Properties

Label 1800.4.f.r.649.2
Level $1800$
Weight $4$
Character 1800.649
Analytic conductor $106.203$
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] = [1800,4,Mod(649,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1800.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1800.f (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: \(106.203438010\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1800.649
Dual form 1800.4.f.r.649.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000i q^{7} +O(q^{10})\) \(q+16.0000i q^{7} +28.0000 q^{11} -26.0000i q^{13} -62.0000i q^{17} +68.0000 q^{19} +208.000i q^{23} -58.0000 q^{29} +160.000 q^{31} -270.000i q^{37} -282.000 q^{41} +76.0000i q^{43} -280.000i q^{47} +87.0000 q^{49} +210.000i q^{53} +196.000 q^{59} +742.000 q^{61} -836.000i q^{67} +504.000 q^{71} -1062.00i q^{73} +448.000i q^{77} -768.000 q^{79} +1052.00i q^{83} -726.000 q^{89} +416.000 q^{91} +1406.00i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 56 q^{11} + 136 q^{19} - 116 q^{29} + 320 q^{31} - 564 q^{41} + 174 q^{49} + 392 q^{59} + 1484 q^{61} + 1008 q^{71} - 1536 q^{79} - 1452 q^{89} + 832 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(5\) 0 0
\(6\) 0 0
\(7\) 16.0000i 0.863919i 0.901893 + 0.431959i \(0.142178\pi\)
−0.901893 + 0.431959i \(0.857822\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 28.0000 0.767483 0.383742 0.923440i \(-0.374635\pi\)
0.383742 + 0.923440i \(0.374635\pi\)
\(12\) 0 0
\(13\) − 26.0000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 62.0000i − 0.884542i −0.896882 0.442271i \(-0.854173\pi\)
0.896882 0.442271i \(-0.145827\pi\)
\(18\) 0 0
\(19\) 68.0000 0.821067 0.410533 0.911846i \(-0.365343\pi\)
0.410533 + 0.911846i \(0.365343\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 208.000i 1.88570i 0.333224 + 0.942848i \(0.391864\pi\)
−0.333224 + 0.942848i \(0.608136\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −58.0000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 160.000 0.926995 0.463498 0.886098i \(-0.346594\pi\)
0.463498 + 0.886098i \(0.346594\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 270.000i − 1.19967i −0.800124 0.599834i \(-0.795233\pi\)
0.800124 0.599834i \(-0.204767\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −282.000 −1.07417 −0.537085 0.843528i \(-0.680475\pi\)
−0.537085 + 0.843528i \(0.680475\pi\)
\(42\) 0 0
\(43\) 76.0000i 0.269532i 0.990877 + 0.134766i \(0.0430283\pi\)
−0.990877 + 0.134766i \(0.956972\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 280.000i − 0.868983i −0.900676 0.434491i \(-0.856928\pi\)
0.900676 0.434491i \(-0.143072\pi\)
\(48\) 0 0
\(49\) 87.0000 0.253644
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 210.000i 0.544259i 0.962261 + 0.272129i \(0.0877279\pi\)
−0.962261 + 0.272129i \(0.912272\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 196.000 0.432492 0.216246 0.976339i \(-0.430619\pi\)
0.216246 + 0.976339i \(0.430619\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 836.000i − 1.52438i −0.647352 0.762191i \(-0.724123\pi\)
0.647352 0.762191i \(-0.275877\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 504.000 0.842448 0.421224 0.906957i \(-0.361601\pi\)
0.421224 + 0.906957i \(0.361601\pi\)
\(72\) 0 0
\(73\) − 1062.00i − 1.70271i −0.524591 0.851354i \(-0.675782\pi\)
0.524591 0.851354i \(-0.324218\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 448.000i 0.663043i
\(78\) 0 0
\(79\) −768.000 −1.09376 −0.546878 0.837212i \(-0.684184\pi\)
−0.546878 + 0.837212i \(0.684184\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1052.00i 1.39123i 0.718415 + 0.695614i \(0.244868\pi\)
