Properties

Label 2100.4.k.j.1849.2
Level $2100$
Weight $4$
Character 2100.1849
Analytic conductor $123.904$
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] = [2100,4,Mod(1849,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1849");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2100.k (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: \(123.904011012\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1849.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1849
Dual form 2100.4.k.j.1849.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+3.00000i q^{3} +7.00000i q^{7} -9.00000 q^{9} +36.0000 q^{11} -62.0000i q^{13} +114.000i q^{17} +76.0000 q^{19} -21.0000 q^{21} +24.0000i q^{23} -27.0000i q^{27} -54.0000 q^{29} -112.000 q^{31} +108.000i q^{33} -178.000i q^{37} +186.000 q^{39} +378.000 q^{41} +172.000i q^{43} -192.000i q^{47} -49.0000 q^{49} -342.000 q^{51} +402.000i q^{53} +228.000i q^{57} -396.000 q^{59} +254.000 q^{61} -63.0000i q^{63} -1012.00i q^{67} -72.0000 q^{69} +840.000 q^{71} -890.000i q^{73} +252.000i q^{77} -80.0000 q^{79} +81.0000 q^{81} +108.000i q^{83} -162.000i q^{87} +1638.00 q^{89} +434.000 q^{91} -336.000i q^{93} +1010.00i q^{97} -324.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 72 q^{11} + 152 q^{19} - 42 q^{21} - 108 q^{29} - 224 q^{31} + 372 q^{39} + 756 q^{41} - 98 q^{49} - 684 q^{51} - 792 q^{59} + 508 q^{61} - 144 q^{69} + 1680 q^{71} - 160 q^{79} + 162 q^{81} + 3276 q^{89} + 868 q^{91} - 648 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000i 0.377964i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 36.0000 0.986764 0.493382 0.869813i \(-0.335760\pi\)
0.493382 + 0.869813i \(0.335760\pi\)
\(12\) 0 0
\(13\) − 62.0000i − 1.32275i −0.750057 0.661373i \(-0.769974\pi\)
0.750057 0.661373i \(-0.230026\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 114.000i 1.62642i 0.581974 + 0.813208i \(0.302281\pi\)
−0.581974 + 0.813208i \(0.697719\pi\)
\(18\) 0 0
\(19\) 76.0000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 24.0000i 0.217580i 0.994065 + 0.108790i \(0.0346976\pi\)
−0.994065 + 0.108790i \(0.965302\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 27.0000i − 0.192450i
\(28\) 0 0
\(29\) −54.0000 −0.345778 −0.172889 0.984941i \(-0.555310\pi\)
−0.172889 + 0.984941i \(0.555310\pi\)
\(30\) 0 0
\(31\) −112.000 −0.648897 −0.324448 0.945903i \(-0.605179\pi\)
−0.324448 + 0.945903i \(0.605179\pi\)
\(32\) 0 0
\(33\) 108.000i 0.569709i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 178.000i − 0.790892i −0.918489 0.395446i \(-0.870590\pi\)
0.918489 0.395446i \(-0.129410\pi\)
\(38\) 0 0
\(39\) 186.000 0.763688
\(40\) 0 0
\(41\) 378.000 1.43985 0.719923 0.694054i \(-0.244177\pi\)
0.719923 + 0.694054i \(0.244177\pi\)
\(42\) 0 0
\(43\) 172.000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 192.000i − 0.595874i −0.954586 0.297937i \(-0.903701\pi\)
