Properties

Label 2304.3.b.g.127.1
Level $2304$
Weight $3$
Character 2304.127
Analytic conductor $62.779$
Analytic rank $0$
Dimension $2$
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] = [2304,3,Mod(127,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2304.127
Dual form 2304.3.b.g.127.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000i q^{5} -8.00000i q^{7} +O(q^{10})\) \(q-2.00000i q^{5} -8.00000i q^{7} +4.00000 q^{11} +14.0000i q^{13} -18.0000 q^{17} +12.0000 q^{19} -40.0000i q^{23} +21.0000 q^{25} -14.0000i q^{29} +32.0000i q^{31} -16.0000 q^{35} -30.0000i q^{37} -14.0000 q^{41} +28.0000 q^{43} +16.0000i q^{47} -15.0000 q^{49} -66.0000i q^{53} -8.00000i q^{55} +52.0000 q^{59} -82.0000i q^{61} +28.0000 q^{65} -4.00000 q^{67} -56.0000i q^{71} -66.0000 q^{73} -32.0000i q^{77} +16.0000i q^{79} -140.000 q^{83} +36.0000i q^{85} -30.0000 q^{89} +112.000 q^{91} -24.0000i q^{95} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{11} - 36 q^{17} + 24 q^{19} + 42 q^{25} - 32 q^{35} - 28 q^{41} + 56 q^{43} - 30 q^{49} + 104 q^{59} + 56 q^{65} - 8 q^{67} - 132 q^{73} - 280 q^{83} - 60 q^{89} + 224 q^{91} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 2.00000i − 0.400000i −0.979796 0.200000i \(-0.935906\pi\)
0.979796 0.200000i \(-0.0640942\pi\)
\(6\) 0 0
\(7\) − 8.00000i − 1.14286i −0.820652 0.571429i \(-0.806389\pi\)
0.820652 0.571429i \(-0.193611\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.00000 0.363636 0.181818 0.983332i \(-0.441802\pi\)
0.181818 + 0.983332i \(0.441802\pi\)
\(12\) 0 0
\(13\) 14.0000i 1.07692i 0.842650 + 0.538462i \(0.180994\pi\)
−0.842650 + 0.538462i \(0.819006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −18.0000 −1.05882 −0.529412 0.848365i \(-0.677587\pi\)
−0.529412 + 0.848365i \(0.677587\pi\)
\(18\) 0 0
\(19\) 12.0000 0.631579 0.315789 0.948829i \(-0.397731\pi\)
0.315789 + 0.948829i \(0.397731\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 40.0000i − 1.73913i −0.493818 0.869565i \(-0.664399\pi\)
0.493818 0.869565i \(-0.335601\pi\)
\(24\) 0 0
\(25\) 21.0000 0.840000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 14.0000i − 0.482759i −0.970431 0.241379i \(-0.922400\pi\)
0.970431 0.241379i \(-0.0775998\pi\)
\(30\) 0 0
\(31\) 32.0000i 1.03226i 0.856511 + 0.516129i \(0.172628\pi\)
−0.856511 + 0.516129i \(0.827372\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −16.0000 −0.457143
\(36\) 0 0
\(37\) − 30.0000i − 0.810811i −0.914137 0.405405i \(-0.867130\pi\)
0.914137 0.405405i \(-0.132870\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −14.0000 −0.341463 −0.170732 0.985318i \(-0.554613\pi\)
−0.170732 + 0.985318i \(0.554613\pi\)
\(42\) 0 0
\(43\) 28.0000 0.651163 0.325581 0.945514i \(-0.394440\pi\)
0.325581 + 0.945514i \(0.394440\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 16.0000i 0.340426i 0.985407 + 0.170213i \(0.0544455\pi\)
−0.985407 + 0.170213i \(0.945555\pi\)
\(48\) 0 0
\(49\) −15.0000 −0.306122
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 66.0000i − 1.24528i −0.782507 0.622642i \(-0.786060\pi\)
0.782507 0.622642i \(-0.213940\pi\)
\(54\) 0 0
\(55\) − 8.00000i − 0.145455i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 52.0000 0.881356 0.440678 0.897665i \(-0.354738\pi\)
0.440678 + 0.897665i \(0.354738\pi\)
\(60\) 0 0
\(61\) − 82.0000i − 1.34426i −0.740432 0.672131i \(-0.765379\pi\)
0.740432 0.672131i \(-0.234621\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 28.0000 0.430769
\(66\) 0 0
\(67\) −4.00000 −0.0597015 −0.0298507 0.999554i \(-0.509503\pi\)
−0.0298507 + 0.999554i \(0.509503\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 56.0000i − 0.788732i −0.918953 0.394366i \(-0.870964\pi\)
0.918953 0.394366i \(-0.129036\pi\)
\(72\) 0 0
\(73\) −66.0000 −0.904110 −0.452055 0.891990i \(-0.649309\pi\)
−0.452055 + 0.891990i \(0.649309\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 32.0000i − 0.415584i
\(78\) 0 0
\(79\) 16.0000i 0.202532i 0.994859 + 0.101266i \(0.0322893\pi\)
−0.994859 + 0.101266i \(0.967711\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −140.000 −1.68675 −0.843373 0.537328i \(-0.819434\pi\)
−0.843373 + 0.537328i \(0.819434\pi\)
\(84\) 0 0