−0.718415 + 0.695614i \(0.755132\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −726.000 −0.864672 −0.432336 0.901712i \(-0.642311\pi\)
−0.432336 + 0.901712i \(0.642311\pi\)
\(90\) 0 0
\(91\) 416.000 0.479216
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1406.00i 1.47173i 0.677129 + 0.735864i \(0.263224\pi\)
−0.677129 + 0.735864i \(0.736776\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −990.000 −0.975333 −0.487667 0.873030i \(-0.662152\pi\)
−0.487667 + 0.873030i \(0.662152\pi\)
\(102\) 0 0
\(103\) 736.000i 0.704080i 0.935985 + 0.352040i \(0.114512\pi\)
−0.935985 + 0.352040i \(0.885488\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1212.00i 1.09503i 0.836795 + 0.547516i \(0.184427\pi\)
−0.836795 + 0.547516i \(0.815573\pi\)
\(108\) 0 0
\(109\) 1834.00 1.61161 0.805804 0.592182i \(-0.201733\pi\)
0.805804 + 0.592182i \(0.201733\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2046.00i 1.70329i 0.524121 + 0.851644i \(0.324394\pi\)
−0.524121 + 0.851644i \(0.675606\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 992.000 0.764172
\(120\) 0 0
\(121\) −547.000 −0.410969
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 1176.00i − 0.821678i −0.911708 0.410839i \(-0.865236\pi\)
0.911708 0.410839i \(-0.134764\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −12.0000 −0.00800340 −0.00400170 0.999992i \(-0.501274\pi\)
−0.00400170 + 0.999992i \(0.501274\pi\)
\(132\) 0 0
\(133\) 1088.00i 0.709335i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 790.000i − 0.492659i −0.969186 0.246329i \(-0.920775\pi\)
0.969186 0.246329i \(-0.0792245\pi\)
\(138\) 0 0
\(139\) 924.000 0.563832 0.281916 0.959439i \(-0.409030\pi\)
0.281916 + 0.959439i \(0.409030\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 728.000i − 0.425723i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3022.00 1.66156 0.830778 0.556604i \(-0.187896\pi\)
0.830778 + 0.556604i \(0.187896\pi\)
\(150\) 0 0
\(151\) 1736.00 0.935587 0.467794 0.883838i \(-0.345049\pi\)
0.467794 + 0.883838i \(0.345049\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1322.00i 0.672020i 0.941858 + 0.336010i \(0.109078\pi\)
−0.941858 + 0.336010i \(0.890922\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3328.00 −1.62909
\(162\) 0 0
\(163\) − 908.000i − 0.436319i −0.975913 0.218160i \(-0.929995\pi\)
0.975913 0.218160i \(-0.0700054\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1296.00i 0.600524i 0.953857 + 0.300262i \(0.0970741\pi\)
−0.953857 + 0.300262i \(0.902926\pi\)
\(168\) 0 0
\(169\) 1521.00 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 2134.00i − 0.937832i −0.883243 0.468916i \(-0.844645\pi\)
0.883243 0.468916i \(-0.155355\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1612.00 0.673109 0.336555 0.941664i \(-0.390738\pi\)
0.336555 + 0.941664i \(0.390738\pi\)
\(180\) 0 0
\(181\) 3086.00 1.26730 0.633648 0.773621i \(-0.281557\pi\)
0.633648 + 0.773621i \(0.281557\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1736.00i − 0.678871i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4208.00 1.59414 0.797069 0.603889i \(-0.206383\pi\)
0.797069 + 0.603889i \(0.206383\pi\)
\(192\) 0 0
\(193\) 2818.00i 1.05101i 0.850792 + 0.525503i \(0.176123\pi\)
−0.850792 + 0.525503i \(0.823877\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 418.000i − 0.151174i −0.997139 0.0755870i \(-0.975917\pi\)
0.997139 0.0755870i \(-0.0240831\pi\)
\(198\) 0 0
\(199\) 3352.00 1.19406 0.597028 0.802221i \(-0.296348\pi\)
0.597028 + 0.802221i \(0.296348\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 928.000i − 0.320851i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1904.00 0.630155
\(210\) 0 0
\(211\) −4276.00 −1.39513 −0.697564 0.716523i \(-0.745733\pi\)
