Properties

Label 1152.3.b.j.703.1
Level $1152$
Weight $3$
Character 1152.703
Analytic conductor $31.390$
Analytic rank $0$
Dimension $8$
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] = [1152,3,Mod(703,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, 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("1152.703");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.b (of order \(2\), degree \(1\), 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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{22} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.1
Root \(0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 1152.703
Dual form 1152.3.b.j.703.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.89898i q^{5} -2.82843i q^{7} +O(q^{10})\) \(q-8.89898i q^{5} -2.82843i q^{7} -18.2419 q^{11} +5.79796i q^{13} +21.5959 q^{17} -18.2419 q^{19} -33.3697i q^{23} -54.1918 q^{25} -4.49490i q^{29} -2.25697i q^{31} -25.1701 q^{35} +43.1918i q^{37} +1.59592 q^{41} -63.4967 q^{43} +72.3962i q^{47} +41.0000 q^{49} -70.2929i q^{53} +162.334i q^{55} +34.6410 q^{59} +63.5959i q^{61} +51.5959 q^{65} +3.24259 q^{67} +68.4537i q^{71} -10.0000 q^{73} +51.5959i q^{77} -35.0552i q^{79} +42.2691 q^{83} -192.182i q^{85} -5.19184 q^{89} +16.3991 q^{91} +162.334i q^{95} -26.8082 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{17} - 120 q^{25} - 144 q^{41} + 328 q^{49} + 256 q^{65} - 80 q^{73} + 272 q^{89} - 528 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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
\(5\) − 8.89898i − 1.77980i −0.456160 0.889898i \(-0.650775\pi\)
0.456160 0.889898i \(-0.349225\pi\)
\(6\) 0 0
\(7\) − 2.82843i − 0.404061i −0.979379 0.202031i \(-0.935246\pi\)
0.979379 0.202031i \(-0.0647540\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −18.2419 −1.65836 −0.829178 0.558985i \(-0.811191\pi\)
−0.829178 + 0.558985i \(0.811191\pi\)
\(12\) 0 0
\(13\) 5.79796i 0.445997i 0.974819 + 0.222998i \(0.0715845\pi\)
−0.974819 + 0.222998i \(0.928416\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 21.5959 1.27035 0.635174 0.772369i \(-0.280928\pi\)
0.635174 + 0.772369i \(0.280928\pi\)
\(18\) 0 0
\(19\) −18.2419 −0.960101 −0.480050 0.877241i \(-0.659382\pi\)
−0.480050 + 0.877241i \(0.659382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 33.3697i − 1.45086i −0.688299 0.725428i \(-0.741642\pi\)
0.688299 0.725428i \(-0.258358\pi\)
\(24\) 0 0
\(25\) −54.1918 −2.16767
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 4.49490i − 0.154996i −0.996992 0.0774982i \(-0.975307\pi\)
0.996992 0.0774982i \(-0.0246932\pi\)
\(30\) 0 0
\(31\) − 2.25697i − 0.0728054i −0.999337 0.0364027i \(-0.988410\pi\)
0.999337 0.0364027i \(-0.0115899\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −25.1701 −0.719146
\(36\) 0 0
\(37\) 43.1918i 1.16735i 0.811988 + 0.583673i \(0.198385\pi\)
−0.811988 + 0.583673i \(0.801615\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.59592 0.0389248 0.0194624 0.999811i \(-0.493805\pi\)
0.0194624 + 0.999811i \(0.493805\pi\)
\(42\) 0 0
\(43\) −63.4967 −1.47667 −0.738334 0.674435i \(-0.764387\pi\)
−0.738334 + 0.674435i \(0.764387\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 72.3962i 1.54034i 0.637836 + 0.770172i \(0.279830\pi\)
−0.637836 + 0.770172i \(0.720170\pi\)
\(48\) 0 0
\(49\) 41.0000 0.836735
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 70.2929i − 1.32628i −0.748495 0.663140i \(-0.769223\pi\)
0.748495 0.663140i \(-0.230777\pi\)
\(54\) 0 0
\(55\) 162.334i 2.95153i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 34.6410 0.587136 0.293568 0.955938i \(-0.405157\pi\)
0.293568 + 0.955938i \(0.405157\pi\)
\(60\) 0 0
\(61\) 63.5959i 1.04256i 0.853387 + 0.521278i \(0.174545\pi\)
−0.853387 + 0.521278i \(0.825455\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 51.5959 0.793783
\(66\) 0 0
\(67\) 3.24259 0.0483968 0.0241984 0.999707i \(-0.492297\pi\)
0.0241984 + 0.999707i \(0.492297\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 68.4537i 0.964137i 0.876134 + 0.482068i \(0.160114\pi\)
−0.876134 + 0.482068i \(0.839886\pi\)
\(72\) 0 0
\(73\) −10.0000 −0.136986 −0.0684932 0.997652i \(-0.521819\pi\)