\(85\) 36.0000i 0.423529i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −30.0000 −0.337079 −0.168539 0.985695i \(-0.553905\pi\)
−0.168539 + 0.985695i \(0.553905\pi\)
\(90\) 0 0
\(91\) 112.000 1.23077
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 24.0000i − 0.252632i
\(96\) 0 0
\(97\) −14.0000 −0.144330 −0.0721649 0.997393i \(-0.522991\pi\)
−0.0721649 + 0.997393i \(0.522991\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 94.0000i 0.930693i 0.885129 + 0.465347i \(0.154070\pi\)
−0.885129 + 0.465347i \(0.845930\pi\)
\(102\) 0 0
\(103\) 152.000i 1.47573i 0.674949 + 0.737864i \(0.264165\pi\)
−0.674949 + 0.737864i \(0.735835\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −156.000 −1.45794 −0.728972 0.684544i \(-0.760002\pi\)
−0.728972 + 0.684544i \(0.760002\pi\)
\(108\) 0 0
\(109\) − 18.0000i − 0.165138i −0.996585 0.0825688i \(-0.973688\pi\)
0.996585 0.0825688i \(-0.0263124\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −98.0000 −0.867257 −0.433628 0.901092i \(-0.642767\pi\)
−0.433628 + 0.901092i \(0.642767\pi\)
\(114\) 0 0
\(115\) −80.0000 −0.695652
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 144.000i 1.21008i
\(120\) 0 0
\(121\) −105.000 −0.867769
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 92.0000i − 0.736000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 68.0000 0.519084 0.259542 0.965732i \(-0.416428\pi\)
0.259542 + 0.965732i \(0.416428\pi\)
\(132\) 0 0
\(133\) − 96.0000i − 0.721805i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −206.000 −1.50365 −0.751825 0.659363i \(-0.770826\pi\)
−0.751825 + 0.659363i \(0.770826\pi\)
\(138\) 0 0
\(139\) −196.000 −1.41007 −0.705036 0.709172i \(-0.749069\pi\)
−0.705036 + 0.709172i \(0.749069\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 56.0000i 0.391608i
\(144\) 0 0
\(145\) −28.0000 −0.193103
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 226.000i − 1.51678i −0.651802 0.758389i \(-0.725987\pi\)
0.651802 0.758389i \(-0.274013\pi\)
\(150\) 0 0
\(151\) − 88.0000i − 0.582781i −0.956604 0.291391i \(-0.905882\pi\)
0.956604 0.291391i \(-0.0941180\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 64.0000 0.412903
\(156\) 0 0
\(157\) 46.0000i 0.292994i 0.989211 + 0.146497i \(0.0467998\pi\)
−0.989211 + 0.146497i \(0.953200\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −320.000 −1.98758
\(162\) 0 0
\(163\) 156.000 0.957055 0.478528 0.878073i \(-0.341171\pi\)
0.478528 + 0.878073i \(0.341171\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 40.0000i 0.239521i 0.992803 + 0.119760i \(0.0382127\pi\)
−0.992803 + 0.119760i \(0.961787\pi\)
\(168\) 0 0
\(169\) −27.0000 −0.159763
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 142.000i − 0.820809i −0.911904 0.410405i \(-0.865387\pi\)
0.911904 0.410405i \(-0.134613\pi\)
\(174\) 0 0
\(175\) − 168.000i − 0.960000i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 276.000 1.54190 0.770950 0.636896i \(-0.219782\pi\)
0.770950 + 0.636896i \(0.219782\pi\)
\(180\) 0 0
\(181\) − 30.0000i − 0.165746i −0.996560 0.0828729i \(-0.973590\pi\)
0.996560 0.0828729i \(-0.0264096\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −60.0000 −0.324324
\(186\) 0 0
\(187\) −72.0000 −0.385027
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 320.000i − 1.67539i −0.546136 0.837696i \(-0.683902\pi\)
0.546136 0.837696i \(-0.316098\pi\)
\(192\) 0 0
\(193\) −206.000 −1.06736 −0.533679 0.845687i \(-0.679191\pi\)
−0.533679 + 0.845687i \(0.679191\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 322.000i − 1.63452i −0.576271 0.817259i \(-0.695493\pi\)
0.576271 0.817259i \(-0.304507\pi\)
\(198\) 0 0
\(199\) − 200.000i − 1.00503i −0.864570 0.502513i \(-0.832409\pi\)
0.864570 0.502513i \(-0.167591\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −112.000 −0.551724
\(204\) 0 0
\(205\) 28.0000i 0.136585i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 48.0000 0.229665
\(210\) 0 0
\(211\) 140.000 0.663507 0.331754 0.943366i \(-0.392360\pi\)
0.331754 + 0.943366i \(0.392360\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 56.0000i − 0.260465i
\(216\) 0 0
\(217\) 256.000 1.17972
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 252.000i − 1.14027i
