Properties

Label 1536.2.d.c.769.2
Level $1536$
Weight $2$
Character 1536.769
Analytic conductor $12.265$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1536,2,Mod(769,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1536.769");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.d (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.2
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1536.769
Dual form 1536.2.d.c.769.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.41421i q^{5} +4.24264 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.41421i q^{5} +4.24264 q^{7} -1.00000 q^{9} -6.00000i q^{11} -5.65685i q^{13} +1.41421 q^{15} -6.00000 q^{17} +4.00000i q^{19} -4.24264i q^{21} -2.82843 q^{23} +3.00000 q^{25} +1.00000i q^{27} -1.41421i q^{29} -1.41421 q^{31} -6.00000 q^{33} +6.00000i q^{35} -8.48528i q^{37} -5.65685 q^{39} +2.00000 q^{41} -1.41421i q^{45} +2.82843 q^{47} +11.0000 q^{49} +6.00000i q^{51} -9.89949i q^{53} +8.48528 q^{55} +4.00000 q^{57} -4.00000i q^{59} +8.48528i q^{61} -4.24264 q^{63} +8.00000 q^{65} +8.00000i q^{67} +2.82843i q^{69} +2.82843 q^{71} -8.00000 q^{73} -3.00000i q^{75} -25.4558i q^{77} +12.7279 q^{79} +1.00000 q^{81} +2.00000i q^{83} -8.48528i q^{85} -1.41421 q^{87} -2.00000 q^{89} -24.0000i q^{91} +1.41421i q^{93} -5.65685 q^{95} +2.00000 q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} - 24 q^{17} + 12 q^{25} - 24 q^{33} + 8 q^{41} + 44 q^{49} + 16 q^{57} + 32 q^{65} - 32 q^{73} + 4 q^{81} - 8 q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(517\) \(1025\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 1.41421i 0.632456i 0.948683 + 0.316228i \(0.102416\pi\)
−0.948683 + 0.316228i \(0.897584\pi\)
\(6\) 0 0
\(7\) 4.24264 1.60357 0.801784 0.597614i \(-0.203885\pi\)
0.801784 + 0.597614i \(0.203885\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 6.00000i − 1.80907i −0.426401 0.904534i \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) 0 0
\(13\) − 5.65685i − 1.56893i −0.620174 0.784465i \(-0.712938\pi\)
0.620174 0.784465i \(-0.287062\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 4.00000i 0.917663i 0.888523 + 0.458831i \(0.151732\pi\)
−0.888523 + 0.458831i \(0.848268\pi\)
\(20\) 0 0
\(21\) − 4.24264i − 0.925820i
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 1.41421i − 0.262613i −0.991342 0.131306i \(-0.958083\pi\)
0.991342 0.131306i \(-0.0419172\pi\)
\(30\) 0 0
\(31\) −1.41421 −0.254000 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 6.00000i 1.01419i
\(36\) 0 0
\(37\) − 8.48528i − 1.39497i −0.716599 0.697486i \(-0.754302\pi\)
0.716599 0.697486i \(-0.245698\pi\)
\(38\) 0 0
\(39\) −5.65685 −0.905822
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) − 1.41421i − 0.210819i
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 6.00000i 0.840168i
\(52\) 0 0
\(53\) − 9.89949i − 1.35980i −0.733305 0.679900i \(-0.762023\pi\)
0.733305 0.679900i \(-0.237977\pi\)
\(54\) 0 0
\(55\) 8.48528 1.14416
\(56\) 0 0
\(57\) 4.00000 0.529813
\(58\) 0 0
\(59\) − 4.00000i − 0.520756i −0.965507 0.260378i \(-0.916153\pi\)
0.965507 0.260378i \(-0.0838471\pi\)
\(60\) 0 0
\(61\) 8.48528i 1.08643i 0.839594 + 0.543214i \(0.182793\pi\)
−0.839594 + 0.543214i \(0.817207\pi\)
\(62\) 0 0
\(63\) −4.24264 −0.534522
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 8.00000i 0.977356i 0.872464 + 0.488678i \(0.162521\pi\)
−0.872464 + 0.488678i \(0.837479\pi\)
\(68\) 0 0
\(69\) 2.82843i 0.340503i
\(70\) 0 0
\(71\) 2.82843 0.335673 0.167836 0.985815i \(-0.446322\pi\)
0.167836 + 0.985815i \(0.446322\pi\)
\(72\) 0 0
\(73\) −8.00000 −0.936329 −0.468165 0.883641i \(-0.655085\pi\)
−0.468165 + 0.883641i \(0.655085\pi\)
\(74\) 0 0
\(75\) − 3.00000i − 0.346410i
\(76\) 0 0
\(77\) − 25.4558i − 2.90096i
\(78\) 0 0
\(79\) 12.7279 1.43200 0.716002 0.698099i \(-0.245970\pi\)
0.716002 + 0.698099i \(0.245970\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) − 8.48528i − 0.920358i