−0.0684932 + 0.997652i \(0.521819\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 51.5959i 0.670077i
\(78\) 0 0
\(79\) − 35.0552i − 0.443736i −0.975077 0.221868i \(-0.928785\pi\)
0.975077 0.221868i \(-0.0712155\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 42.2691 0.509266 0.254633 0.967038i \(-0.418045\pi\)
0.254633 + 0.967038i \(0.418045\pi\)
\(84\) 0 0
\(85\) − 192.182i − 2.26096i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19184 −0.0583352 −0.0291676 0.999575i \(-0.509286\pi\)
−0.0291676 + 0.999575i \(0.509286\pi\)
\(90\) 0 0
\(91\) 16.3991 0.180210
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 162.334i 1.70878i
\(96\) 0 0
\(97\) −26.8082 −0.276373 −0.138186 0.990406i \(-0.544127\pi\)
−0.138186 + 0.990406i \(0.544127\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 50.8786i 0.503748i 0.967760 + 0.251874i \(0.0810469\pi\)
−0.967760 + 0.251874i \(0.918953\pi\)
\(102\) 0 0
\(103\) 148.764i 1.44431i 0.691732 + 0.722154i \(0.256848\pi\)
−0.691732 + 0.722154i \(0.743152\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −116.380 −1.08766 −0.543830 0.839195i \(-0.683026\pi\)
−0.543830 + 0.839195i \(0.683026\pi\)
\(108\) 0 0
\(109\) − 44.1816i − 0.405336i −0.979248 0.202668i \(-0.935039\pi\)
0.979248 0.202668i \(-0.0649612\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −199.576 −1.76615 −0.883077 0.469227i \(-0.844533\pi\)
−0.883077 + 0.469227i \(0.844533\pi\)
\(114\) 0 0
\(115\) −296.956 −2.58223
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 61.0825i − 0.513298i
\(120\) 0 0
\(121\) 211.767 1.75014
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 259.778i 2.07822i
\(126\) 0 0
\(127\) 183.276i 1.44312i 0.692352 + 0.721560i \(0.256575\pi\)
−0.692352 + 0.721560i \(0.743425\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −176.891 −1.35031 −0.675155 0.737676i \(-0.735923\pi\)
−0.675155 + 0.737676i \(0.735923\pi\)
\(132\) 0 0
\(133\) 51.5959i 0.387939i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −147.980 −1.08014 −0.540071 0.841619i \(-0.681603\pi\)
−0.540071 + 0.841619i \(0.681603\pi\)
\(138\) 0 0
\(139\) −114.980 −0.827193 −0.413597 0.910460i \(-0.635728\pi\)
−0.413597 + 0.910460i \(0.635728\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 105.766i − 0.739621i
\(144\) 0 0
\(145\) −40.0000 −0.275862
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 229.303i − 1.53895i −0.638679 0.769473i \(-0.720519\pi\)
0.638679 0.769473i \(-0.279481\pi\)
\(150\) 0 0
\(151\) − 225.674i − 1.49453i −0.664527 0.747264i \(-0.731367\pi\)
0.664527 0.747264i \(-0.268633\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −20.0847 −0.129579
\(156\) 0 0
\(157\) 36.8082i 0.234447i 0.993106 + 0.117223i \(0.0373994\pi\)
−0.993106 + 0.117223i \(0.962601\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −94.3837 −0.586234
\(162\) 0 0
\(163\) 180.833 1.10941 0.554703 0.832048i \(-0.312832\pi\)
0.554703 + 0.832048i \(0.312832\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 174.791i − 1.04665i −0.852132 0.523326i \(-0.824691\pi\)
0.852132 0.523326i \(-0.175309\pi\)
\(168\) 0 0
\(169\) 135.384 0.801087
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 158.111i 0.913938i 0.889483 + 0.456969i \(0.151065\pi\)
−0.889483 + 0.456969i \(0.848935\pi\)
\(174\) 0 0
\(175\) 153.278i 0.875872i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −45.6979 −0.255295 −0.127648 0.991820i \(-0.540743\pi\)
−0.127648 + 0.991820i \(0.540743\pi\)
\(180\) 0 0
\(181\) − 263.778i − 1.45733i −0.684868 0.728667i \(-0.740140\pi\)
0.684868 0.728667i \(-0.259860\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 384.363 2.07764
\(186\) 0 0
\(187\) −393.951 −2.10669
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 154.963i − 0.811325i −0.914023 0.405663i \(-0.867041\pi\)
0.914023 0.405663i \(-0.132959\pi\)
\(192\) 0 0
\(193\) −182.767 −0.946981 −0.473491 0.880799i \(-0.657006\pi\)
−0.473491 + 0.880799i \(0.657006\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 178.697i − 0.907091i −0.891233 0.453546i \(-0.850159\pi\)
0.891233 0.453546i \(-0.149841\pi\)
\(198\) 0 0
\(199\) 37.9125i 0.190515i 0.995453 + 0.0952575i \(0.0303674\pi\)