−0.697564 + 0.716523i \(0.745733\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2560.00i 0.800848i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1612.00 −0.490655
\(222\) 0 0
\(223\) 4712.00i 1.41497i 0.706727 + 0.707486i \(0.250171\pi\)
−0.706727 + 0.707486i \(0.749829\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 732.000i − 0.214029i −0.994257 0.107014i \(-0.965871\pi\)
0.994257 0.107014i \(-0.0341291\pi\)
\(228\) 0 0
\(229\) 5186.00 1.49651 0.748254 0.663412i \(-0.230892\pi\)
0.748254 + 0.663412i \(0.230892\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3798.00i 1.06788i 0.845523 + 0.533938i \(0.179289\pi\)
−0.845523 + 0.533938i \(0.820711\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −3120.00 −0.844419 −0.422209 0.906498i \(-0.638745\pi\)
−0.422209 + 0.906498i \(0.638745\pi\)
\(240\) 0 0
\(241\) 1490.00 0.398255 0.199127 0.979974i \(-0.436189\pi\)
0.199127 + 0.979974i \(0.436189\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1768.00i − 0.455446i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5292.00 1.33079 0.665395 0.746492i \(-0.268263\pi\)
0.665395 + 0.746492i \(0.268263\pi\)
\(252\) 0 0
\(253\) 5824.00i 1.44724i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3918.00i − 0.950965i −0.879725 0.475483i \(-0.842273\pi\)
0.879725 0.475483i \(-0.157727\pi\)
\(258\) 0 0
\(259\) 4320.00 1.03642
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 6624.00i − 1.55305i −0.630084 0.776527i \(-0.716979\pi\)
0.630084 0.776527i \(-0.283021\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2954.00 −0.669549 −0.334774 0.942298i \(-0.608660\pi\)
−0.334774 + 0.942298i \(0.608660\pi\)
\(270\) 0 0
\(271\) −6576.00 −1.47404 −0.737018 0.675874i \(-0.763767\pi\)
−0.737018 + 0.675874i \(0.763767\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4478.00i − 0.971325i −0.874146 0.485662i \(-0.838578\pi\)
0.874146 0.485662i \(-0.161422\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6358.00 1.34977 0.674887 0.737921i \(-0.264192\pi\)
0.674887 + 0.737921i \(0.264192\pi\)
\(282\) 0 0
\(283\) 860.000i 0.180642i 0.995913 + 0.0903210i \(0.0287893\pi\)
−0.995913 + 0.0903210i \(0.971211\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4512.00i − 0.927996i
\(288\) 0 0
\(289\) 1069.00 0.217586
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5794.00i 1.15525i 0.816301 + 0.577626i \(0.196021\pi\)
−0.816301 + 0.577626i \(0.803979\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5408.00 1.04600
\(300\) 0 0
\(301\) −1216.00 −0.232854
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6860.00i 1.27531i 0.770321 + 0.637656i \(0.220096\pi\)
−0.770321 + 0.637656i \(0.779904\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6248.00 1.13920 0.569601 0.821922i \(-0.307098\pi\)
0.569601 + 0.821922i \(0.307098\pi\)
\(312\) 0 0
\(313\) 11018.0i 1.98969i 0.101388 + 0.994847i \(0.467672\pi\)
−0.101388 + 0.994847i \(0.532328\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 954.000i − 0.169028i −0.996422 0.0845142i \(-0.973066\pi\)
0.996422 0.0845142i \(-0.0269338\pi\)
\(318\) 0 0
\(319\) −1624.00 −0.285036
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 4216.00i − 0.726268i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4480.00 0.750731
\(330\) 0 0
\(331\) 9396.00 1.56027 0.780137 0.625608i \(-0.215149\pi\)
0.780137 + 0.625608i \(0.215149\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 5074.00i − 0.820173i −0.912047 0.410087i \(-0.865498\pi\)
0.912047 0.410087i \(-0.134502\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 4480.00 0.711453
\(342\) 0 0
\(343\) 6880.00i 1.08305i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3916.00i 0.605827i 0.953018 + 0.302913i \(0.0979593\pi\)