\(86\) 0 0
\(87\) −1.41421 −0.151620
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) − 24.0000i − 2.51588i
\(92\) 0 0
\(93\) 1.41421i 0.146647i
\(94\) 0 0
\(95\) −5.65685 −0.580381
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 6.00000i 0.603023i
\(100\) 0 0
\(101\) − 1.41421i − 0.140720i −0.997522 0.0703598i \(-0.977585\pi\)
0.997522 0.0703598i \(-0.0224147\pi\)
\(102\) 0 0
\(103\) −7.07107 −0.696733 −0.348367 0.937358i \(-0.613264\pi\)
−0.348367 + 0.937358i \(0.613264\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) 0 0
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 0 0
\(109\) − 16.9706i − 1.62549i −0.582623 0.812743i \(-0.697974\pi\)
0.582623 0.812743i \(-0.302026\pi\)
\(110\) 0 0
\(111\) −8.48528 −0.805387
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) − 4.00000i − 0.373002i
\(116\) 0 0
\(117\) 5.65685i 0.522976i
\(118\) 0 0
\(119\) −25.4558 −2.33353
\(120\) 0 0
\(121\) −25.0000 −2.27273
\(122\) 0 0
\(123\) − 2.00000i − 0.180334i
\(124\) 0 0
\(125\) 11.3137i 1.01193i
\(126\) 0 0
\(127\) 12.7279 1.12942 0.564710 0.825289i \(-0.308988\pi\)
0.564710 + 0.825289i \(0.308988\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 12.0000i − 1.04844i −0.851581 0.524222i \(-0.824356\pi\)
0.851581 0.524222i \(-0.175644\pi\)
\(132\) 0 0
\(133\) 16.9706i 1.47153i
\(134\) 0 0
\(135\) −1.41421 −0.121716
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) − 2.82843i − 0.238197i
\(142\) 0 0
\(143\) −33.9411 −2.83830
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) − 11.0000i − 0.907265i
\(148\) 0 0
\(149\) − 4.24264i − 0.347571i −0.984784 0.173785i \(-0.944400\pi\)
0.984784 0.173785i \(-0.0555999\pi\)
\(150\) 0 0
\(151\) 12.7279 1.03578 0.517892 0.855446i \(-0.326717\pi\)
0.517892 + 0.855446i \(0.326717\pi\)
\(152\) 0 0
\(153\) 6.00000 0.485071
\(154\) 0 0
\(155\) − 2.00000i − 0.160644i
\(156\) 0 0
\(157\) − 2.82843i − 0.225733i −0.993610 0.112867i \(-0.963997\pi\)
0.993610 0.112867i \(-0.0360032\pi\)
\(158\) 0 0
\(159\) −9.89949 −0.785081
\(160\) 0 0
\(161\) −12.0000 −0.945732
\(162\) 0 0
\(163\) 8.00000i 0.626608i 0.949653 + 0.313304i \(0.101436\pi\)
−0.949653 + 0.313304i \(0.898564\pi\)
\(164\) 0 0
\(165\) − 8.48528i − 0.660578i
\(166\) 0 0
\(167\) 22.6274 1.75096 0.875481 0.483252i \(-0.160545\pi\)
0.875481 + 0.483252i \(0.160545\pi\)
\(168\) 0 0
\(169\) −19.0000 −1.46154
\(170\) 0 0
\(171\) − 4.00000i − 0.305888i
\(172\) 0 0
\(173\) 7.07107i 0.537603i 0.963196 + 0.268802i \(0.0866276\pi\)
−0.963196 + 0.268802i \(0.913372\pi\)
\(174\) 0 0
\(175\) 12.7279 0.962140
\(176\) 0 0
\(177\) −4.00000 −0.300658
\(178\) 0 0
\(179\) 20.0000i 1.49487i 0.664335 + 0.747435i \(0.268715\pi\)
−0.664335 + 0.747435i \(0.731285\pi\)
\(180\) 0 0
\(181\) − 16.9706i − 1.26141i −0.776022 0.630706i \(-0.782765\pi\)
0.776022 0.630706i \(-0.217235\pi\)
\(182\) 0 0
\(183\) 8.48528 0.627250
\(184\) 0 0
\(185\) 12.0000 0.882258
\(186\) 0 0
\(187\) 36.0000i 2.63258i
\(188\) 0 0
\(189\) 4.24264i 0.308607i
\(190\) 0 0
\(191\) 11.3137 0.818631 0.409316 0.912393i \(-0.365768\pi\)
0.409316 + 0.912393i \(0.365768\pi\)
\(192\) 0 0
\(193\) 12.0000 0.863779 0.431889 0.901927i \(-0.357847\pi\)
0.431889 + 0.901927i \(0.357847\pi\)
\(194\) 0 0
\(195\) − 8.00000i − 0.572892i
\(196\) 0 0
\(197\) − 4.24264i − 0.302276i −0.988513 0.151138i \(-0.951706\pi\)
0.988513 0.151138i \(-0.0482937\pi\)
\(198\) 0 0
\(199\) 4.24264 0.300753 0.150376 0.988629i \(-0.451951\pi\)
0.150376 + 0.988629i \(0.451951\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) 0 0
\(203\) − 6.00000i − 0.421117i
\(204\) 0 0
\(205\) 2.82843i 0.197546i
\(206\) 0 0
\(207\) 2.82843 0.196589
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) − 2.82843i − 0.193801i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) 8.00000i 0.540590i
\(220\) 0 0
\(221\) 33.9411i 2.28313i
\(222\) 0 0
\(223\) 15.5563 1.04173 0.520865 0.853639i \(-0.325609\pi\)