\(222\) 0 0
\(223\) 224.000i 1.00448i 0.864727 + 0.502242i \(0.167491\pi\)
−0.864727 + 0.502242i \(0.832509\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 164.000 0.722467 0.361233 0.932475i \(-0.382356\pi\)
0.361233 + 0.932475i \(0.382356\pi\)
\(228\) 0 0
\(229\) 2.00000i 0.00873362i 0.999990 + 0.00436681i \(0.00139000\pi\)
−0.999990 + 0.00436681i \(0.998610\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.00000 0.00858369 0.00429185 0.999991i \(-0.498634\pi\)
0.00429185 + 0.999991i \(0.498634\pi\)
\(234\) 0 0
\(235\) 32.0000 0.136170
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 208.000i 0.870293i 0.900360 + 0.435146i \(0.143303\pi\)
−0.900360 + 0.435146i \(0.856697\pi\)
\(240\) 0 0
\(241\) −46.0000 −0.190871 −0.0954357 0.995436i \(-0.530424\pi\)
−0.0954357 + 0.995436i \(0.530424\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 30.0000i 0.122449i
\(246\) 0 0
\(247\) 168.000i 0.680162i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.0478088 −0.0239044 0.999714i \(-0.507610\pi\)
−0.0239044 + 0.999714i \(0.507610\pi\)
\(252\) 0 0
\(253\) − 160.000i − 0.632411i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.00000 −0.00778210 −0.00389105 0.999992i \(-0.501239\pi\)
−0.00389105 + 0.999992i \(0.501239\pi\)
\(258\) 0 0
\(259\) −240.000 −0.926641
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 136.000i 0.517110i 0.965997 + 0.258555i \(0.0832464\pi\)
−0.965997 + 0.258555i \(0.916754\pi\)
\(264\) 0 0
\(265\) −132.000 −0.498113
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 402.000i 1.49442i 0.664586 + 0.747212i \(0.268608\pi\)
−0.664586 + 0.747212i \(0.731392\pi\)
\(270\) 0 0
\(271\) − 432.000i − 1.59410i −0.603916 0.797048i \(-0.706394\pi\)
0.603916 0.797048i \(-0.293606\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 84.0000 0.305455
\(276\) 0 0
\(277\) − 382.000i − 1.37906i −0.724256 0.689531i \(-0.757817\pi\)
0.724256 0.689531i \(-0.242183\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −350.000 −1.24555 −0.622776 0.782400i \(-0.713995\pi\)
−0.622776 + 0.782400i \(0.713995\pi\)
\(282\) 0 0
\(283\) −340.000 −1.20141 −0.600707 0.799469i \(-0.705114\pi\)
−0.600707 + 0.799469i \(0.705114\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 112.000i 0.390244i
\(288\) 0 0
\(289\) 35.0000 0.121107
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 2.00000i − 0.00682594i −0.999994 0.00341297i \(-0.998914\pi\)
0.999994 0.00341297i \(-0.00108638\pi\)
\(294\) 0 0
\(295\) − 104.000i − 0.352542i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 560.000 1.87291
\(300\) 0 0
\(301\) − 224.000i − 0.744186i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −164.000 −0.537705
\(306\) 0 0
\(307\) −84.0000 −0.273616 −0.136808 0.990598i \(-0.543684\pi\)
−0.136808 + 0.990598i \(0.543684\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 248.000i 0.797428i 0.917075 + 0.398714i \(0.130543\pi\)
−0.917075 + 0.398714i \(0.869457\pi\)
\(312\) 0 0
\(313\) −530.000 −1.69329 −0.846645 0.532158i \(-0.821381\pi\)
−0.846645 + 0.532158i \(0.821381\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 590.000i − 1.86120i −0.366039 0.930599i \(-0.619286\pi\)
0.366039 0.930599i \(-0.380714\pi\)
\(318\) 0 0
\(319\) − 56.0000i − 0.175549i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −216.000 −0.668731
\(324\) 0 0
\(325\) 294.000i 0.904615i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 128.000 0.389058
\(330\) 0 0
\(331\) 572.000 1.72810 0.864048 0.503409i \(-0.167921\pi\)
0.864048 + 0.503409i \(0.167921\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000i 0.0238806i
\(336\) 0 0
\(337\) 98.0000 0.290801 0.145401 0.989373i \(-0.453553\pi\)
0.145401 + 0.989373i \(0.453553\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 128.000i 0.375367i
\(342\) 0 0
\(343\) − 272.000i − 0.793003i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −172.000 −0.495677 −0.247839 0.968801i \(-0.579720\pi\)
−0.247839 + 0.968801i \(0.579720\pi\)
\(348\) 0 0
\(349\) − 434.000i − 1.24355i −0.783195 0.621777i \(-0.786411\pi\)
0.783195 0.621777i \(-0.213589\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −130.000 −0.368272 −0.184136 0.982901i \(-0.558949\pi\)