−0.995453 + 0.0952575i \(0.969633\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.7135 −0.0626280
\(204\) 0 0
\(205\) − 14.2020i − 0.0692782i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 332.767 1.59219
\(210\) 0 0
\(211\) −19.8986 −0.0943060 −0.0471530 0.998888i \(-0.515015\pi\)
−0.0471530 + 0.998888i \(0.515015\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 565.056i 2.62817i
\(216\) 0 0
\(217\) −6.38367 −0.0294178
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 125.212i 0.566571i
\(222\) 0 0
\(223\) − 212.703i − 0.953827i −0.878950 0.476914i \(-0.841755\pi\)
0.878950 0.476914i \(-0.158245\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −104.180 −0.458942 −0.229471 0.973315i \(-0.573700\pi\)
−0.229471 + 0.973315i \(0.573700\pi\)
\(228\) 0 0
\(229\) 316.545i 1.38229i 0.722715 + 0.691146i \(0.242894\pi\)
−0.722715 + 0.691146i \(0.757106\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −120.424 −0.516843 −0.258422 0.966032i \(-0.583202\pi\)
−0.258422 + 0.966032i \(0.583202\pi\)
\(234\) 0 0
\(235\) 644.252 2.74150
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 178.734i − 0.747839i −0.927461 0.373919i \(-0.878014\pi\)
0.927461 0.373919i \(-0.121986\pi\)
\(240\) 0 0
\(241\) 46.8082 0.194225 0.0971124 0.995273i \(-0.469039\pi\)
0.0971124 + 0.995273i \(0.469039\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 364.858i − 1.48922i
\(246\) 0 0
\(247\) − 105.766i − 0.428202i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 39.9833 0.159296 0.0796480 0.996823i \(-0.474620\pi\)
0.0796480 + 0.996823i \(0.474620\pi\)
\(252\) 0 0
\(253\) 608.727i 2.40603i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −290.000 −1.12840 −0.564202 0.825637i \(-0.690816\pi\)
−0.564202 + 0.825637i \(0.690816\pi\)
\(258\) 0 0
\(259\) 122.165 0.471679
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 249.415i 0.948347i 0.880431 + 0.474174i \(0.157253\pi\)
−0.880431 + 0.474174i \(0.842747\pi\)
\(264\) 0 0
\(265\) −625.535 −2.36051
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 100.858i − 0.374937i −0.982271 0.187469i \(-0.939972\pi\)
0.982271 0.187469i \(-0.0600283\pi\)
\(270\) 0 0
\(271\) 222.817i 0.822201i 0.911590 + 0.411101i \(0.134856\pi\)
−0.911590 + 0.411101i \(0.865144\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 988.563 3.59477
\(276\) 0 0
\(277\) − 194.202i − 0.701090i −0.936546 0.350545i \(-0.885996\pi\)
0.936546 0.350545i \(-0.114004\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −202.767 −0.721592 −0.360796 0.932645i \(-0.617495\pi\)
−0.360796 + 0.932645i \(0.617495\pi\)
\(282\) 0 0
\(283\) 156.549 0.553177 0.276589 0.960988i \(-0.410796\pi\)
0.276589 + 0.960988i \(0.410796\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4.51394i − 0.0157280i
\(288\) 0 0
\(289\) 177.384 0.613784
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 116.677i − 0.398213i −0.979978 0.199107i \(-0.936196\pi\)
0.979978 0.199107i \(-0.0638040\pi\)
\(294\) 0 0
\(295\) − 308.270i − 1.04498i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 193.476 0.647077
\(300\) 0 0
\(301\) 179.596i 0.596664i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 565.939 1.85554
\(306\) 0 0
\(307\) 17.9850 0.0585832 0.0292916 0.999571i \(-0.490675\pi\)
0.0292916 + 0.999571i \(0.490675\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 290.214i − 0.933164i −0.884478 0.466582i \(-0.845485\pi\)
0.884478 0.466582i \(-0.154515\pi\)
\(312\) 0 0
\(313\) −255.535 −0.816405 −0.408202 0.912891i \(-0.633844\pi\)
−0.408202 + 0.912891i \(0.633844\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 93.4847i 0.294904i 0.989069 + 0.147452i \(0.0471073\pi\)
−0.989069 + 0.147452i \(0.952893\pi\)
\(318\) 0 0
\(319\) 81.9955i 0.257039i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −393.951 −1.21966
\(324\) 0 0
\(325\) − 314.202i − 0.966776i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 204.767 0.622393
\(330\) 0 0
\(331\) 151.464 0.457594 0.228797 0.973474i \(-0.426521\pi\)
0.228797 + 0.973474i \(0.426521\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 28.8557i − 0.0861365i
\(336\) 0 0