−0.953018 + 0.302913i \(0.902041\pi\)
\(348\) 0 0
\(349\) 1818.00 0.278840 0.139420 0.990233i \(-0.455476\pi\)
0.139420 + 0.990233i \(0.455476\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 7118.00i 1.07324i 0.843825 + 0.536619i \(0.180299\pi\)
−0.843825 + 0.536619i \(0.819701\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5304.00 0.779762 0.389881 0.920865i \(-0.372516\pi\)
0.389881 + 0.920865i \(0.372516\pi\)
\(360\) 0 0
\(361\) −2235.00 −0.325849
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 5672.00i − 0.806747i −0.915036 0.403373i \(-0.867838\pi\)
0.915036 0.403373i \(-0.132162\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3360.00 −0.470195
\(372\) 0 0
\(373\) 7774.00i 1.07915i 0.841938 + 0.539574i \(0.181415\pi\)
−0.841938 + 0.539574i \(0.818585\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1508.00i 0.206010i
\(378\) 0 0
\(379\) 5516.00 0.747593 0.373797 0.927511i \(-0.378056\pi\)
0.373797 + 0.927511i \(0.378056\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7128.00i 0.950976i 0.879722 + 0.475488i \(0.157728\pi\)
−0.879722 + 0.475488i \(0.842272\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10722.0 −1.39750 −0.698749 0.715367i \(-0.746260\pi\)
−0.698749 + 0.715367i \(0.746260\pi\)
\(390\) 0 0
\(391\) 12896.0 1.66798
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 12122.0i 1.53246i 0.642568 + 0.766229i \(0.277869\pi\)
−0.642568 + 0.766229i \(0.722131\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10482.0 −1.30535 −0.652676 0.757637i \(-0.726354\pi\)
−0.652676 + 0.757637i \(0.726354\pi\)
\(402\) 0 0
\(403\) − 4160.00i − 0.514204i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 7560.00i − 0.920726i
\(408\) 0 0
\(409\) −3850.00 −0.465453 −0.232726 0.972542i \(-0.574765\pi\)
−0.232726 + 0.972542i \(0.574765\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3136.00i 0.373638i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −5796.00 −0.675783 −0.337892 0.941185i \(-0.609714\pi\)
−0.337892 + 0.941185i \(0.609714\pi\)
\(420\) 0 0
\(421\) 3294.00 0.381330 0.190665 0.981655i \(-0.438936\pi\)
0.190665 + 0.981655i \(0.438936\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 11872.0i 1.34549i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1696.00 −0.189544 −0.0947720 0.995499i \(-0.530212\pi\)
−0.0947720 + 0.995499i \(0.530212\pi\)
\(432\) 0 0
\(433\) − 12334.0i − 1.36890i −0.729059 0.684451i \(-0.760042\pi\)
0.729059 0.684451i \(-0.239958\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14144.0i 1.54828i
\(438\) 0 0
\(439\) −376.000 −0.0408781 −0.0204391 0.999791i \(-0.506506\pi\)
−0.0204391 + 0.999791i \(0.506506\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 8028.00i − 0.860997i −0.902591 0.430499i \(-0.858338\pi\)
0.902591 0.430499i \(-0.141662\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8898.00 0.935240 0.467620 0.883930i \(-0.345112\pi\)
0.467620 + 0.883930i \(0.345112\pi\)
\(450\) 0 0
\(451\) −7896.00 −0.824408
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 10330.0i − 1.05737i −0.848819 0.528684i \(-0.822686\pi\)
0.848819 0.528684i \(-0.177314\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1878.00 −0.189734 −0.0948668 0.995490i \(-0.530243\pi\)
−0.0948668 + 0.995490i \(0.530243\pi\)
\(462\) 0 0
\(463\) − 13224.0i − 1.32737i −0.748013 0.663684i \(-0.768992\pi\)
0.748013 0.663684i \(-0.231008\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8012.00i − 0.793900i −0.917840 0.396950i \(-0.870069\pi\)
0.917840 0.396950i \(-0.129931\pi\)
\(468\) 0 0
\(469\) 13376.0 1.31694
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2128.00i 0.206862i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1792.00 −0.170936 −0.0854682 0.996341i \(-0.527239\pi\)