0.520865 + 0.853639i \(0.325609\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 14.0000i 0.929213i 0.885517 + 0.464606i \(0.153804\pi\)
−0.885517 + 0.464606i \(0.846196\pi\)
\(228\) 0 0
\(229\) 22.6274i 1.49526i 0.664114 + 0.747631i \(0.268809\pi\)
−0.664114 + 0.747631i \(0.731191\pi\)
\(230\) 0 0
\(231\) −25.4558 −1.67487
\(232\) 0 0
\(233\) −22.0000 −1.44127 −0.720634 0.693316i \(-0.756149\pi\)
−0.720634 + 0.693316i \(0.756149\pi\)
\(234\) 0 0
\(235\) 4.00000i 0.260931i
\(236\) 0 0
\(237\) − 12.7279i − 0.826767i
\(238\) 0 0
\(239\) 11.3137 0.731823 0.365911 0.930650i \(-0.380757\pi\)
0.365911 + 0.930650i \(0.380757\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) 15.5563i 0.993859i
\(246\) 0 0
\(247\) 22.6274 1.43975
\(248\) 0 0
\(249\) 2.00000 0.126745
\(250\) 0 0
\(251\) 18.0000i 1.13615i 0.822977 + 0.568075i \(0.192312\pi\)
−0.822977 + 0.568075i \(0.807688\pi\)
\(252\) 0 0
\(253\) 16.9706i 1.06693i
\(254\) 0 0
\(255\) −8.48528 −0.531369
\(256\) 0 0
\(257\) −30.0000 −1.87135 −0.935674 0.352865i \(-0.885208\pi\)
−0.935674 + 0.352865i \(0.885208\pi\)
\(258\) 0 0
\(259\) − 36.0000i − 2.23693i
\(260\) 0 0
\(261\) 1.41421i 0.0875376i
\(262\) 0 0
\(263\) −16.9706 −1.04645 −0.523225 0.852195i \(-0.675271\pi\)
−0.523225 + 0.852195i \(0.675271\pi\)
\(264\) 0 0
\(265\) 14.0000 0.860013
\(266\) 0 0
\(267\) 2.00000i 0.122398i
\(268\) 0 0
\(269\) 12.7279i 0.776035i 0.921652 + 0.388018i \(0.126840\pi\)
−0.921652 + 0.388018i \(0.873160\pi\)
\(270\) 0 0
\(271\) 4.24264 0.257722 0.128861 0.991663i \(-0.458868\pi\)
0.128861 + 0.991663i \(0.458868\pi\)
\(272\) 0 0
\(273\) −24.0000 −1.45255
\(274\) 0 0
\(275\) − 18.0000i − 1.08544i
\(276\) 0 0
\(277\) 16.9706i 1.01966i 0.860274 + 0.509831i \(0.170292\pi\)
−0.860274 + 0.509831i \(0.829708\pi\)
\(278\) 0 0
\(279\) 1.41421 0.0846668
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) − 4.00000i − 0.237775i −0.992908 0.118888i \(-0.962067\pi\)
0.992908 0.118888i \(-0.0379328\pi\)
\(284\) 0 0
\(285\) 5.65685i 0.335083i
\(286\) 0 0
\(287\) 8.48528 0.500870
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) − 2.00000i − 0.117242i
\(292\) 0 0
\(293\) 24.0416i 1.40453i 0.711917 + 0.702264i \(0.247827\pi\)
−0.711917 + 0.702264i \(0.752173\pi\)
\(294\) 0 0
\(295\) 5.65685 0.329355
\(296\) 0 0
\(297\) 6.00000 0.348155
\(298\) 0 0
\(299\) 16.0000i 0.925304i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.41421 −0.0812444
\(304\) 0 0
\(305\) −12.0000 −0.687118
\(306\) 0 0
\(307\) − 16.0000i − 0.913168i −0.889680 0.456584i \(-0.849073\pi\)
0.889680 0.456584i \(-0.150927\pi\)
\(308\) 0 0
\(309\) 7.07107i 0.402259i
\(310\) 0 0
\(311\) −22.6274 −1.28308 −0.641542 0.767088i \(-0.721705\pi\)
−0.641542 + 0.767088i \(0.721705\pi\)
\(312\) 0 0
\(313\) 20.0000 1.13047 0.565233 0.824931i \(-0.308786\pi\)
0.565233 + 0.824931i \(0.308786\pi\)
\(314\) 0 0
\(315\) − 6.00000i − 0.338062i
\(316\) 0 0
\(317\) − 21.2132i − 1.19145i −0.803188 0.595726i \(-0.796864\pi\)
0.803188 0.595726i \(-0.203136\pi\)
\(318\) 0 0
\(319\) −8.48528 −0.475085
\(320\) 0 0
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) − 24.0000i − 1.33540i
\(324\) 0 0
\(325\) − 16.9706i − 0.941357i
\(326\) 0 0
\(327\) −16.9706 −0.938474
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) − 16.0000i − 0.879440i −0.898135 0.439720i \(-0.855078\pi\)
0.898135 0.439720i \(-0.144922\pi\)
\(332\) 0 0
\(333\) 8.48528i 0.464991i
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 10.0000i 0.543125i
\(340\) 0 0
\(341\) 8.48528i 0.459504i
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −4.00000 −0.215353
\(346\) 0 0
\(347\) − 2.00000i − 0.107366i −0.998558 0.0536828i \(-0.982904\pi\)
0.998558 0.0536828i \(-0.0170960\pi\)
\(348\) 0 0
\(349\) 14.1421i 0.757011i 0.925599 + 0.378506i \(0.123562\pi\)
−0.925599 + 0.378506i \(0.876438\pi\)
\(350\) 0 0
\(351\) 5.65685 0.301941
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) 4.00000i 0.212298i