0.954586 0.297937i \(-0.0962985\pi\)
\(48\) 0 0
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −342.000 −0.939011
\(52\) 0 0
\(53\) 402.000i 1.04187i 0.853597 + 0.520933i \(0.174416\pi\)
−0.853597 + 0.520933i \(0.825584\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 228.000i 0.529813i
\(58\) 0 0
\(59\) −396.000 −0.873810 −0.436905 0.899508i \(-0.643925\pi\)
−0.436905 + 0.899508i \(0.643925\pi\)
\(60\) 0 0
\(61\) 254.000 0.533137 0.266569 0.963816i \(-0.414110\pi\)
0.266569 + 0.963816i \(0.414110\pi\)
\(62\) 0 0
\(63\) − 63.0000i − 0.125988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 1012.00i − 1.84531i −0.385632 0.922653i \(-0.626016\pi\)
0.385632 0.922653i \(-0.373984\pi\)
\(68\) 0 0
\(69\) −72.0000 −0.125620
\(70\) 0 0
\(71\) 840.000 1.40408 0.702040 0.712138i \(-0.252273\pi\)
0.702040 + 0.712138i \(0.252273\pi\)
\(72\) 0 0
\(73\) − 890.000i − 1.42694i −0.700686 0.713470i \(-0.747122\pi\)
0.700686 0.713470i \(-0.252878\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 252.000i 0.372962i
\(78\) 0 0
\(79\) −80.0000 −0.113933 −0.0569665 0.998376i \(-0.518143\pi\)
−0.0569665 + 0.998376i \(0.518143\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 108.000i 0.142826i 0.997447 + 0.0714129i \(0.0227508\pi\)
−0.997447 + 0.0714129i \(0.977249\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 162.000i − 0.199635i
\(88\) 0 0
\(89\) 1638.00 1.95087 0.975436 0.220282i \(-0.0706977\pi\)
0.975436 + 0.220282i \(0.0706977\pi\)
\(90\) 0 0
\(91\) 434.000 0.499951
\(92\) 0 0
\(93\) − 336.000i − 0.374641i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1010.00i 1.05722i 0.848866 + 0.528608i \(0.177286\pi\)
−0.848866 + 0.528608i \(0.822714\pi\)
\(98\) 0 0
\(99\) −324.000 −0.328921
\(100\) 0 0
\(101\) 6.00000 0.00591111 0.00295556 0.999996i \(-0.499059\pi\)
0.00295556 + 0.999996i \(0.499059\pi\)
\(102\) 0 0
\(103\) 472.000i 0.451530i 0.974182 + 0.225765i \(0.0724881\pi\)
−0.974182 + 0.225765i \(0.927512\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 972.000i − 0.878194i −0.898440 0.439097i \(-0.855298\pi\)
0.898440 0.439097i \(-0.144702\pi\)
\(108\) 0 0
\(109\) 1786.00 1.56943 0.784715 0.619857i \(-0.212810\pi\)
0.784715 + 0.619857i \(0.212810\pi\)
\(110\) 0 0
\(111\) 534.000 0.456622
\(112\) 0 0
\(113\) 2286.00i 1.90309i 0.307515 + 0.951543i \(0.400503\pi\)
−0.307515 + 0.951543i \(0.599497\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 558.000i 0.440916i
\(118\) 0 0
\(119\) −798.000 −0.614727
\(120\) 0 0
\(121\) −35.0000 −0.0262960
\(122\) 0 0
\(123\) 1134.00i 0.831295i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1328.00i 0.927881i 0.885866 + 0.463941i \(0.153565\pi\)
−0.885866 + 0.463941i \(0.846435\pi\)
\(128\) 0 0
\(129\) −516.000 −0.352180
\(130\) 0 0
\(131\) −1212.00 −0.808343 −0.404171 0.914683i \(-0.632440\pi\)
−0.404171 + 0.914683i \(0.632440\pi\)
\(132\) 0 0