−0.0854682 + 0.996341i \(0.527239\pi\)
\(480\) 0 0
\(481\) −7020.00 −0.665456
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 8272.00i − 0.769692i −0.922981 0.384846i \(-0.874255\pi\)
0.922981 0.384846i \(-0.125745\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −516.000 −0.0474272 −0.0237136 0.999719i \(-0.507549\pi\)
−0.0237136 + 0.999719i \(0.507549\pi\)
\(492\) 0 0
\(493\) 3596.00i 0.328511i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8064.00i 0.727807i
\(498\) 0 0
\(499\) 14020.0 1.25776 0.628879 0.777503i \(-0.283514\pi\)
0.628879 + 0.777503i \(0.283514\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 1872.00i − 0.165941i −0.996552 0.0829705i \(-0.973559\pi\)
0.996552 0.0829705i \(-0.0264407\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8678.00 0.755689 0.377844 0.925869i \(-0.376665\pi\)
0.377844 + 0.925869i \(0.376665\pi\)
\(510\) 0 0
\(511\) 16992.0 1.47100
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 7840.00i − 0.666930i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −18074.0 −1.51984 −0.759920 0.650017i \(-0.774762\pi\)
−0.759920 + 0.650017i \(0.774762\pi\)
\(522\) 0 0
\(523\) − 20852.0i − 1.74339i −0.490047 0.871696i \(-0.663020\pi\)
0.490047 0.871696i \(-0.336980\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 9920.00i − 0.819966i
\(528\) 0 0
\(529\) −31097.0 −2.55585
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7332.00i 0.595843i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 2436.00 0.194668
\(540\) 0 0
\(541\) −12410.0 −0.986225 −0.493112 0.869966i \(-0.664141\pi\)
−0.493112 + 0.869966i \(0.664141\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 3620.00i − 0.282962i −0.989941 0.141481i \(-0.954814\pi\)
0.989941 0.141481i \(-0.0451864\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3944.00 −0.304937
\(552\) 0 0
\(553\) − 12288.0i − 0.944917i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11734.0i 0.892613i 0.894880 + 0.446307i \(0.147261\pi\)
−0.894880 + 0.446307i \(0.852739\pi\)
\(558\) 0 0
\(559\) 1976.00 0.149510
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1372.00i 0.102705i 0.998681 + 0.0513525i \(0.0163532\pi\)
−0.998681 + 0.0513525i \(0.983647\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 18922.0 1.39412 0.697058 0.717015i \(-0.254492\pi\)
0.697058 + 0.717015i \(0.254492\pi\)
\(570\) 0 0
\(571\) 14596.0 1.06974 0.534872 0.844933i \(-0.320360\pi\)
0.534872 + 0.844933i \(0.320360\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 2302.00i 0.166089i 0.996546 + 0.0830446i \(0.0264644\pi\)
−0.996546 + 0.0830446i \(0.973536\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16832.0 −1.20191
\(582\) 0 0
\(583\) 5880.00i 0.417710i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 23292.0i 1.63776i 0.573966 + 0.818879i \(0.305404\pi\)
−0.573966 + 0.818879i \(0.694596\pi\)
\(588\) 0 0
\(589\) 10880.0 0.761125
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16542.0i 1.14553i 0.819720 + 0.572764i \(0.194129\pi\)
−0.819720 + 0.572764i \(0.805871\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 7464.00 0.509133 0.254567 0.967055i \(-0.418067\pi\)
0.254567 + 0.967055i \(0.418067\pi\)
\(600\) 0 0
\(601\) −17270.0 −1.17214 −0.586072 0.810259i \(-0.699326\pi\)
−0.586072 + 0.810259i \(0.699326\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 984.000i − 0.0657979i −0.999459 0.0328990i \(-0.989526\pi\)
0.999459 0.0328990i \(-0.0104740\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −7280.00 −0.482025
\(612\) 0 0
\(613\) 7278.00i 0.479536i 0.970830 + 0.239768i \(0.0770714\pi\)