−0.184136 + 0.982901i \(0.558949\pi\)
\(354\) 0 0
\(355\) −112.000 −0.315493
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 408.000i − 1.13649i −0.822859 0.568245i \(-0.807623\pi\)
0.822859 0.568245i \(-0.192377\pi\)
\(360\) 0 0
\(361\) −217.000 −0.601108
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 132.000i 0.361644i
\(366\) 0 0
\(367\) 304.000i 0.828338i 0.910200 + 0.414169i \(0.135928\pi\)
−0.910200 + 0.414169i \(0.864072\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −528.000 −1.42318
\(372\) 0 0
\(373\) − 254.000i − 0.680965i −0.940251 0.340483i \(-0.889410\pi\)
0.940251 0.340483i \(-0.110590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 196.000 0.519894
\(378\) 0 0
\(379\) 268.000 0.707124 0.353562 0.935411i \(-0.384970\pi\)
0.353562 + 0.935411i \(0.384970\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 128.000i 0.334204i 0.985940 + 0.167102i \(0.0534409\pi\)
−0.985940 + 0.167102i \(0.946559\pi\)
\(384\) 0 0
\(385\) −64.0000 −0.166234
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 94.0000i 0.241645i 0.992674 + 0.120823i \(0.0385532\pi\)
−0.992674 + 0.120823i \(0.961447\pi\)
\(390\) 0 0
\(391\) 720.000i 1.84143i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 32.0000 0.0810127
\(396\) 0 0
\(397\) 558.000i 1.40554i 0.711416 + 0.702771i \(0.248054\pi\)
−0.711416 + 0.702771i \(0.751946\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 110.000 0.274314 0.137157 0.990549i \(-0.456203\pi\)
0.137157 + 0.990549i \(0.456203\pi\)
\(402\) 0 0
\(403\) −448.000 −1.11166
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 120.000i − 0.294840i
\(408\) 0 0
\(409\) 430.000 1.05134 0.525672 0.850687i \(-0.323814\pi\)
0.525672 + 0.850687i \(0.323814\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 416.000i − 1.00726i
\(414\) 0 0
\(415\) 280.000i 0.674699i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −476.000 −1.13604 −0.568019 0.823015i \(-0.692290\pi\)
−0.568019 + 0.823015i \(0.692290\pi\)
\(420\) 0 0
\(421\) 322.000i 0.764846i 0.923987 + 0.382423i \(0.124910\pi\)
−0.923987 + 0.382423i \(0.875090\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −378.000 −0.889412
\(426\) 0 0
\(427\) −656.000 −1.53630
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 528.000i 1.22506i 0.790448 + 0.612529i \(0.209848\pi\)
−0.790448 + 0.612529i \(0.790152\pi\)
\(432\) 0 0
\(433\) 210.000 0.484988 0.242494 0.970153i \(-0.422034\pi\)
0.242494 + 0.970153i \(0.422034\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 480.000i − 1.09840i
\(438\) 0 0
\(439\) − 376.000i − 0.856492i −0.903662 0.428246i \(-0.859132\pi\)
0.903662 0.428246i \(-0.140868\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −268.000 −0.604966 −0.302483 0.953155i \(-0.597816\pi\)
−0.302483 + 0.953155i \(0.597816\pi\)
\(444\) 0 0
\(445\) 60.0000i 0.134831i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 462.000 1.02895 0.514477 0.857504i \(-0.327986\pi\)
0.514477 + 0.857504i \(0.327986\pi\)
\(450\) 0 0
\(451\) −56.0000 −0.124169
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 224.000i − 0.492308i
\(456\) 0 0
\(457\) 590.000 1.29103 0.645514 0.763748i \(-0.276643\pi\)
0.645514 + 0.763748i \(0.276643\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 658.000i 1.42733i 0.700486 + 0.713666i \(0.252967\pi\)
−0.700486 + 0.713666i \(0.747033\pi\)
\(462\) 0 0
\(463\) − 112.000i − 0.241901i −0.992659 0.120950i \(-0.961406\pi\)
0.992659 0.120950i \(-0.0385942\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 180.000 0.385439 0.192719 0.981254i \(-0.438269\pi\)
0.192719 + 0.981254i \(0.438269\pi\)
\(468\) 0 0
\(469\) 32.0000i 0.0682303i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 112.000 0.236786
\(474\) 0 0
\(475\) 252.000 0.530526
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 32.0000i 0.0668058i 0.999442 + 0.0334029i \(0.0106345\pi\)
−0.999442 + 0.0334029i \(0.989366\pi\)
\(480\) 0 0
\(481\) 420.000 0.873181
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.0000i 0.0577320i
\(486\) 0 0
\(487\) − 360.000i − 0.739220i −0.929187 0.369610i \(-0.879491\pi\)
0.929187 0.369610i \(-0.120509\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 292.000 0.594705 0.297352 0.954768i \(-0.403896\pi\)