\(337\) 102.767 0.304948 0.152474 0.988308i \(-0.451276\pi\)
0.152474 + 0.988308i \(0.451276\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 41.1714i 0.120737i
\(342\) 0 0
\(343\) − 254.558i − 0.742153i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 333.626 0.961458 0.480729 0.876869i \(-0.340372\pi\)
0.480729 + 0.876869i \(0.340372\pi\)
\(348\) 0 0
\(349\) − 553.939i − 1.58722i −0.608429 0.793609i \(-0.708200\pi\)
0.608429 0.793609i \(-0.291800\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 398.727 1.12954 0.564768 0.825249i \(-0.308965\pi\)
0.564768 + 0.825249i \(0.308965\pi\)
\(354\) 0 0
\(355\) 609.168 1.71597
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 110.280i − 0.307186i −0.988134 0.153593i \(-0.950916\pi\)
0.988134 0.153593i \(-0.0490845\pi\)
\(360\) 0 0
\(361\) −28.2327 −0.0782068
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 88.9898i 0.243808i
\(366\) 0 0
\(367\) − 372.181i − 1.01412i −0.861912 0.507058i \(-0.830733\pi\)
0.861912 0.507058i \(-0.169267\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −198.818 −0.535898
\(372\) 0 0
\(373\) 277.980i 0.745254i 0.927981 + 0.372627i \(0.121543\pi\)
−0.927981 + 0.372627i \(0.878457\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 26.0612 0.0691279
\(378\) 0 0
\(379\) −320.797 −0.846430 −0.423215 0.906029i \(-0.639099\pi\)
−0.423215 + 0.906029i \(0.639099\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 609.797i − 1.59216i −0.605191 0.796080i \(-0.706903\pi\)
0.605191 0.796080i \(-0.293097\pi\)
\(384\) 0 0
\(385\) 459.151 1.19260
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 126.111i − 0.324193i −0.986775 0.162097i \(-0.948174\pi\)
0.986775 0.162097i \(-0.0518256\pi\)
\(390\) 0 0
\(391\) − 720.649i − 1.84309i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −311.955 −0.789760
\(396\) 0 0
\(397\) − 4.36326i − 0.0109906i −0.999985 0.00549529i \(-0.998251\pi\)
0.999985 0.00549529i \(-0.00174921\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −691.049 −1.72331 −0.861657 0.507491i \(-0.830573\pi\)
−0.861657 + 0.507491i \(0.830573\pi\)
\(402\) 0 0
\(403\) 13.0858 0.0324710
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 787.902i − 1.93588i
\(408\) 0 0
\(409\) 485.959 1.18816 0.594082 0.804404i \(-0.297515\pi\)
0.594082 + 0.804404i \(0.297515\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 97.9796i − 0.237239i
\(414\) 0 0
\(415\) − 376.152i − 0.906390i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −499.531 −1.19220 −0.596098 0.802911i \(-0.703283\pi\)
−0.596098 + 0.802911i \(0.703283\pi\)
\(420\) 0 0
\(421\) 21.8796i 0.0519705i 0.999662 + 0.0259853i \(0.00827230\pi\)
−0.999662 + 0.0259853i \(0.991728\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1170.32 −2.75370
\(426\) 0 0
\(427\) 179.876 0.421256
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 513.631i 1.19172i 0.803089 + 0.595859i \(0.203188\pi\)
−0.803089 + 0.595859i \(0.796812\pi\)
\(432\) 0 0
\(433\) −16.3837 −0.0378376 −0.0189188 0.999821i \(-0.506022\pi\)
−0.0189188 + 0.999821i \(0.506022\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 608.727i 1.39297i
\(438\) 0 0
\(439\) 324.068i 0.738197i 0.929390 + 0.369098i \(0.120333\pi\)
−0.929390 + 0.369098i \(0.879667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 221.003 0.498877 0.249439 0.968391i \(-0.419754\pi\)
0.249439 + 0.968391i \(0.419754\pi\)
\(444\) 0 0
\(445\) 46.2020i 0.103825i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −690.322 −1.53747 −0.768733 0.639570i \(-0.779113\pi\)
−0.768733 + 0.639570i \(0.779113\pi\)
\(450\) 0 0
\(451\) −29.1126 −0.0645512
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 145.935i − 0.320737i
\(456\) 0 0
\(457\) 397.878 0.870629 0.435315 0.900278i \(-0.356637\pi\)
0.435315 + 0.900278i \(0.356637\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 707.160i 1.53397i 0.641665 + 0.766985i \(0.278244\pi\)
−0.641665 + 0.766985i \(0.721756\pi\)
\(462\) 0 0
\(463\) − 869.926i − 1.87889i −0.342700 0.939445i \(-0.611341\pi\)
0.342700 0.939445i \(-0.388659\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −75.4396 −0.161541 −0.0807705 0.996733i \(-0.525738\pi\)