\(356\) 0 0
\(357\) 25.4558i 1.34727i
\(358\) 0 0
\(359\) 31.1127 1.64207 0.821033 0.570881i \(-0.193398\pi\)
0.821033 + 0.570881i \(0.193398\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 25.0000i 1.31216i
\(364\) 0 0
\(365\) − 11.3137i − 0.592187i
\(366\) 0 0
\(367\) −12.7279 −0.664392 −0.332196 0.943210i \(-0.607790\pi\)
−0.332196 + 0.943210i \(0.607790\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) − 42.0000i − 2.18053i
\(372\) 0 0
\(373\) 19.7990i 1.02515i 0.858642 + 0.512576i \(0.171309\pi\)
−0.858642 + 0.512576i \(0.828691\pi\)
\(374\) 0 0
\(375\) 11.3137 0.584237
\(376\) 0 0
\(377\) −8.00000 −0.412021
\(378\) 0 0
\(379\) − 4.00000i − 0.205466i −0.994709 0.102733i \(-0.967241\pi\)
0.994709 0.102733i \(-0.0327588\pi\)
\(380\) 0 0
\(381\) − 12.7279i − 0.652071i
\(382\) 0 0
\(383\) 5.65685 0.289052 0.144526 0.989501i \(-0.453834\pi\)
0.144526 + 0.989501i \(0.453834\pi\)
\(384\) 0 0
\(385\) 36.0000 1.83473
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 1.41421i − 0.0717035i −0.999357 0.0358517i \(-0.988586\pi\)
0.999357 0.0358517i \(-0.0114144\pi\)
\(390\) 0 0
\(391\) 16.9706 0.858238
\(392\) 0 0
\(393\) −12.0000 −0.605320
\(394\) 0 0
\(395\) 18.0000i 0.905678i
\(396\) 0 0
\(397\) 31.1127i 1.56150i 0.624843 + 0.780751i \(0.285163\pi\)
−0.624843 + 0.780751i \(0.714837\pi\)
\(398\) 0 0
\(399\) 16.9706 0.849591
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) 0 0
\(405\) 1.41421i 0.0702728i
\(406\) 0 0
\(407\) −50.9117 −2.52360
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) − 18.0000i − 0.887875i
\(412\) 0 0
\(413\) − 16.9706i − 0.835067i
\(414\) 0 0
\(415\) −2.82843 −0.138842
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 22.0000i 1.07477i 0.843337 + 0.537385i \(0.180588\pi\)
−0.843337 + 0.537385i \(0.819412\pi\)
\(420\) 0 0
\(421\) − 11.3137i − 0.551396i −0.961244 0.275698i \(-0.911091\pi\)
0.961244 0.275698i \(-0.0889090\pi\)
\(422\) 0 0
\(423\) −2.82843 −0.137523
\(424\) 0 0
\(425\) −18.0000 −0.873128
\(426\) 0 0
\(427\) 36.0000i 1.74216i
\(428\) 0 0
\(429\) 33.9411i 1.63869i
\(430\) 0 0
\(431\) −19.7990 −0.953684 −0.476842 0.878989i \(-0.658219\pi\)
−0.476842 + 0.878989i \(0.658219\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) − 2.00000i − 0.0958927i
\(436\) 0 0
\(437\) − 11.3137i − 0.541208i
\(438\) 0 0
\(439\) 9.89949 0.472477 0.236239 0.971695i \(-0.424085\pi\)
0.236239 + 0.971695i \(0.424085\pi\)
\(440\) 0 0
\(441\) −11.0000 −0.523810
\(442\) 0 0
\(443\) 22.0000i 1.04525i 0.852562 + 0.522626i \(0.175047\pi\)
−0.852562 + 0.522626i \(0.824953\pi\)
\(444\) 0 0
\(445\) − 2.82843i − 0.134080i
\(446\) 0 0
\(447\) −4.24264 −0.200670
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) − 12.0000i − 0.565058i
\(452\) 0 0
\(453\) − 12.7279i − 0.598010i
\(454\) 0 0
\(455\) 33.9411 1.59118
\(456\) 0 0
\(457\) −42.0000 −1.96468 −0.982339 0.187112i \(-0.940087\pi\)
−0.982339 + 0.187112i \(0.940087\pi\)
\(458\) 0 0
\(459\) − 6.00000i − 0.280056i
\(460\) 0 0
\(461\) 35.3553i 1.64666i 0.567561 + 0.823331i \(0.307887\pi\)
−0.567561 + 0.823331i \(0.692113\pi\)
\(462\) 0 0
\(463\) 29.6985 1.38021 0.690103 0.723711i \(-0.257565\pi\)
0.690103 + 0.723711i \(0.257565\pi\)
\(464\) 0 0
\(465\) −2.00000 −0.0927478
\(466\) 0 0
\(467\) − 18.0000i − 0.832941i −0.909149 0.416470i \(-0.863267\pi\)
0.909149 0.416470i \(-0.136733\pi\)
\(468\) 0 0
\(469\) 33.9411i 1.56726i
\(470\) 0 0
\(471\) −2.82843 −0.130327
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 12.0000i 0.550598i
\(476\) 0 0
\(477\) 9.89949i 0.453267i
\(478\) 0 0
\(479\) −8.48528 −0.387702 −0.193851 0.981031i \(-0.562098\pi\)
−0.193851 + 0.981031i \(0.562098\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 0 0
\(483\) 12.0000i 0.546019i
\(484\) 0 0
\(485\) 2.82843i 0.128432i
\(486\) 0 0
\(487\) −35.3553 −1.60210 −0.801052 0.598595i \(-0.795726\pi\)