0.297352 + 0.954768i \(0.403896\pi\)
\(492\) 0 0
\(493\) 252.000i 0.511156i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −448.000 −0.901408
\(498\) 0 0
\(499\) 940.000 1.88377 0.941884 0.335939i \(-0.109054\pi\)
0.941884 + 0.335939i \(0.109054\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 56.0000i 0.111332i 0.998449 + 0.0556660i \(0.0177282\pi\)
−0.998449 + 0.0556660i \(0.982272\pi\)
\(504\) 0 0
\(505\) 188.000 0.372277
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 110.000i − 0.216110i −0.994145 0.108055i \(-0.965538\pi\)
0.994145 0.108055i \(-0.0344623\pi\)
\(510\) 0 0
\(511\) 528.000i 1.03327i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 304.000 0.590291
\(516\) 0 0
\(517\) 64.0000i 0.123791i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 178.000 0.341651 0.170825 0.985301i \(-0.445357\pi\)
0.170825 + 0.985301i \(0.445357\pi\)
\(522\) 0 0
\(523\) 380.000 0.726577 0.363289 0.931677i \(-0.381654\pi\)
0.363289 + 0.931677i \(0.381654\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 576.000i − 1.09298i
\(528\) 0 0
\(529\) −1071.00 −2.02457
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 196.000i − 0.367730i
\(534\) 0 0
\(535\) 312.000i 0.583178i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −60.0000 −0.111317
\(540\) 0 0
\(541\) 110.000i 0.203327i 0.994819 + 0.101664i \(0.0324165\pi\)
−0.994819 + 0.101664i \(0.967583\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −36.0000 −0.0660550
\(546\) 0 0
\(547\) 604.000 1.10420 0.552102 0.833776i \(-0.313826\pi\)
0.552102 + 0.833776i \(0.313826\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 168.000i − 0.304900i
\(552\) 0 0
\(553\) 128.000 0.231465
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 690.000i 1.23878i 0.785084 + 0.619390i \(0.212620\pi\)
−0.785084 + 0.619390i \(0.787380\pi\)
\(558\) 0 0
\(559\) 392.000i 0.701252i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 340.000 0.603908 0.301954 0.953323i \(-0.402361\pi\)
0.301954 + 0.953323i \(0.402361\pi\)
\(564\) 0 0
\(565\) 196.000i 0.346903i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 530.000 0.931459 0.465729 0.884927i \(-0.345792\pi\)
0.465729 + 0.884927i \(0.345792\pi\)
\(570\) 0 0
\(571\) −116.000 −0.203152 −0.101576 0.994828i \(-0.532389\pi\)
−0.101576 + 0.994828i \(0.532389\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 840.000i − 1.46087i
\(576\) 0 0
\(577\) 66.0000 0.114385 0.0571924 0.998363i \(-0.481785\pi\)
0.0571924 + 0.998363i \(0.481785\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1120.00i 1.92771i
\(582\) 0 0
\(583\) − 264.000i − 0.452830i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −700.000 −1.19250 −0.596252 0.802797i \(-0.703344\pi\)
−0.596252 + 0.802797i \(0.703344\pi\)
\(588\) 0 0
\(589\) 384.000i 0.651952i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 222.000 0.374368 0.187184 0.982325i \(-0.440064\pi\)
0.187184 + 0.982325i \(0.440064\pi\)
\(594\) 0 0
\(595\) 288.000 0.484034
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 872.000i − 1.45576i −0.685705 0.727880i \(-0.740506\pi\)
0.685705 0.727880i \(-0.259494\pi\)
\(600\) 0 0
\(601\) −994.000 −1.65391 −0.826955 0.562268i \(-0.809929\pi\)
−0.826955 + 0.562268i \(0.809929\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 210.000i 0.347107i
\(606\) 0 0
\(607\) − 800.000i − 1.31796i −0.752162 0.658979i \(-0.770989\pi\)
0.752162 0.658979i \(-0.229011\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −224.000 −0.366612
\(612\) 0 0
\(613\) − 318.000i − 0.518760i −0.965775 0.259380i \(-0.916482\pi\)
0.965775 0.259380i \(-0.0835182\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 322.000 0.521880 0.260940 0.965355i \(-0.415968\pi\)
0.260940 + 0.965355i \(0.415968\pi\)
\(618\) 0 0
\(619\) 284.000 0.458805 0.229402 0.973332i \(-0.426323\pi\)
0.229402 + 0.973332i \(0.426323\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 240.000i 0.385233i
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 540.000i 0.858506i
\(630\) 0 0
\(631\) 840.000i 1.33122i 0.746300 + 0.665610i \(0.231829\pi\)