−0.0807705 + 0.996733i \(0.525738\pi\)
\(468\) 0 0
\(469\) − 9.17143i − 0.0195553i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1158.30 2.44884
\(474\) 0 0
\(475\) 988.563 2.08118
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 90.5674i − 0.189076i −0.995521 0.0945380i \(-0.969863\pi\)
0.995521 0.0945380i \(-0.0301374\pi\)
\(480\) 0 0
\(481\) −250.424 −0.520633
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 238.565i 0.491887i
\(486\) 0 0
\(487\) 137.392i 0.282120i 0.990001 + 0.141060i \(0.0450510\pi\)
−0.990001 + 0.141060i \(0.954949\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 89.5529 0.182389 0.0911944 0.995833i \(-0.470932\pi\)
0.0911944 + 0.995833i \(0.470932\pi\)
\(492\) 0 0
\(493\) − 97.0714i − 0.196899i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 193.616 0.389570
\(498\) 0 0
\(499\) −268.286 −0.537648 −0.268824 0.963189i \(-0.586635\pi\)
−0.268824 + 0.963189i \(0.586635\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 93.2515i − 0.185391i −0.995695 0.0926953i \(-0.970452\pi\)
0.995695 0.0926953i \(-0.0295483\pi\)
\(504\) 0 0
\(505\) 452.767 0.896569
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 168.272i 0.330594i 0.986244 + 0.165297i \(0.0528583\pi\)
−0.986244 + 0.165297i \(0.947142\pi\)
\(510\) 0 0
\(511\) 28.2843i 0.0553508i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1323.85 2.57057
\(516\) 0 0
\(517\) − 1320.64i − 2.55444i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −316.788 −0.608038 −0.304019 0.952666i \(-0.598329\pi\)
−0.304019 + 0.952666i \(0.598329\pi\)
\(522\) 0 0
\(523\) −443.591 −0.848167 −0.424083 0.905623i \(-0.639404\pi\)
−0.424083 + 0.905623i \(0.639404\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 48.7413i − 0.0924883i
\(528\) 0 0
\(529\) −584.535 −1.10498
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 9.25307i 0.0173604i
\(534\) 0 0
\(535\) 1035.66i 1.93581i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −747.918 −1.38760
\(540\) 0 0
\(541\) − 966.443i − 1.78640i −0.449659 0.893200i \(-0.648454\pi\)
0.449659 0.893200i \(-0.351546\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −393.171 −0.721415
\(546\) 0 0
\(547\) 578.470 1.05753 0.528766 0.848768i \(-0.322655\pi\)
0.528766 + 0.848768i \(0.322655\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 81.9955i 0.148812i
\(552\) 0 0
\(553\) −99.1510 −0.179297
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 590.293i − 1.05977i −0.848069 0.529886i \(-0.822235\pi\)
0.848069 0.529886i \(-0.177765\pi\)
\(558\) 0 0
\(559\) − 368.152i − 0.658589i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 468.877 0.832818 0.416409 0.909177i \(-0.363288\pi\)
0.416409 + 0.909177i \(0.363288\pi\)
\(564\) 0 0
\(565\) 1776.02i 3.14340i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 548.829 0.964549 0.482275 0.876020i \(-0.339811\pi\)
0.482275 + 0.876020i \(0.339811\pi\)
\(570\) 0 0
\(571\) −133.036 −0.232987 −0.116494 0.993191i \(-0.537165\pi\)
−0.116494 + 0.993191i \(0.537165\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1808.36i 3.14498i
\(576\) 0 0
\(577\) −735.535 −1.27476 −0.637378 0.770551i \(-0.719981\pi\)
−0.637378 + 0.770551i \(0.719981\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 119.555i − 0.205775i
\(582\) 0 0
\(583\) 1282.28i 2.19944i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 178.290 0.303732 0.151866 0.988401i \(-0.451472\pi\)
0.151866 + 0.988401i \(0.451472\pi\)
\(588\) 0 0
\(589\) 41.1714i 0.0699006i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 584.261 0.985263 0.492632 0.870238i \(-0.336035\pi\)
0.492632 + 0.870238i \(0.336035\pi\)
\(594\) 0 0
\(595\) −543.572 −0.913566
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 134.050i 0.223790i 0.993720 + 0.111895i \(0.0356920\pi\)
−0.993720 + 0.111895i \(0.964308\pi\)
\(600\) 0 0
\(601\) 621.918 1.03481 0.517403 0.855742i \(-0.326899\pi\)
0.517403 + 0.855742i \(0.326899\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 1884.51i − 3.11490i
\(606\) 0 0
\(607\) − 36.1404i − 0.0595393i −0.999557 0.0297697i \(-0.990523\pi\)
0.999557 0.0297697i \(-0.00947738\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −419.750 −0.686989