\(133\) 532.000i 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1254.00i − 0.782018i −0.920387 0.391009i \(-0.872126\pi\)
0.920387 0.391009i \(-0.127874\pi\)
\(138\) 0 0
\(139\) 340.000 0.207471 0.103735 0.994605i \(-0.466921\pi\)
0.103735 + 0.994605i \(0.466921\pi\)
\(140\) 0 0
\(141\) 576.000 0.344028
\(142\) 0 0
\(143\) − 2232.00i − 1.30524i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 147.000i − 0.0824786i
\(148\) 0 0
\(149\) −1038.00 −0.570713 −0.285357 0.958421i \(-0.592112\pi\)
−0.285357 + 0.958421i \(0.592112\pi\)
\(150\) 0 0
\(151\) 2936.00 1.58231 0.791153 0.611618i \(-0.209481\pi\)
0.791153 + 0.611618i \(0.209481\pi\)
\(152\) 0 0
\(153\) − 1026.00i − 0.542138i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 1330.00i − 0.676086i −0.941131 0.338043i \(-0.890235\pi\)
0.941131 0.338043i \(-0.109765\pi\)
\(158\) 0 0
\(159\) −1206.00 −0.601522
\(160\) 0 0
\(161\) −168.000 −0.0822376
\(162\) 0 0
\(163\) 3364.00i 1.61650i 0.588842 + 0.808248i \(0.299584\pi\)
−0.588842 + 0.808248i \(0.700416\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3048.00i 1.41234i 0.708041 + 0.706172i \(0.249579\pi\)
−0.708041 + 0.706172i \(0.750421\pi\)
\(168\) 0 0
\(169\) −1647.00 −0.749659
\(170\) 0 0
\(171\) −684.000 −0.305888
\(172\) 0 0
\(173\) 2706.00i 1.18921i 0.804018 + 0.594605i \(0.202692\pi\)
−0.804018 + 0.594605i \(0.797308\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1188.00i − 0.504495i
\(178\) 0 0
\(179\) −4716.00 −1.96922 −0.984610 0.174766i \(-0.944083\pi\)
−0.984610 + 0.174766i \(0.944083\pi\)
\(180\) 0 0
\(181\) 1910.00 0.784360 0.392180 0.919888i \(-0.371721\pi\)
0.392180 + 0.919888i \(0.371721\pi\)
\(182\) 0 0
\(183\) 762.000i 0.307807i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4104.00i 1.60489i
\(188\) 0 0
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) 4080.00 1.54565 0.772823 0.634621i \(-0.218844\pi\)
0.772823 + 0.634621i \(0.218844\pi\)
\(192\) 0 0
\(193\) 2686.00i 1.00177i 0.865512 + 0.500887i \(0.166993\pi\)
−0.865512 + 0.500887i \(0.833007\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 510.000i 0.184447i 0.995738 + 0.0922233i \(0.0293974\pi\)
−0.995738 + 0.0922233i \(0.970603\pi\)
\(198\) 0 0
\(199\) −1352.00 −0.481612 −0.240806 0.970573i \(-0.577412\pi\)
−0.240806 + 0.970573i \(0.577412\pi\)
\(200\) 0 0
\(201\) 3036.00 1.06539
\(202\) 0 0
\(203\) − 378.000i − 0.130692i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 216.000i − 0.0725268i
\(208\) 0 0
\(209\) 2736.00 0.905517
\(210\) 0 0
\(211\) −3364.00 −1.09757 −0.548785 0.835963i \(-0.684909\pi\)
−0.548785 + 0.835963i \(0.684909\pi\)
\(212\) 0 0
\(213\) 2520.00i 0.810646i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 784.000i − 0.245260i
\(218\) 0 0
\(219\) 2670.00 0.823844
\(220\) 0 0
\(221\) 7068.00 2.15134
\(222\) 0 0
\(223\) 4768.00i 1.43179i 0.698209 + 0.715894i \(0.253981\pi\)