−0.801052 + 0.598595i \(0.795726\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 12.0000i 0.541552i 0.962642 + 0.270776i \(0.0872803\pi\)
−0.962642 + 0.270776i \(0.912720\pi\)
\(492\) 0 0
\(493\) 8.48528i 0.382158i
\(494\) 0 0
\(495\) −8.48528 −0.381385
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 28.0000i 1.25345i 0.779240 + 0.626726i \(0.215605\pi\)
−0.779240 + 0.626726i \(0.784395\pi\)
\(500\) 0 0
\(501\) − 22.6274i − 1.01092i
\(502\) 0 0
\(503\) −14.1421 −0.630567 −0.315283 0.948998i \(-0.602100\pi\)
−0.315283 + 0.948998i \(0.602100\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 19.0000i 0.843820i
\(508\) 0 0
\(509\) 9.89949i 0.438787i 0.975636 + 0.219394i \(0.0704079\pi\)
−0.975636 + 0.219394i \(0.929592\pi\)
\(510\) 0 0
\(511\) −33.9411 −1.50147
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) − 10.0000i − 0.440653i
\(516\) 0 0
\(517\) − 16.9706i − 0.746364i
\(518\) 0 0
\(519\) 7.07107 0.310385
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) − 4.00000i − 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) − 12.7279i − 0.555492i
\(526\) 0 0
\(527\) 8.48528 0.369625
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 4.00000i 0.173585i
\(532\) 0 0
\(533\) − 11.3137i − 0.490051i
\(534\) 0 0
\(535\) −5.65685 −0.244567
\(536\) 0 0
\(537\) 20.0000 0.863064
\(538\) 0 0
\(539\) − 66.0000i − 2.84282i
\(540\) 0 0
\(541\) − 22.6274i − 0.972829i −0.873728 0.486414i \(-0.838305\pi\)
0.873728 0.486414i \(-0.161695\pi\)
\(542\) 0 0
\(543\) −16.9706 −0.728277
\(544\) 0 0
\(545\) 24.0000 1.02805
\(546\) 0 0
\(547\) 40.0000i 1.71028i 0.518400 + 0.855138i \(0.326528\pi\)
−0.518400 + 0.855138i \(0.673472\pi\)
\(548\) 0 0
\(549\) − 8.48528i − 0.362143i
\(550\) 0 0
\(551\) 5.65685 0.240990
\(552\) 0 0
\(553\) 54.0000 2.29631
\(554\) 0 0
\(555\) − 12.0000i − 0.509372i
\(556\) 0 0
\(557\) − 7.07107i − 0.299611i −0.988716 0.149805i \(-0.952135\pi\)
0.988716 0.149805i \(-0.0478647\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 36.0000 1.51992
\(562\) 0 0
\(563\) 46.0000i 1.93867i 0.245745 + 0.969334i \(0.420967\pi\)
−0.245745 + 0.969334i \(0.579033\pi\)
\(564\) 0 0
\(565\) − 14.1421i − 0.594964i
\(566\) 0 0
\(567\) 4.24264 0.178174
\(568\) 0 0
\(569\) −22.0000 −0.922288 −0.461144 0.887325i \(-0.652561\pi\)
−0.461144 + 0.887325i \(0.652561\pi\)
\(570\) 0 0
\(571\) 28.0000i 1.17176i 0.810397 + 0.585882i \(0.199252\pi\)
−0.810397 + 0.585882i \(0.800748\pi\)
\(572\) 0 0
\(573\) − 11.3137i − 0.472637i
\(574\) 0 0
\(575\) −8.48528 −0.353861
\(576\) 0 0
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 0 0
\(579\) − 12.0000i − 0.498703i
\(580\) 0 0
\(581\) 8.48528i 0.352029i
\(582\) 0 0
\(583\) −59.3970 −2.45997
\(584\) 0 0
\(585\) −8.00000 −0.330759
\(586\) 0 0
\(587\) − 44.0000i − 1.81607i −0.418890 0.908037i \(-0.637581\pi\)
0.418890 0.908037i \(-0.362419\pi\)
\(588\) 0 0
\(589\) − 5.65685i − 0.233087i
\(590\) 0 0
\(591\) −4.24264 −0.174519
\(592\) 0 0
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) 0 0
\(595\) − 36.0000i − 1.47586i
\(596\) 0 0
\(597\) − 4.24264i − 0.173640i
\(598\) 0 0
\(599\) 31.1127 1.27123 0.635615 0.772006i \(-0.280747\pi\)
0.635615 + 0.772006i \(0.280747\pi\)
\(600\) 0 0
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 0 0
\(603\) − 8.00000i − 0.325785i
\(604\) 0 0
\(605\) − 35.3553i − 1.43740i
\(606\) 0 0
\(607\) −9.89949 −0.401808 −0.200904 0.979611i \(-0.564388\pi\)
−0.200904 + 0.979611i \(0.564388\pi\)
\(608\) 0 0
\(609\) −6.00000 −0.243132
\(610\) 0 0
\(611\) − 16.0000i − 0.647291i
\(612\) 0 0
\(613\) 2.82843i 0.114239i 0.998367 + 0.0571195i \(0.0181916\pi\)
−0.998367 + 0.0571195i \(0.981808\pi\)
\(614\) 0 0
\(615\) 2.82843 0.114053
\(616\) 0 0
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) − 28.0000i − 1.12542i −0.826656 0.562708i \(-0.809760\pi\)
0.826656 0.562708i \(-0.190240\pi\)
\(620\) 0 0
\(621\) − 2.82843i − 0.113501i
\(622\) 0 0
\(623\) −8.48528 −0.339956
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) − 24.0000i − 0.958468i