−0.970830 + 0.239768i \(0.922929\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 18090.0i 1.18035i 0.807275 + 0.590175i \(0.200941\pi\)
−0.807275 + 0.590175i \(0.799059\pi\)
\(618\) 0 0
\(619\) −24740.0 −1.60644 −0.803219 0.595684i \(-0.796881\pi\)
−0.803219 + 0.595684i \(0.796881\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 11616.0i − 0.747007i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −16740.0 −1.06116
\(630\) 0 0
\(631\) 19720.0 1.24412 0.622061 0.782969i \(-0.286296\pi\)
0.622061 + 0.782969i \(0.286296\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 2262.00i − 0.140697i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 16542.0 1.01930 0.509649 0.860383i \(-0.329775\pi\)
0.509649 + 0.860383i \(0.329775\pi\)
\(642\) 0 0
\(643\) − 10092.0i − 0.618957i −0.950906 0.309479i \(-0.899845\pi\)
0.950906 0.309479i \(-0.100155\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 14544.0i − 0.883746i −0.897078 0.441873i \(-0.854314\pi\)
0.897078 0.441873i \(-0.145686\pi\)
\(648\) 0 0
\(649\) 5488.00 0.331930
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 23062.0i − 1.38206i −0.722826 0.691030i \(-0.757157\pi\)
0.722826 0.691030i \(-0.242843\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −28020.0 −1.65630 −0.828152 0.560504i \(-0.810608\pi\)
−0.828152 + 0.560504i \(0.810608\pi\)
\(660\) 0 0
\(661\) −6738.00 −0.396487 −0.198243 0.980153i \(-0.563524\pi\)
−0.198243 + 0.980153i \(0.563524\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 12064.0i − 0.700330i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 20776.0 1.19530
\(672\) 0 0
\(673\) − 14430.0i − 0.826502i −0.910617 0.413251i \(-0.864393\pi\)
0.910617 0.413251i \(-0.135607\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 17890.0i − 1.01561i −0.861472 0.507805i \(-0.830457\pi\)
0.861472 0.507805i \(-0.169543\pi\)
\(678\) 0 0
\(679\) −22496.0 −1.27145
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 10860.0i − 0.608413i −0.952606 0.304207i \(-0.901609\pi\)
0.952606 0.304207i \(-0.0983914\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5460.00 0.301900
\(690\) 0 0
\(691\) −8692.00 −0.478523 −0.239261 0.970955i \(-0.576905\pi\)
−0.239261 + 0.970955i \(0.576905\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 17484.0i 0.950149i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 698.000 0.0376078 0.0188039 0.999823i \(-0.494014\pi\)
0.0188039 + 0.999823i \(0.494014\pi\)
\(702\) 0 0
\(703\) − 18360.0i − 0.985008i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 15840.0i − 0.842609i
\(708\) 0 0
\(709\) −2654.00 −0.140583 −0.0702913 0.997527i \(-0.522393\pi\)
−0.0702913 + 0.997527i \(0.522393\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33280.0i 1.74803i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −28240.0 −1.46478 −0.732388 0.680887i \(-0.761594\pi\)
−0.732388 + 0.680887i \(0.761594\pi\)
\(720\) 0 0
\(721\) −11776.0 −0.608268
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8320.00i 0.424445i 0.977221 + 0.212223i \(0.0680702\pi\)
−0.977221 + 0.212223i \(0.931930\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4712.00 0.238413
\(732\) 0 0
\(733\) − 2154.00i − 0.108540i −0.998526 0.0542700i \(-0.982717\pi\)
0.998526 0.0542700i \(-0.0172832\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 23408.0i − 1.16994i
\(738\) 0 0
\(739\) −22380.0 −1.11402 −0.557011 0.830505i \(-0.688052\pi\)
−0.557011 + 0.830505i \(0.688052\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 5760.00i 0.284406i 0.989837 + 0.142203i \(0.0454186\pi\)
−0.989837 + 0.142203i \(0.954581\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −19392.0 −0.946019
\(750\) 0 0
\(751\) −6192.00 −0.300865 −0.150432 0.988620i \(-0.548067\pi\)