−0.698209 + 0.715894i \(0.746019\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 420.000i 0.122803i 0.998113 + 0.0614017i \(0.0195571\pi\)
−0.998113 + 0.0614017i \(0.980443\pi\)
\(228\) 0 0
\(229\) 1882.00 0.543083 0.271542 0.962427i \(-0.412467\pi\)
0.271542 + 0.962427i \(0.412467\pi\)
\(230\) 0 0
\(231\) −756.000 −0.215330
\(232\) 0 0
\(233\) − 5082.00i − 1.42890i −0.699688 0.714448i \(-0.746678\pi\)
0.699688 0.714448i \(-0.253322\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 240.000i − 0.0657792i
\(238\) 0 0
\(239\) 5424.00 1.46799 0.733995 0.679155i \(-0.237654\pi\)
0.733995 + 0.679155i \(0.237654\pi\)
\(240\) 0 0
\(241\) −2590.00 −0.692268 −0.346134 0.938185i \(-0.612506\pi\)
−0.346134 + 0.938185i \(0.612506\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 4712.00i − 1.21384i
\(248\) 0 0
\(249\) −324.000 −0.0824605
\(250\) 0 0
\(251\) −4932.00 −1.24026 −0.620130 0.784499i \(-0.712920\pi\)
−0.620130 + 0.784499i \(0.712920\pi\)
\(252\) 0 0
\(253\) 864.000i 0.214700i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3438.00i − 0.834461i −0.908801 0.417231i \(-0.863001\pi\)
0.908801 0.417231i \(-0.136999\pi\)
\(258\) 0 0
\(259\) 1246.00 0.298929
\(260\) 0 0
\(261\) 486.000 0.115259
\(262\) 0 0
\(263\) − 6120.00i − 1.43489i −0.696617 0.717444i \(-0.745312\pi\)
0.696617 0.717444i \(-0.254688\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 4914.00i 1.12634i
\(268\) 0 0
\(269\) 18.0000 0.00407985 0.00203992 0.999998i \(-0.499351\pi\)
0.00203992 + 0.999998i \(0.499351\pi\)
\(270\) 0 0
\(271\) 6896.00 1.54576 0.772882 0.634549i \(-0.218814\pi\)
0.772882 + 0.634549i \(0.218814\pi\)
\(272\) 0 0
\(273\) 1302.00i 0.288647i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 6254.00i 1.35656i 0.734805 + 0.678279i \(0.237274\pi\)
−0.734805 + 0.678279i \(0.762726\pi\)
\(278\) 0 0
\(279\) 1008.00 0.216299
\(280\) 0 0
\(281\) 1194.00 0.253481 0.126740 0.991936i \(-0.459549\pi\)
0.126740 + 0.991936i \(0.459549\pi\)
\(282\) 0 0
\(283\) 7156.00i 1.50311i 0.659671 + 0.751555i \(0.270696\pi\)
−0.659671 + 0.751555i \(0.729304\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2646.00i 0.544211i
\(288\) 0 0
\(289\) −8083.00 −1.64523
\(290\) 0 0
\(291\) −3030.00 −0.610384
\(292\) 0 0
\(293\) 3738.00i 0.745312i 0.927970 + 0.372656i \(0.121553\pi\)
−0.927970 + 0.372656i \(0.878447\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 972.000i − 0.189903i
\(298\) 0 0
\(299\) 1488.00 0.287804
\(300\) 0 0
\(301\) −1204.00 −0.230556
\(302\) 0 0
\(303\) 18.0000i 0.00341278i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 844.000i − 0.156904i −0.996918 0.0784522i \(-0.975002\pi\)
0.996918 0.0784522i \(-0.0249978\pi\)
\(308\) 0 0
\(309\) −1416.00 −0.260691
\(310\) 0 0
\(311\) 6312.00 1.15087 0.575435 0.817847i \(-0.304833\pi\)
0.575435 + 0.817847i \(0.304833\pi\)
\(312\) 0 0
\(313\) − 8282.00i − 1.49561i −0.663918 0.747806i \(-0.731108\pi\)