\(628\) 0 0
\(629\) 50.9117i 2.02998i
\(630\) 0 0
\(631\) −1.41421 −0.0562990 −0.0281495 0.999604i \(-0.508961\pi\)
−0.0281495 + 0.999604i \(0.508961\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 18.0000i 0.714308i
\(636\) 0 0
\(637\) − 62.2254i − 2.46546i
\(638\) 0 0
\(639\) −2.82843 −0.111891
\(640\) 0 0
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) − 32.0000i − 1.26196i −0.775800 0.630978i \(-0.782654\pi\)
0.775800 0.630978i \(-0.217346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 14.1421 0.555985 0.277992 0.960583i \(-0.410331\pi\)
0.277992 + 0.960583i \(0.410331\pi\)
\(648\) 0 0
\(649\) −24.0000 −0.942082
\(650\) 0 0
\(651\) 6.00000i 0.235159i
\(652\) 0 0
\(653\) 29.6985i 1.16219i 0.813835 + 0.581096i \(0.197376\pi\)
−0.813835 + 0.581096i \(0.802624\pi\)
\(654\) 0 0
\(655\) 16.9706 0.663095
\(656\) 0 0
\(657\) 8.00000 0.312110
\(658\) 0 0
\(659\) 36.0000i 1.40236i 0.712984 + 0.701180i \(0.247343\pi\)
−0.712984 + 0.701180i \(0.752657\pi\)
\(660\) 0 0
\(661\) − 8.48528i − 0.330039i −0.986290 0.165020i \(-0.947231\pi\)
0.986290 0.165020i \(-0.0527687\pi\)
\(662\) 0 0
\(663\) 33.9411 1.31816
\(664\) 0 0
\(665\) −24.0000 −0.930680
\(666\) 0 0
\(667\) 4.00000i 0.154881i
\(668\) 0 0
\(669\) − 15.5563i − 0.601443i
\(670\) 0 0
\(671\) 50.9117 1.96542
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) 0 0
\(675\) 3.00000i 0.115470i
\(676\) 0 0
\(677\) − 43.8406i − 1.68493i −0.538750 0.842466i \(-0.681103\pi\)
0.538750 0.842466i \(-0.318897\pi\)
\(678\) 0 0
\(679\) 8.48528 0.325635
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) − 30.0000i − 1.14792i −0.818884 0.573959i \(-0.805407\pi\)
0.818884 0.573959i \(-0.194593\pi\)
\(684\) 0 0
\(685\) 25.4558i 0.972618i
\(686\) 0 0
\(687\) 22.6274 0.863290
\(688\) 0 0
\(689\) −56.0000 −2.13343
\(690\) 0 0
\(691\) 32.0000i 1.21734i 0.793424 + 0.608669i \(0.208296\pi\)
−0.793424 + 0.608669i \(0.791704\pi\)
\(692\) 0 0
\(693\) 25.4558i 0.966988i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) 22.0000i 0.832116i
\(700\) 0 0
\(701\) − 38.1838i − 1.44218i −0.692841 0.721090i \(-0.743641\pi\)
0.692841 0.721090i \(-0.256359\pi\)
\(702\) 0 0
\(703\) 33.9411 1.28011
\(704\) 0 0
\(705\) 4.00000 0.150649
\(706\) 0 0
\(707\) − 6.00000i − 0.225653i
\(708\) 0 0
\(709\) − 16.9706i − 0.637343i −0.947865 0.318671i \(-0.896763\pi\)
0.947865 0.318671i \(-0.103237\pi\)
\(710\) 0 0
\(711\) −12.7279 −0.477334
\(712\) 0 0
\(713\) 4.00000 0.149801
\(714\) 0 0
\(715\) − 48.0000i − 1.79510i
\(716\) 0 0
\(717\) − 11.3137i − 0.422518i
\(718\) 0 0
\(719\) 31.1127 1.16031 0.580154 0.814507i \(-0.302992\pi\)
0.580154 + 0.814507i \(0.302992\pi\)
\(720\) 0 0
\(721\) −30.0000 −1.11726
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 4.24264i − 0.157568i
\(726\) 0 0
\(727\) −26.8701 −0.996555 −0.498278 0.867018i \(-0.666034\pi\)
−0.498278 + 0.867018i \(0.666034\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 28.2843i − 1.04470i −0.852730 0.522352i \(-0.825055\pi\)
0.852730 0.522352i \(-0.174945\pi\)
\(734\) 0 0
\(735\) 15.5563 0.573805
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) 20.0000i 0.735712i 0.929883 + 0.367856i \(0.119908\pi\)
−0.929883 + 0.367856i \(0.880092\pi\)
\(740\) 0 0
\(741\) − 22.6274i − 0.831239i
\(742\) 0 0
\(743\) 28.2843 1.03765 0.518825 0.854881i \(-0.326370\pi\)
0.518825 + 0.854881i \(0.326370\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) − 2.00000i − 0.0731762i
\(748\) 0 0
\(749\) 16.9706i 0.620091i
\(750\) 0 0
\(751\) 7.07107 0.258027 0.129013 0.991643i \(-0.458819\pi\)
0.129013 + 0.991643i \(0.458819\pi\)
\(752\) 0 0
\(753\) 18.0000 0.655956
\(754\) 0 0
\(755\) 18.0000i 0.655087i
\(756\) 0 0
\(757\) − 45.2548i − 1.64481i −0.568899 0.822407i \(-0.692630\pi\)
0.568899 0.822407i \(-0.307370\pi\)
\(758\) 0 0
\(759\) 16.9706 0.615992
\(760\) 0 0
\(761\) 46.0000 1.66750 0.833749 0.552143i \(-0.186190\pi\)