\(612\) 0 0
\(613\) 385.857i 0.629457i 0.949182 + 0.314728i \(0.101913\pi\)
−0.949182 + 0.314728i \(0.898087\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −290.849 −0.471392 −0.235696 0.971827i \(-0.575737\pi\)
−0.235696 + 0.971827i \(0.575737\pi\)
\(618\) 0 0
\(619\) −275.658 −0.445327 −0.222664 0.974895i \(-0.571475\pi\)
−0.222664 + 0.974895i \(0.571475\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.6847i 0.0235710i
\(624\) 0 0
\(625\) 956.959 1.53113
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 932.767i 1.48294i
\(630\) 0 0
\(631\) − 940.608i − 1.49066i −0.666694 0.745331i \(-0.732291\pi\)
0.666694 0.745331i \(-0.267709\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1630.97 2.56846
\(636\) 0 0
\(637\) 237.716i 0.373181i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 34.3633 0.0536088 0.0268044 0.999641i \(-0.491467\pi\)
0.0268044 + 0.999641i \(0.491467\pi\)
\(642\) 0 0
\(643\) −308.340 −0.479534 −0.239767 0.970830i \(-0.577071\pi\)
−0.239767 + 0.970830i \(0.577071\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 190.503i − 0.294441i −0.989104 0.147220i \(-0.952967\pi\)
0.989104 0.147220i \(-0.0470327\pi\)
\(648\) 0 0
\(649\) −631.918 −0.973680
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1103.38i − 1.68971i −0.534993 0.844857i \(-0.679686\pi\)
0.534993 0.844857i \(-0.320314\pi\)
\(654\) 0 0
\(655\) 1574.15i 2.40328i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −815.544 −1.23755 −0.618774 0.785569i \(-0.712370\pi\)
−0.618774 + 0.785569i \(0.712370\pi\)
\(660\) 0 0
\(661\) 121.576i 0.183927i 0.995762 + 0.0919633i \(0.0293143\pi\)
−0.995762 + 0.0919633i \(0.970686\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 459.151 0.690453
\(666\) 0 0
\(667\) −149.993 −0.224877
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 1160.11i − 1.72893i
\(672\) 0 0
\(673\) −1326.60 −1.97118 −0.985590 0.169152i \(-0.945897\pi\)
−0.985590 + 0.169152i \(0.945897\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 657.526i 0.971234i 0.874172 + 0.485617i \(0.161405\pi\)
−0.874172 + 0.485617i \(0.838595\pi\)
\(678\) 0 0
\(679\) 75.8249i 0.111671i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −772.318 −1.13077 −0.565386 0.824826i \(-0.691273\pi\)
−0.565386 + 0.824826i \(0.691273\pi\)
\(684\) 0 0
\(685\) 1316.87i 1.92243i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 407.555 0.591517
\(690\) 0 0
\(691\) 1269.00 1.83647 0.918237 0.396030i \(-0.129613\pi\)
0.918237 + 0.396030i \(0.129613\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1023.20i 1.47224i
\(696\) 0 0
\(697\) 34.4653 0.0494481
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 535.383i 0.763741i 0.924216 + 0.381871i \(0.124720\pi\)
−0.924216 + 0.381871i \(0.875280\pi\)
\(702\) 0 0
\(703\) − 787.902i − 1.12077i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 143.906 0.203545
\(708\) 0 0
\(709\) − 56.8673i − 0.0802078i −0.999196 0.0401039i \(-0.987231\pi\)
0.999196 0.0401039i \(-0.0127689\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −75.3143 −0.105630
\(714\) 0 0
\(715\) −941.208 −1.31638
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 980.922i 1.36429i 0.731219 + 0.682143i \(0.238952\pi\)
−0.731219 + 0.682143i \(0.761048\pi\)
\(720\) 0 0
\(721\) 420.767 0.583589
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 243.587i 0.335982i
\(726\) 0 0
\(727\) − 83.1673i − 0.114398i −0.998363 0.0571990i \(-0.981783\pi\)
0.998363 0.0571990i \(-0.0182169\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −1371.27 −1.87588
\(732\) 0 0
\(733\) − 345.476i − 0.471317i −0.971836 0.235659i \(-0.924275\pi\)
0.971836 0.235659i \(-0.0757247\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −59.1510 −0.0802592
\(738\) 0 0
\(739\) −1388.23 −1.87852 −0.939261 0.343203i \(-0.888488\pi\)
−0.939261 + 0.343203i \(0.888488\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 885.269i − 1.19148i −0.803178 0.595739i \(-0.796859\pi\)
0.803178 0.595739i \(-0.203141\pi\)
\(744\) 0 0
\(745\) −2040.56 −2.73901
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 329.171i 0.439481i
\(750\) 0 0
\(751\) 1225.11i 1.63130i 0.578545 + 0.815651i \(0.303621\pi\)