0.663918 0.747806i \(-0.268892\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9318.00i 1.65095i 0.564439 + 0.825475i \(0.309093\pi\)
−0.564439 + 0.825475i \(0.690907\pi\)
\(318\) 0 0
\(319\) −1944.00 −0.341201
\(320\) 0 0
\(321\) 2916.00 0.507026
\(322\) 0 0
\(323\) 8664.00i 1.49250i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 5358.00i 0.906110i
\(328\) 0 0
\(329\) 1344.00 0.225219
\(330\) 0 0
\(331\) 1652.00 0.274327 0.137163 0.990548i \(-0.456201\pi\)
0.137163 + 0.990548i \(0.456201\pi\)
\(332\) 0 0
\(333\) 1602.00i 0.263631i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 1294.00i − 0.209165i −0.994516 0.104583i \(-0.966649\pi\)
0.994516 0.104583i \(-0.0333507\pi\)
\(338\) 0 0
\(339\) −6858.00 −1.09875
\(340\) 0 0
\(341\) −4032.00 −0.640308
\(342\) 0 0
\(343\) − 343.000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3636.00i 0.562509i 0.959633 + 0.281255i \(0.0907505\pi\)
−0.959633 + 0.281255i \(0.909249\pi\)
\(348\) 0 0
\(349\) −10478.0 −1.60709 −0.803545 0.595244i \(-0.797055\pi\)
−0.803545 + 0.595244i \(0.797055\pi\)
\(350\) 0 0
\(351\) −1674.00 −0.254563
\(352\) 0 0
\(353\) 7566.00i 1.14079i 0.821372 + 0.570393i \(0.193209\pi\)
−0.821372 + 0.570393i \(0.806791\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 2394.00i − 0.354913i
\(358\) 0 0
\(359\) 8040.00 1.18199 0.590996 0.806675i \(-0.298735\pi\)
0.590996 + 0.806675i \(0.298735\pi\)
\(360\) 0 0
\(361\) −1083.00 −0.157895
\(362\) 0 0
\(363\) − 105.000i − 0.0151820i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7568.00i 1.07642i 0.842811 + 0.538210i \(0.180899\pi\)
−0.842811 + 0.538210i \(0.819101\pi\)
\(368\) 0 0
\(369\) −3402.00 −0.479949
\(370\) 0 0
\(371\) −2814.00 −0.393789
\(372\) 0 0
\(373\) 13522.0i 1.87706i 0.345200 + 0.938529i \(0.387811\pi\)
−0.345200 + 0.938529i \(0.612189\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3348.00i 0.457376i
\(378\) 0 0
\(379\) −2468.00 −0.334492 −0.167246 0.985915i \(-0.553487\pi\)
−0.167246 + 0.985915i \(0.553487\pi\)
\(380\) 0 0
\(381\) −3984.00 −0.535713
\(382\) 0 0
\(383\) 12336.0i 1.64580i 0.568189 + 0.822898i \(0.307644\pi\)
−0.568189 + 0.822898i \(0.692356\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 1548.00i − 0.203331i
\(388\) 0 0
\(389\) 3762.00 0.490337 0.245168 0.969481i \(-0.421157\pi\)
0.245168 + 0.969481i \(0.421157\pi\)
\(390\) 0 0
\(391\) −2736.00 −0.353876
\(392\) 0 0
\(393\) − 3636.00i − 0.466697i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 8770.00i − 1.10870i −0.832284 0.554350i \(-0.812967\pi\)
0.832284 0.554350i \(-0.187033\pi\)
\(398\) 0 0
\(399\) −1596.00 −0.200250
\(400\) 0 0
\(401\) 6642.00 0.827146 0.413573 0.910471i \(-0.364281\pi\)
0.413573 + 0.910471i \(0.364281\pi\)
\(402\) 0 0
\(403\) 6944.00i 0.858326i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 6408.00i − 0.780424i
\(408\) 0 0
\(409\) 1510.00 0.182554 0.0912771 0.995826i \(-0.470905\pi\)
0.0912771 + 0.995826i \(0.470905\pi\)