0.833749 + 0.552143i \(0.186190\pi\)
\(762\) 0 0
\(763\) − 72.0000i − 2.60658i
\(764\) 0 0
\(765\) 8.48528i 0.306786i
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) −28.0000 −1.00971 −0.504853 0.863205i \(-0.668453\pi\)
−0.504853 + 0.863205i \(0.668453\pi\)
\(770\) 0 0
\(771\) 30.0000i 1.08042i
\(772\) 0 0
\(773\) − 18.3848i − 0.661254i −0.943761 0.330627i \(-0.892740\pi\)
0.943761 0.330627i \(-0.107260\pi\)
\(774\) 0 0
\(775\) −4.24264 −0.152400
\(776\) 0 0
\(777\) −36.0000 −1.29149
\(778\) 0 0
\(779\) 8.00000i 0.286630i
\(780\) 0 0
\(781\) − 16.9706i − 0.607254i
\(782\) 0 0
\(783\) 1.41421 0.0505399
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 0 0
\(789\) 16.9706i 0.604168i
\(790\) 0 0
\(791\) −42.4264 −1.50851
\(792\) 0 0
\(793\) 48.0000 1.70453
\(794\) 0 0
\(795\) − 14.0000i − 0.496529i
\(796\) 0 0
\(797\) − 24.0416i − 0.851598i −0.904818 0.425799i \(-0.859993\pi\)
0.904818 0.425799i \(-0.140007\pi\)
\(798\) 0 0
\(799\) −16.9706 −0.600375
\(800\) 0 0
\(801\) 2.00000 0.0706665
\(802\) 0 0
\(803\) 48.0000i 1.69388i
\(804\) 0 0
\(805\) − 16.9706i − 0.598134i
\(806\) 0 0
\(807\) 12.7279 0.448044
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 0 0
\(813\) − 4.24264i − 0.148796i
\(814\) 0 0
\(815\) −11.3137 −0.396302
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 24.0000i 0.838628i
\(820\) 0 0
\(821\) − 1.41421i − 0.0493564i −0.999695 0.0246782i \(-0.992144\pi\)
0.999695 0.0246782i \(-0.00785611\pi\)
\(822\) 0 0
\(823\) −41.0122 −1.42960 −0.714798 0.699331i \(-0.753481\pi\)
−0.714798 + 0.699331i \(0.753481\pi\)
\(824\) 0 0
\(825\) −18.0000 −0.626680
\(826\) 0 0
\(827\) 28.0000i 0.973655i 0.873498 + 0.486828i \(0.161846\pi\)
−0.873498 + 0.486828i \(0.838154\pi\)
\(828\) 0 0
\(829\) 28.2843i 0.982353i 0.871060 + 0.491177i \(0.163433\pi\)
−0.871060 + 0.491177i \(0.836567\pi\)
\(830\) 0 0
\(831\) 16.9706 0.588702
\(832\) 0 0
\(833\) −66.0000 −2.28676
\(834\) 0 0
\(835\) 32.0000i 1.10741i
\(836\) 0 0
\(837\) − 1.41421i − 0.0488824i
\(838\) 0 0
\(839\) 19.7990 0.683537 0.341769 0.939784i \(-0.388974\pi\)
0.341769 + 0.939784i \(0.388974\pi\)
\(840\) 0 0
\(841\) 27.0000 0.931034
\(842\) 0 0
\(843\) − 18.0000i − 0.619953i
\(844\) 0 0
\(845\) − 26.8701i − 0.924358i
\(846\) 0 0
\(847\) −106.066 −3.64447
\(848\) 0 0
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) 24.0000i 0.822709i
\(852\) 0 0
\(853\) − 42.4264i − 1.45265i −0.687350 0.726326i \(-0.741226\pi\)
0.687350 0.726326i \(-0.258774\pi\)
\(854\) 0 0
\(855\) 5.65685 0.193460
\(856\) 0 0
\(857\) −46.0000 −1.57133 −0.785665 0.618652i \(-0.787679\pi\)
−0.785665 + 0.618652i \(0.787679\pi\)
\(858\) 0 0
\(859\) − 32.0000i − 1.09183i −0.837842 0.545913i \(-0.816183\pi\)
0.837842 0.545913i \(-0.183817\pi\)
\(860\) 0 0
\(861\) − 8.48528i − 0.289178i
\(862\) 0 0
\(863\) −11.3137 −0.385123 −0.192562 0.981285i \(-0.561680\pi\)
−0.192562 + 0.981285i \(0.561680\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) − 19.0000i − 0.645274i
\(868\) 0 0
\(869\) − 76.3675i − 2.59059i
\(870\) 0 0
\(871\) 45.2548 1.53340
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) 48.0000i 1.62270i
\(876\) 0 0
\(877\) 36.7696i 1.24162i 0.783961 + 0.620810i \(0.213196\pi\)
−0.783961 + 0.620810i \(0.786804\pi\)
\(878\) 0 0
\(879\) 24.0416 0.810904
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 24.0000i 0.807664i 0.914833 + 0.403832i \(0.132322\pi\)
−0.914833 + 0.403832i \(0.867678\pi\)
\(884\) 0 0
\(885\) − 5.65685i − 0.190153i
\(886\) 0 0
\(887\) 39.5980 1.32957 0.664785 0.747035i \(-0.268523\pi\)
0.664785 + 0.747035i \(0.268523\pi\)
\(888\) 0 0
\(889\) 54.0000 1.81110
\(890\) 0 0
\(891\) − 6.00000i − 0.201008i
\(892\) 0 0
\(893\) 11.3137i 0.378599i
\(894\) 0 0
\(895\) −28.2843 −0.945439
\(896\) 0 0
\(897\) 16.0000 0.534224
\(898\) 0 0
\(899\) 2.00000i 0.0667037i
\(900\) 0 0
\(901\) 59.3970i 1.97880i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 24.0000 0.797787