−0.578545 + 0.815651i \(0.696379\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2008.27 −2.65996
\(756\) 0 0
\(757\) 86.9694i 0.114887i 0.998349 + 0.0574435i \(0.0182949\pi\)
−0.998349 + 0.0574435i \(0.981705\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 764.020 1.00397 0.501985 0.864877i \(-0.332603\pi\)
0.501985 + 0.864877i \(0.332603\pi\)
\(762\) 0 0
\(763\) −124.965 −0.163781
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 200.847i 0.261861i
\(768\) 0 0
\(769\) 269.918 0.350999 0.175500 0.984480i \(-0.443846\pi\)
0.175500 + 0.984480i \(0.443846\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1109.24i − 1.43498i −0.696569 0.717490i \(-0.745291\pi\)
0.696569 0.717490i \(-0.254709\pi\)
\(774\) 0 0
\(775\) 122.309i 0.157818i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −29.1126 −0.0373718
\(780\) 0 0
\(781\) − 1248.73i − 1.59888i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 327.555 0.417268
\(786\) 0 0
\(787\) −1260.98 −1.60226 −0.801129 0.598491i \(-0.795767\pi\)
−0.801129 + 0.598491i \(0.795767\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 564.485i 0.713634i
\(792\) 0 0
\(793\) −368.727 −0.464977
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 386.779i − 0.485293i −0.970115 0.242647i \(-0.921984\pi\)
0.970115 0.242647i \(-0.0780155\pi\)
\(798\) 0 0
\(799\) 1563.46i 1.95677i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 182.419 0.227172
\(804\) 0 0
\(805\) 839.918i 1.04338i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1139.98 −1.40912 −0.704561 0.709643i \(-0.748856\pi\)
−0.704561 + 0.709643i \(0.748856\pi\)
\(810\) 0 0
\(811\) −959.450 −1.18305 −0.591523 0.806288i \(-0.701473\pi\)
−0.591523 + 0.806288i \(0.701473\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1609.23i − 1.97452i
\(816\) 0 0
\(817\) 1158.30 1.41775
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 918.674i 1.11897i 0.828840 + 0.559485i \(0.189001\pi\)
−0.828840 + 0.559485i \(0.810999\pi\)
\(822\) 0 0
\(823\) − 678.107i − 0.823945i −0.911196 0.411972i \(-0.864840\pi\)
0.911196 0.411972i \(-0.135160\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1237.58 −1.49647 −0.748234 0.663434i \(-0.769098\pi\)
−0.748234 + 0.663434i \(0.769098\pi\)
\(828\) 0 0
\(829\) 778.120i 0.938625i 0.883032 + 0.469313i \(0.155498\pi\)
−0.883032 + 0.469313i \(0.844502\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 885.433 1.06294
\(834\) 0 0
\(835\) −1555.46 −1.86283
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 345.582i − 0.411897i −0.978563 0.205949i \(-0.933972\pi\)
0.978563 0.205949i \(-0.0660280\pi\)
\(840\) 0 0
\(841\) 820.796 0.975976
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1204.78i − 1.42577i
\(846\) 0 0
\(847\) − 598.968i − 0.707165i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1441.30 1.69365
\(852\) 0 0
\(853\) 1026.38i 1.20326i 0.798774 + 0.601632i \(0.205482\pi\)
−0.798774 + 0.601632i \(0.794518\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 296.061 0.345462 0.172731 0.984969i \(-0.444741\pi\)
0.172731 + 0.984969i \(0.444741\pi\)
\(858\) 0 0
\(859\) 167.118 0.194550 0.0972748 0.995258i \(-0.468987\pi\)
0.0972748 + 0.995258i \(0.468987\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 10.0553i − 0.0116516i −0.999983 0.00582580i \(-0.998146\pi\)
0.999983 0.00582580i \(-0.00185442\pi\)
\(864\) 0 0
\(865\) 1407.03 1.62662
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 639.473i 0.735873i
\(870\) 0 0
\(871\) 18.8004i 0.0215848i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 734.762 0.839728
\(876\) 0 0
\(877\) − 1154.58i − 1.31651i −0.752793 0.658257i \(-0.771294\pi\)
0.752793 0.658257i \(-0.228706\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −160.220 −0.181862 −0.0909310 0.995857i \(-0.528984\pi\)
−0.0909310 + 0.995857i \(0.528984\pi\)
\(882\) 0 0
\(883\) 1311.46 1.48523 0.742616 0.669718i \(-0.233585\pi\)
0.742616 + 0.669718i \(0.233585\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 234.673i 0.264569i 0.991212 + 0.132285i \(0.0422313\pi\)
−0.991212 + 0.132285i \(0.957769\pi\)
\(888\) 0 0
\(889\) 518.384 0.583109