\(410\) 0 0
\(411\) 3762.00 0.451498
\(412\) 0 0
\(413\) − 2772.00i − 0.330269i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1020.00i 0.119783i
\(418\) 0 0
\(419\) 1260.00 0.146909 0.0734547 0.997299i \(-0.476598\pi\)
0.0734547 + 0.997299i \(0.476598\pi\)
\(420\) 0 0
\(421\) 3998.00 0.462828 0.231414 0.972855i \(-0.425665\pi\)
0.231414 + 0.972855i \(0.425665\pi\)
\(422\) 0 0
\(423\) 1728.00i 0.198625i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1778.00i 0.201507i
\(428\) 0 0
\(429\) 6696.00 0.753580
\(430\) 0 0
\(431\) −2736.00 −0.305774 −0.152887 0.988244i \(-0.548857\pi\)
−0.152887 + 0.988244i \(0.548857\pi\)
\(432\) 0 0
\(433\) − 2690.00i − 0.298552i −0.988796 0.149276i \(-0.952306\pi\)
0.988796 0.149276i \(-0.0476943\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1824.00i 0.199665i
\(438\) 0 0
\(439\) 1240.00 0.134811 0.0674054 0.997726i \(-0.478528\pi\)
0.0674054 + 0.997726i \(0.478528\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 3900.00i 0.418272i 0.977887 + 0.209136i \(0.0670652\pi\)
−0.977887 + 0.209136i \(0.932935\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 3114.00i − 0.329501i
\(448\) 0 0
\(449\) 10878.0 1.14335 0.571675 0.820480i \(-0.306294\pi\)
0.571675 + 0.820480i \(0.306294\pi\)
\(450\) 0 0
\(451\) 13608.0 1.42079
\(452\) 0 0
\(453\) 8808.00i 0.913545i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2330.00i 0.238496i 0.992864 + 0.119248i \(0.0380484\pi\)
−0.992864 + 0.119248i \(0.961952\pi\)
\(458\) 0 0
\(459\) 3078.00 0.313004
\(460\) 0 0
\(461\) 15150.0 1.53060 0.765299 0.643675i \(-0.222591\pi\)
0.765299 + 0.643675i \(0.222591\pi\)
\(462\) 0 0
\(463\) 2992.00i 0.300324i 0.988661 + 0.150162i \(0.0479795\pi\)
−0.988661 + 0.150162i \(0.952020\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8724.00i 0.864451i 0.901766 + 0.432225i \(0.142272\pi\)
−0.901766 + 0.432225i \(0.857728\pi\)
\(468\) 0 0
\(469\) 7084.00 0.697460
\(470\) 0 0
\(471\) 3990.00 0.390339
\(472\) 0 0
\(473\) 6192.00i 0.601921i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 3618.00i − 0.347289i
\(478\) 0 0
\(479\) −9744.00 −0.929467 −0.464734 0.885451i \(-0.653850\pi\)
−0.464734 + 0.885451i \(0.653850\pi\)
\(480\) 0 0
\(481\) −11036.0 −1.04615
\(482\) 0 0
\(483\) − 504.000i − 0.0474799i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 4136.00i 0.384846i 0.981312 + 0.192423i \(0.0616346\pi\)
−0.981312 + 0.192423i \(0.938365\pi\)
\(488\) 0 0
\(489\) −10092.0 −0.933284
\(490\) 0 0
\(491\) 16212.0 1.49010 0.745048 0.667011i \(-0.232426\pi\)
0.745048 + 0.667011i \(0.232426\pi\)
\(492\) 0 0
\(493\) − 6156.00i − 0.562378i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5880.00i 0.530692i
\(498\) 0 0
\(499\) −2396.00 −0.214949 −0.107475 0.994208i \(-0.534276\pi\)
−0.107475 + 0.994208i \(0.534276\pi\)
\(500\) 0 0
\(501\) −9144.00 −0.815417
\(502\) 0 0
\(503\) − 13128.0i − 1.16371i −0.813291 0.581857i \(-0.802326\pi\)