\(906\) 0 0
\(907\) 32.0000i 1.06254i 0.847202 + 0.531271i \(0.178286\pi\)
−0.847202 + 0.531271i \(0.821714\pi\)
\(908\) 0 0
\(909\) 1.41421i 0.0469065i
\(910\) 0 0
\(911\) −28.2843 −0.937100 −0.468550 0.883437i \(-0.655223\pi\)
−0.468550 + 0.883437i \(0.655223\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 12.0000i 0.396708i
\(916\) 0 0
\(917\) − 50.9117i − 1.68125i
\(918\) 0 0
\(919\) 26.8701 0.886361 0.443181 0.896432i \(-0.353850\pi\)
0.443181 + 0.896432i \(0.353850\pi\)
\(920\) 0 0
\(921\) −16.0000 −0.527218
\(922\) 0 0
\(923\) − 16.0000i − 0.526646i
\(924\) 0 0
\(925\) − 25.4558i − 0.836983i
\(926\) 0 0
\(927\) 7.07107 0.232244
\(928\) 0 0
\(929\) 18.0000 0.590561 0.295280 0.955411i \(-0.404587\pi\)
0.295280 + 0.955411i \(0.404587\pi\)
\(930\) 0 0
\(931\) 44.0000i 1.44204i
\(932\) 0 0
\(933\) 22.6274i 0.740788i
\(934\) 0 0
\(935\) −50.9117 −1.66499
\(936\) 0 0
\(937\) 18.0000 0.588034 0.294017 0.955800i \(-0.405008\pi\)
0.294017 + 0.955800i \(0.405008\pi\)
\(938\) 0 0
\(939\) − 20.0000i − 0.652675i
\(940\) 0 0
\(941\) 1.41421i 0.0461020i 0.999734 + 0.0230510i \(0.00733802\pi\)
−0.999734 + 0.0230510i \(0.992662\pi\)
\(942\) 0 0
\(943\) −5.65685 −0.184213
\(944\) 0 0
\(945\) −6.00000 −0.195180
\(946\) 0 0
\(947\) − 52.0000i − 1.68977i −0.534946 0.844886i \(-0.679668\pi\)
0.534946 0.844886i \(-0.320332\pi\)
\(948\) 0 0
\(949\) 45.2548i 1.46903i
\(950\) 0 0
\(951\) −21.2132 −0.687885
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 16.0000i 0.517748i
\(956\) 0 0
\(957\) 8.48528i 0.274290i
\(958\) 0 0
\(959\) 76.3675 2.46604
\(960\) 0 0
\(961\) −29.0000 −0.935484
\(962\) 0 0
\(963\) − 4.00000i − 0.128898i
\(964\) 0 0
\(965\) 16.9706i 0.546302i
\(966\) 0 0
\(967\) −21.2132 −0.682171 −0.341085 0.940032i \(-0.610795\pi\)
−0.341085 + 0.940032i \(0.610795\pi\)
\(968\) 0 0
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) − 14.0000i − 0.449281i −0.974442 0.224641i \(-0.927879\pi\)
0.974442 0.224641i \(-0.0721208\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −16.9706 −0.543493
\(976\) 0 0
\(977\) 42.0000 1.34370 0.671850 0.740688i \(-0.265500\pi\)
0.671850 + 0.740688i \(0.265500\pi\)
\(978\) 0 0
\(979\) 12.0000i 0.383522i
\(980\) 0 0
\(981\) 16.9706i 0.541828i
\(982\) 0 0
\(983\) 28.2843 0.902128 0.451064 0.892492i \(-0.351045\pi\)
0.451064 + 0.892492i \(0.351045\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) − 12.0000i − 0.381964i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −4.24264 −0.134772 −0.0673860 0.997727i \(-0.521466\pi\)
−0.0673860 + 0.997727i \(0.521466\pi\)
\(992\) 0 0
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 6.00000i 0.190213i
\(996\) 0 0
\(997\) − 31.1127i − 0.985349i −0.870214 0.492675i \(-0.836019\pi\)
0.870214 0.492675i \(-0.163981\pi\)
\(998\) 0 0
\(999\) 8.48528 0.268462
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 1536.2.d.c.769.2 4
3.2 odd 2 4608.2.d.n.2305.2 4
4.3 odd 2 inner 1536.2.d.c.769.4 4
8.3 odd 2 inner 1536.2.d.c.769.1 4
8.5 even 2 inner 1536.2.d.c.769.3 4
12.11 even 2 4608.2.d.n.2305.1 4
16.3 odd 4 1536.2.a.d.1.2 yes 2
16.5 even 4 1536.2.a.d.1.1 2
16.11 odd 4 1536.2.a.i.1.1 yes 2
16.13 even 4 1536.2.a.i.1.2 yes 2
24.5 odd 2 4608.2.d.n.2305.4 4
24.11 even 2 4608.2.d.n.2305.3 4
48.5 odd 4 4608.2.a.g.1.2 2
48.11 even 4 4608.2.a.l.1.2 2
48.29 odd 4 4608.2.a.l.1.1 2
48.35 even 4 4608.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.d.1.1 2 16.5 even 4
1536.2.a.d.1.2 yes 2 16.3 odd 4
1536.2.a.i.1.1 yes 2 16.11 odd 4
1536.2.a.i.1.2 yes 2 16.13 even 4
1536.2.d.c.769.1 4 8.3 odd 2 inner
1536.2.d.c.769.2 4 1.1 even 1 trivial
1536.2.d.c.769.3 4 8.5 even 2 inner
1536.2.d.c.769.4 4 4.3 odd 2 inner
4608.2.a.g.1.1 2 48.35 even 4
4608.2.a.g.1.2 2 48.5 odd 4
4608.2.a.l.1.1 2 48.29 odd 4
4608.2.a.l.1.2 2 48.11 even 4
4608.2.d.n.2305.1 4 12.11 even 2
4608.2.d.n.2305.2 4 3.2 odd 2
4608.2.d.n.2305.3 4 24.11 even 2
4608.2.d.n.2305.4 4 24.5 odd 2