−0.150432 + 0.988620i \(0.548067\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13666.0i 0.656142i 0.944653 + 0.328071i \(0.106398\pi\)
−0.944653 + 0.328071i \(0.893602\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 32022.0 1.52536 0.762678 0.646778i \(-0.223884\pi\)
0.762678 + 0.646778i \(0.223884\pi\)
\(762\) 0 0
\(763\) 29344.0i 1.39230i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5096.00i − 0.239903i
\(768\) 0 0
\(769\) −22786.0 −1.06851 −0.534255 0.845323i \(-0.679408\pi\)
−0.534255 + 0.845323i \(0.679408\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8286.00i − 0.385546i −0.981243 0.192773i \(-0.938252\pi\)
0.981243 0.192773i \(-0.0617480\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −19176.0 −0.881966
\(780\) 0 0
\(781\) 14112.0 0.646565
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 25804.0i 1.16876i 0.811481 + 0.584379i \(0.198662\pi\)
−0.811481 + 0.584379i \(0.801338\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −32736.0 −1.47150
\(792\) 0 0
\(793\) − 19292.0i − 0.863908i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17670.0i 0.785324i 0.919683 + 0.392662i \(0.128446\pi\)
−0.919683 + 0.392662i \(0.871554\pi\)
\(798\) 0 0
\(799\) −17360.0 −0.768652
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 29736.0i − 1.30680i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −7398.00 −0.321508 −0.160754 0.986995i \(-0.551393\pi\)
−0.160754 + 0.986995i \(0.551393\pi\)
\(810\) 0 0
\(811\) −28108.0 −1.21702 −0.608511 0.793545i \(-0.708233\pi\)
−0.608511 + 0.793545i \(0.708233\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5168.00i 0.221304i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −30830.0 −1.31057 −0.655283 0.755384i \(-0.727451\pi\)
−0.655283 + 0.755384i \(0.727451\pi\)
\(822\) 0 0
\(823\) 5872.00i 0.248706i 0.992238 + 0.124353i \(0.0396855\pi\)
−0.992238 + 0.124353i \(0.960314\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 16308.0i − 0.685713i −0.939388 0.342857i \(-0.888606\pi\)
0.939388 0.342857i \(-0.111394\pi\)
\(828\) 0 0
\(829\) −28294.0 −1.18539 −0.592697 0.805426i \(-0.701937\pi\)
−0.592697 + 0.805426i \(0.701937\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5394.00i − 0.224359i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 20536.0 0.845032 0.422516 0.906356i \(-0.361147\pi\)
0.422516 + 0.906356i \(0.361147\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 8752.00i − 0.355044i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 56160.0 2.26221
\(852\) 0 0
\(853\) 27710.0i 1.11228i 0.831090 + 0.556139i \(0.187718\pi\)
−0.831090 + 0.556139i \(0.812282\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12858.0i 0.512510i 0.966609 + 0.256255i \(0.0824887\pi\)
−0.966609 + 0.256255i \(0.917511\pi\)
\(858\) 0 0
\(859\) 3148.00 0.125039 0.0625194 0.998044i \(-0.480086\pi\)
0.0625194 + 0.998044i \(0.480086\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 48456.0i − 1.91131i −0.294487 0.955656i \(-0.595149\pi\)
0.294487 0.955656i \(-0.404851\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −21504.0 −0.839440
\(870\) 0 0
\(871\) −21736.0 −0.845576
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 9478.00i − 0.364937i −0.983212 0.182468i \(-0.941591\pi\)
0.983212 0.182468i \(-0.0584087\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8178.00 −0.312740 −0.156370 0.987699i \(-0.549979\pi\)
−0.156370 + 0.987699i \(0.549979\pi\)
\(882\) 0 0
\(883\) − 316.000i − 0.0120433i −0.999982 0.00602166i \(-0.998083\pi\)
0.999982 0.00602166i \(-0.00191676\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 6304.00i − 0.238633i −0.992856 0.119317i \(-0.961930\pi\)
0.992856 0.119317i \(-0.0380703\pi\)
\(888\) 0 0
\(889\) 18816.0 0.709863
