.h.1537.5 8 48.35 even 4
3072.2.d.h.1537.8 8 48.5 odd 4
9216.2.a.bc.1.1 4 4.3 odd 2
9216.2.a.bc.1.4 4 8.5 even 2
9216.2.a.bi.1.1 4 1.1 even 1 trivial
9216.2.a.bi.1.4 4 8.3 odd 2 inner