−0.746300 + 0.665610i \(0.768171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 210.000i − 0.329670i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 718.000 1.12012 0.560062 0.828450i \(-0.310777\pi\)
0.560062 + 0.828450i \(0.310777\pi\)
\(642\) 0 0
\(643\) −196.000 −0.304821 −0.152411 0.988317i \(-0.548704\pi\)
−0.152411 + 0.988317i \(0.548704\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1032.00i 1.59505i 0.603283 + 0.797527i \(0.293859\pi\)
−0.603283 + 0.797527i \(0.706141\pi\)
\(648\) 0 0
\(649\) 208.000 0.320493
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 530.000i 0.811639i 0.913953 + 0.405819i \(0.133014\pi\)
−0.913953 + 0.405819i \(0.866986\pi\)
\(654\) 0 0
\(655\) − 136.000i − 0.207634i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 308.000 0.467375 0.233687 0.972312i \(-0.424921\pi\)
0.233687 + 0.972312i \(0.424921\pi\)
\(660\) 0 0
\(661\) 290.000i 0.438729i 0.975643 + 0.219365i \(0.0703984\pi\)
−0.975643 + 0.219365i \(0.929602\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −192.000 −0.288722
\(666\) 0 0
\(667\) −560.000 −0.839580
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 328.000i − 0.488823i
\(672\) 0 0
\(673\) −14.0000 −0.0208024 −0.0104012 0.999946i \(-0.503311\pi\)
−0.0104012 + 0.999946i \(0.503311\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 670.000i 0.989660i 0.868990 + 0.494830i \(0.164770\pi\)
−0.868990 + 0.494830i \(0.835230\pi\)
\(678\) 0 0
\(679\) 112.000i 0.164948i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 228.000 0.333821 0.166911 0.985972i \(-0.446621\pi\)
0.166911 + 0.985972i \(0.446621\pi\)
\(684\) 0 0
\(685\) 412.000i 0.601460i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 924.000 1.34107
\(690\) 0 0
\(691\) −1300.00 −1.88133 −0.940666 0.339335i \(-0.889798\pi\)
−0.940666 + 0.339335i \(0.889798\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 392.000i 0.564029i
\(696\) 0 0
\(697\) 252.000 0.361549
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 558.000i − 0.796006i −0.917384 0.398003i \(-0.869703\pi\)
0.917384 0.398003i \(-0.130297\pi\)
\(702\) 0 0
\(703\) − 360.000i − 0.512091i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 752.000 1.06365
\(708\) 0 0
\(709\) − 478.000i − 0.674189i −0.941471 0.337094i \(-0.890556\pi\)
0.941471 0.337094i \(-0.109444\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1280.00 1.79523
\(714\) 0 0
\(715\) 112.000 0.156643
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 912.000i − 1.26843i −0.773157 0.634214i \(-0.781324\pi\)
0.773157 0.634214i \(-0.218676\pi\)
\(720\) 0 0
\(721\) 1216.00 1.68655
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 294.000i − 0.405517i
\(726\) 0 0
\(727\) − 280.000i − 0.385144i −0.981283 0.192572i \(-0.938317\pi\)
0.981283 0.192572i \(-0.0616830\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −504.000 −0.689466
\(732\) 0 0
\(733\) − 754.000i − 1.02865i −0.857596 0.514325i \(-0.828043\pi\)
0.857596 0.514325i \(-0.171957\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −16.0000 −0.0217096
\(738\) 0 0
\(739\) −164.000 −0.221922 −0.110961 0.993825i \(-0.535393\pi\)
−0.110961 + 0.993825i \(0.535393\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 616.000i 0.829071i 0.910033 + 0.414536i \(0.136056\pi\)
−0.910033 + 0.414536i \(0.863944\pi\)
\(744\) 0 0
\(745\) −452.000 −0.606711
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1248.00i 1.66622i
\(750\) 0 0
\(751\) − 464.000i − 0.617843i −0.951088 0.308921i \(-0.900032\pi\)
0.951088 0.308921i \(-0.0999680\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −176.000 −0.233113
\(756\) 0 0
\(757\) 1442.00i 1.90489i 0.304714 + 0.952444i \(0.401439\pi\)
−0.304714 + 0.952444i \(0.598561\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −110.000 −0.144547 −0.0722733 0.997385i \(-0.523025\pi\)
−0.0722733 + 0.997385i \(0.523025\pi\)
\(762\) 0 0
\(763\) −144.000 −0.188729
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 728.000i 0.949153i
\(768\) 0 0
\(769\) 434.000 0.564369 0.282185 0.959360i \(-0.408941\pi\)
0.282185 + 0.959360i \(0.408941\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 990.000i 1.28072i 0.768073 + 0.640362i \(0.221216\pi\)
−0.768073 + 0.640362i \(0.778784\pi\)
\(774\) 0 0