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 19040.0i − 0.713493i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −9280.00 −0.344277
\(900\) 0 0
\(901\) 13020.0 0.481420
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 1596.00i − 0.0584281i −0.999573 0.0292141i \(-0.990700\pi\)
0.999573 0.0292141i \(-0.00930045\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 25792.0 0.938010 0.469005 0.883196i \(-0.344613\pi\)
0.469005 + 0.883196i \(0.344613\pi\)
\(912\) 0 0
\(913\) 29456.0i 1.06775i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 192.000i − 0.00691428i
\(918\) 0 0
\(919\) 9736.00 0.349468 0.174734 0.984616i \(-0.444093\pi\)
0.174734 + 0.984616i \(0.444093\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 13104.0i − 0.467306i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −94.0000 −0.00331974 −0.00165987 0.999999i \(-0.500528\pi\)
−0.00165987 + 0.999999i \(0.500528\pi\)
\(930\) 0 0
\(931\) 5916.00 0.208259
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 8678.00i 0.302559i 0.988491 + 0.151280i \(0.0483394\pi\)
−0.988491 + 0.151280i \(0.951661\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −28406.0 −0.984069 −0.492035 0.870576i \(-0.663747\pi\)
−0.492035 + 0.870576i \(0.663747\pi\)
\(942\) 0 0
\(943\) − 58656.0i − 2.02556i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31988.0i 1.09765i 0.835939 + 0.548823i \(0.184924\pi\)
−0.835939 + 0.548823i \(0.815076\pi\)
\(948\) 0 0
\(949\) −27612.0 −0.944493
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 6714.00i − 0.228214i −0.993468 0.114107i \(-0.963599\pi\)
0.993468 0.114107i \(-0.0364006\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12640.0 0.425617
\(960\) 0 0
\(961\) −4191.00 −0.140680
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 15312.0i − 0.509204i −0.967046 0.254602i \(-0.918055\pi\)
0.967046 0.254602i \(-0.0819445\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8540.00 0.282247 0.141123 0.989992i \(-0.454929\pi\)
0.141123 + 0.989992i \(0.454929\pi\)
\(972\) 0 0
\(973\) 14784.0i 0.487105i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 8126.00i − 0.266094i −0.991110 0.133047i \(-0.957524\pi\)
0.991110 0.133047i \(-0.0424761\pi\)
\(978\) 0 0
\(979\) −20328.0 −0.663622
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1392.00i − 0.0451657i −0.999745 0.0225829i \(-0.992811\pi\)
0.999745 0.0225829i \(-0.00718896\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −15808.0 −0.508256
\(990\) 0 0
\(991\) −48832.0 −1.56529 −0.782644 0.622470i \(-0.786129\pi\)
−0.782644 + 0.622470i \(0.786129\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 46926.0i − 1.49063i −0.666711 0.745317i \(-0.732298\pi\)
0.666711 0.745317i \(-0.267702\pi\)
\(998\) 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 1800.4.f.r.649.2 2
3.2 odd 2 600.4.f.c.49.1 2
5.2 odd 4 360.4.a.b.1.1 1
5.3 odd 4 1800.4.a.bb.1.1 1
5.4 even 2 inner 1800.4.f.r.649.1 2
12.11 even 2 1200.4.f.o.49.2 2
15.2 even 4 120.4.a.c.1.1 1
15.8 even 4 600.4.a.q.1.1 1
15.14 odd 2 600.4.f.c.49.2 2
20.7 even 4 720.4.a.l.1.1 1
60.23 odd 4 1200.4.a.c.1.1 1
60.47 odd 4 240.4.a.l.1.1 1
60.59 even 2 1200.4.f.o.49.1 2
120.77 even 4 960.4.a.u.1.1 1
120.107 odd 4 960.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.a.c.1.1 1 15.2 even 4
240.4.a.l.1.1 1 60.47 odd 4
360.4.a.b.1.1 1 5.2 odd 4
600.4.a.q.1.1 1 15.8 even 4
600.4.f.c.49.1 2 3.2 odd 2
600.4.f.c.49.2 2 15.14 odd 2
720.4.a.l.1.1 1 20.7 even 4
960.4.a.h.1.1 1 120.107 odd 4
960.4.a.u.1.1 1 120.77 even 4
1200.4.a.c.1.1 1 60.23 odd 4
1200.4.f.o.49.1 2 60.59 even 2
1200.4.f.o.49.2 2 12.11 even 2
1800.4.a.bb.1.1 1 5.3 odd 4
1800.4.f.r.649.1 2 5.4 even 2 inner
1800.4.f.r.649.2 2 1.1 even 1 trivial