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1320.64i − 1.47889i
\(894\) 0 0
\(895\) 406.664i 0.454374i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10.1448 −0.0112846
\(900\) 0 0
\(901\) − 1518.04i − 1.68484i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2347.35 −2.59376
\(906\) 0 0
\(907\) 1161.70 1.28081 0.640406 0.768036i \(-0.278766\pi\)
0.640406 + 0.768036i \(0.278766\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1382.33i − 1.51737i −0.651456 0.758687i \(-0.725841\pi\)
0.651456 0.758687i \(-0.274159\pi\)
\(912\) 0 0
\(913\) −771.069 −0.844545
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 500.322i 0.545608i
\(918\) 0 0
\(919\) 1206.60i 1.31294i 0.754350 + 0.656472i \(0.227952\pi\)
−0.754350 + 0.656472i \(0.772048\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −396.892 −0.430002
\(924\) 0 0
\(925\) − 2340.64i − 2.53043i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1159.90 −1.24854 −0.624272 0.781207i \(-0.714604\pi\)
−0.624272 + 0.781207i \(0.714604\pi\)
\(930\) 0 0
\(931\) −747.918 −0.803349
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3505.76i 3.74948i
\(936\) 0 0
\(937\) 173.837 0.185525 0.0927624 0.995688i \(-0.470430\pi\)
0.0927624 + 0.995688i \(0.470430\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 603.968i 0.641837i 0.947107 + 0.320918i \(0.103992\pi\)
−0.947107 + 0.320918i \(0.896008\pi\)
\(942\) 0 0
\(943\) − 53.2553i − 0.0564743i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −951.077 −1.00431 −0.502153 0.864779i \(-0.667458\pi\)
−0.502153 + 0.864779i \(0.667458\pi\)
\(948\) 0 0
\(949\) − 57.9796i − 0.0610955i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1407.98 1.47742 0.738709 0.674024i \(-0.235436\pi\)
0.738709 + 0.674024i \(0.235436\pi\)
\(954\) 0 0
\(955\) −1379.01 −1.44399
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 418.549i 0.436444i
\(960\) 0 0
\(961\) 955.906 0.994699
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1626.44i 1.68543i
\(966\) 0 0
\(967\) 467.718i 0.483679i 0.970316 + 0.241840i \(0.0777508\pi\)
−0.970316 + 0.241840i \(0.922249\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1115.70 1.14902 0.574510 0.818498i \(-0.305193\pi\)
0.574510 + 0.818498i \(0.305193\pi\)
\(972\) 0 0
\(973\) 325.212i 0.334237i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −494.363 −0.506001 −0.253001 0.967466i \(-0.581417\pi\)
−0.253001 + 0.967466i \(0.581417\pi\)
\(978\) 0 0
\(979\) 94.7090 0.0967406
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 188.391i − 0.191649i −0.995398 0.0958243i \(-0.969451\pi\)
0.995398 0.0958243i \(-0.0305487\pi\)
\(984\) 0 0
\(985\) −1590.22 −1.61444
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2118.87i 2.14243i
\(990\) 0 0
\(991\) 1895.81i 1.91303i 0.291681 + 0.956516i \(0.405785\pi\)
−0.291681 + 0.956516i \(0.594215\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 337.382 0.339078
\(996\) 0 0
\(997\) 1317.98i 1.32195i 0.750410 + 0.660973i \(0.229856\pi\)
−0.750410 + 0.660973i \(0.770144\pi\)
\(998\) 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 1152.3.b.j.703.1 8
3.2 odd 2 384.3.b.c.319.4 yes 8
4.3 odd 2 inner 1152.3.b.j.703.2 8
8.3 odd 2 inner 1152.3.b.j.703.8 8
8.5 even 2 inner 1152.3.b.j.703.7 8
12.11 even 2 384.3.b.c.319.8 yes 8
16.3 odd 4 2304.3.g.o.1279.1 4
16.5 even 4 2304.3.g.x.1279.4 4
16.11 odd 4 2304.3.g.x.1279.3 4
16.13 even 4 2304.3.g.o.1279.2 4
24.5 odd 2 384.3.b.c.319.5 yes 8
24.11 even 2 384.3.b.c.319.1 8
48.5 odd 4 768.3.g.c.511.3 4
48.11 even 4 768.3.g.c.511.1 4
48.29 odd 4 768.3.g.g.511.2 4
48.35 even 4 768.3.g.g.511.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.3.b.c.319.1 8 24.11 even 2
384.3.b.c.319.4 yes 8 3.2 odd 2
384.3.b.c.319.5 yes 8 24.5 odd 2
384.3.b.c.319.8 yes 8 12.11 even 2
768.3.g.c.511.1 4 48.11 even 4
768.3.g.c.511.3 4 48.5 odd 4
768.3.g.g.511.2 4 48.29 odd 4
768.3.g.g.511.4 4 48.35 even 4
1152.3.b.j.703.1 8 1.1 even 1 trivial
1152.3.b.j.703.2 8 4.3 odd 2 inner
1152.3.b.j.703.7 8 8.5 even 2 inner
1152.3.b.j.703.8 8 8.3 odd 2 inner
2304.3.g.o.1279.1 4 16.3 odd 4
2304.3.g.o.1279.2 4 16.13 even 4
2304.3.g.x.1279.3 4 16.11 odd 4
2304.3.g.x.1279.4 4 16.5 even 4