\(775\) 672.000i 0.867097i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −168.000 −0.215661
\(780\) 0 0
\(781\) − 224.000i − 0.286812i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 92.0000 0.117197
\(786\) 0 0
\(787\) 1228.00 1.56036 0.780178 0.625558i \(-0.215129\pi\)
0.780178 + 0.625558i \(0.215129\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 784.000i 0.991150i
\(792\) 0 0
\(793\) 1148.00 1.44767
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 686.000i − 0.860728i −0.902655 0.430364i \(-0.858385\pi\)
0.902655 0.430364i \(-0.141615\pi\)
\(798\) 0 0
\(799\) − 288.000i − 0.360451i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −264.000 −0.328767
\(804\) 0 0
\(805\) 640.000i 0.795031i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −270.000 −0.333745 −0.166873 0.985978i \(-0.553367\pi\)
−0.166873 + 0.985978i \(0.553367\pi\)
\(810\) 0 0
\(811\) −420.000 −0.517879 −0.258940 0.965894i \(-0.583373\pi\)
−0.258940 + 0.965894i \(0.583373\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 312.000i − 0.382822i
\(816\) 0 0
\(817\) 336.000 0.411261
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 898.000i − 1.09379i −0.837202 0.546894i \(-0.815810\pi\)
0.837202 0.546894i \(-0.184190\pi\)
\(822\) 0 0
\(823\) − 248.000i − 0.301337i −0.988584 0.150668i \(-0.951857\pi\)
0.988584 0.150668i \(-0.0481425\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 756.000 0.914148 0.457074 0.889429i \(-0.348898\pi\)
0.457074 + 0.889429i \(0.348898\pi\)
\(828\) 0 0
\(829\) − 18.0000i − 0.0217129i −0.999941 0.0108565i \(-0.996544\pi\)
0.999941 0.0108565i \(-0.00345578\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 270.000 0.324130
\(834\) 0 0
\(835\) 80.0000 0.0958084
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 952.000i − 1.13468i −0.823482 0.567342i \(-0.807972\pi\)
0.823482 0.567342i \(-0.192028\pi\)
\(840\) 0 0
\(841\) 645.000 0.766944
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 54.0000i 0.0639053i
\(846\) 0 0
\(847\) 840.000i 0.991736i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1200.00 −1.41011
\(852\) 0 0
\(853\) − 1022.00i − 1.19812i −0.800703 0.599062i \(-0.795540\pi\)
0.800703 0.599062i \(-0.204460\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 850.000 0.991832 0.495916 0.868371i \(-0.334832\pi\)
0.495916 + 0.868371i \(0.334832\pi\)
\(858\) 0 0
\(859\) 940.000 1.09430 0.547148 0.837036i \(-0.315714\pi\)
0.547148 + 0.837036i \(0.315714\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1440.00i 1.66860i 0.551312 + 0.834299i \(0.314127\pi\)
−0.551312 + 0.834299i \(0.685873\pi\)
\(864\) 0 0
\(865\) −284.000 −0.328324
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 64.0000i 0.0736479i
\(870\) 0 0
\(871\) − 56.0000i − 0.0642939i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −736.000 −0.841143
\(876\) 0 0
\(877\) 1454.00i 1.65792i 0.559304 + 0.828962i \(0.311068\pi\)
−0.559304 + 0.828962i \(0.688932\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1374.00 1.55959 0.779796 0.626034i \(-0.215323\pi\)
0.779796 + 0.626034i \(0.215323\pi\)
\(882\) 0 0
\(883\) −1428.00 −1.61721 −0.808607 0.588349i \(-0.799778\pi\)
−0.808607 + 0.588349i \(0.799778\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 824.000i 0.928974i 0.885580 + 0.464487i \(0.153761\pi\)
−0.885580 + 0.464487i \(0.846239\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 192.000i 0.215006i
\(894\) 0 0
\(895\) − 552.000i − 0.616760i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 448.000 0.498331
\(900\) 0 0
\(901\) 1188.00i 1.31853i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −60.0000 −0.0662983
\(906\) 0 0
\(907\) −836.000 −0.921720 −0.460860 0.887473i \(-0.652459\pi\)
−0.460860 + 0.887473i \(0.652459\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 560.000i 0.614709i 0.951595 + 0.307355i \(0.0994438\pi\)
−0.951595 + 0.307355i \(0.900556\pi\)
\(912\) 0 0
\(913\) −560.000 −0.613363
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 544.000i − 0.593239i
\(918\) 0 0
\(919\) − 600.000i − 0.652884i −0.945217 0.326442i \(-0.894150\pi\)
0.945217 0.326442i \(-0.105850\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 784.000 0.849404
\(924\) 0 0
\(925\) − 630.000i − 0.681081i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −754.000 −0.811625 −0.405813 0.913956i \(-0.633011\pi\)
−0.405813 + 0.913956i \(0.633011\pi\)
\(930\) 0 0
\(931\) −180.000 −0.193340
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 144.000i 0.154011i
\(936\) 0 0
\(937\) 894.000 0.954109 0.477054 0.878874i \(-0.341704\pi\)
0.477054 + 0.878874i \(0.341704\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1298.00i 1.37938i 0.724103 + 0.689692i \(0.242254\pi\)
−0.724103 + 0.689692i \(0.757746\pi\)
\(942\) 0 0
\(943\) 560.000i 0.593849i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1748.00 1.84583 0.922914 0.385005i \(-0.125800\pi\)
0.922914 + 0.385005i \(0.125800\pi\)
\(948\) 0 0
\(949\) − 924.000i − 0.973656i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1134.00 −1.18993 −0.594963 0.803753i \(-0.702833\pi\)
−0.594963 + 0.803753i \(0.702833\pi\)
\(954\) 0 0
\(955\) −640.000 −0.670157
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1648.00i 1.71846i
\(960\) 0 0
\(961\) −63.0000 −0.0655567
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 412.000i 0.426943i
\(966\) 0 0
\(967\) − 1736.00i − 1.79524i −0.440767 0.897622i \(-0.645294\pi\)
0.440767 0.897622i \(-0.354706\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −124.000 −0.127703 −0.0638517 0.997959i \(-0.520338\pi\)
−0.0638517 + 0.997959i \(0.520338\pi\)
\(972\) 0 0
\(973\) 1568.00i 1.61151i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −914.000 −0.935517 −0.467758 0.883856i \(-0.654938\pi\)
−0.467758 + 0.883856i \(0.654938\pi\)
\(978\) 0 0
\(979\) −120.000 −0.122574
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1256.00i − 1.27772i −0.769322 0.638861i \(-0.779406\pi\)
0.769322 0.638861i \(-0.220594\pi\)
\(984\) 0 0
\(985\) −644.000 −0.653807
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1120.00i − 1.13246i
\(990\) 0 0
\(991\) 864.000i 0.871847i 0.899984 + 0.435923i \(0.143578\pi\)
−0.899984 + 0.435923i \(0.856422\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −400.000 −0.402010
\(996\) 0 0
\(997\) − 1054.00i − 1.05717i −0.848880 0.528586i \(-0.822723\pi\)
0.848880 0.528586i \(-0.177277\pi\)
\(998\) 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 2304.3.b.g.127.1 2
3.2 odd 2 256.3.d.a.127.2 2
4.3 odd 2 2304.3.b.c.127.1 2
8.3 odd 2 inner 2304.3.b.g.127.2 2
8.5 even 2 2304.3.b.c.127.2 2
12.11 even 2 256.3.d.c.127.2 2
16.3 odd 4 288.3.g.b.127.1 2
16.5 even 4 576.3.g.g.127.2 2
16.11 odd 4 576.3.g.g.127.1 2
16.13 even 4 288.3.g.b.127.2 2
24.5 odd 2 256.3.d.c.127.1 2
24.11 even 2 256.3.d.a.127.1 2
48.5 odd 4 64.3.c.b.63.2 2
48.11 even 4 64.3.c.b.63.1 2
48.29 odd 4 32.3.c.a.31.1 2
48.35 even 4 32.3.c.a.31.2 yes 2
240.29 odd 4 800.3.b.a.351.2 2
240.53 even 4 1600.3.h.a.1599.1 2
240.59 even 4 1600.3.b.e.1151.2 2
240.77 even 4 800.3.h.a.799.2 2
240.83 odd 4 800.3.h.a.799.1 2
240.107 odd 4 1600.3.h.a.1599.2 2
240.149 odd 4 1600.3.b.e.1151.1 2
240.173 even 4 800.3.h.b.799.2 2
240.179 even 4 800.3.b.a.351.1 2
240.197 even 4 1600.3.h.c.1599.1 2
240.203 odd 4 1600.3.h.c.1599.2 2
240.227 odd 4 800.3.h.b.799.1 2
336.83 odd 4 1568.3.d.b.1471.1 2
336.125 even 4 1568.3.d.b.1471.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.3.c.a.31.1 2 48.29 odd 4
32.3.c.a.31.2 yes 2 48.35 even 4
64.3.c.b.63.1 2 48.11 even 4
64.3.c.b.63.2 2 48.5 odd 4
256.3.d.a.127.1 2 24.11 even 2
256.3.d.a.127.2 2 3.2 odd 2
256.3.d.c.127.1 2 24.5 odd 2
256.3.d.c.127.2 2 12.11 even 2
288.3.g.b.127.1 2 16.3 odd 4
288.3.g.b.127.2 2 16.13 even 4
576.3.g.g.127.1 2 16.11 odd 4
576.3.g.g.127.2 2 16.5 even 4
800.3.b.a.351.1 2 240.179 even 4
800.3.b.a.351.2 2 240.29 odd 4
800.3.h.a.799.1 2 240.83 odd 4
800.3.h.a.799.2 2 240.77 even 4
800.3.h.b.799.1 2 240.227 odd 4
800.3.h.b.799.2 2 240.173 even 4
1568.3.d.b.1471.1 2 336.83 odd 4
1568.3.d.b.1471.2 2 336.125 even 4
1600.3.b.e.1151.1 2 240.149 odd 4
1600.3.b.e.1151.2 2 240.59 even 4
1600.3.h.a.1599.1 2 240.53 even 4
1600.3.h.a.1599.2 2 240.107 odd 4
1600.3.h.c.1599.1 2 240.197 even 4
1600.3.h.c.1599.2 2 240.203 odd 4
2304.3.b.c.127.1 2 4.3 odd 2
2304.3.b.c.127.2 2 8.5 even 2
2304.3.b.g.127.1 2 1.1 even 1 trivial
2304.3.b.g.127.2 2 8.3 odd 2 inner