0.813291 0.581857i \(-0.197674\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 4941.00i − 0.432816i
\(508\) 0 0
\(509\) −12798.0 −1.11446 −0.557231 0.830357i \(-0.688136\pi\)
−0.557231 + 0.830357i \(0.688136\pi\)
\(510\) 0 0
\(511\) 6230.00 0.539333
\(512\) 0 0
\(513\) − 2052.00i − 0.176604i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 6912.00i − 0.587987i
\(518\) 0 0
\(519\) −8118.00 −0.686591
\(520\) 0 0
\(521\) 7386.00 0.621087 0.310544 0.950559i \(-0.399489\pi\)
0.310544 + 0.950559i \(0.399489\pi\)
\(522\) 0 0
\(523\) − 5180.00i − 0.433089i −0.976273 0.216545i \(-0.930521\pi\)
0.976273 0.216545i \(-0.0694787\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 12768.0i − 1.05538i
\(528\) 0 0
\(529\) 11591.0 0.952659
\(530\) 0 0
\(531\) 3564.00 0.291270
\(532\) 0 0
\(533\) − 23436.0i − 1.90455i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 14148.0i − 1.13693i
\(538\) 0 0
\(539\) −1764.00 −0.140966
\(540\) 0 0
\(541\) 4070.00 0.323444 0.161722 0.986836i \(-0.448295\pi\)
0.161722 + 0.986836i \(0.448295\pi\)
\(542\) 0 0
\(543\) 5730.00i 0.452851i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 14780.0i 1.15530i 0.816286 + 0.577648i \(0.196029\pi\)
−0.816286 + 0.577648i \(0.803971\pi\)
\(548\) 0 0
\(549\) −2286.00 −0.177712
\(550\) 0 0
\(551\) −4104.00 −0.317307
\(552\) 0 0
\(553\) − 560.000i − 0.0430626i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 6858.00i − 0.521693i −0.965380 0.260846i \(-0.915998\pi\)
0.965380 0.260846i \(-0.0840016\pi\)
\(558\) 0 0
\(559\) 10664.0 0.806868
\(560\) 0 0
\(561\) −12312.0 −0.926583
\(562\) 0 0
\(563\) − 6660.00i − 0.498553i −0.968432 0.249277i \(-0.919807\pi\)
0.968432 0.249277i \(-0.0801929\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 567.000i 0.0419961i
\(568\) 0 0
\(569\) 150.000 0.0110515 0.00552577 0.999985i \(-0.498241\pi\)
0.00552577 + 0.999985i \(0.498241\pi\)
\(570\) 0 0
\(571\) −8188.00 −0.600100 −0.300050 0.953923i \(-0.597003\pi\)
−0.300050 + 0.953923i \(0.597003\pi\)
\(572\) 0 0
\(573\) 12240.0i 0.892379i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 5854.00i − 0.422366i −0.977447 0.211183i \(-0.932268\pi\)
0.977447 0.211183i \(-0.0677316\pi\)
\(578\) 0 0
\(579\) −8058.00 −0.578375
\(580\) 0 0
\(581\) −756.000 −0.0539831
\(582\) 0 0
\(583\) 14472.0i 1.02808i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 17580.0i 1.23612i 0.786130 + 0.618062i \(0.212082\pi\)
−0.786130 + 0.618062i \(0.787918\pi\)
\(588\) 0 0
\(589\) −8512.00 −0.595468
\(590\) 0 0
\(591\) −1530.00 −0.106490
\(592\) 0 0
\(593\) − 17154.0i − 1.18791i −0.804498 0.593955i \(-0.797566\pi\)
0.804498 0.593955i \(-0.202434\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 4056.00i − 0.278059i
\(598\) 0 0
\(599\) 18120.0 1.23600 0.617999 0.786179i \(-0.287943\pi\)
0.617999 + 0.786179i \(0.287943\pi\)
\(600\) 0 0
\(601\) 17546.0 1.19088 0.595438 0.803401i \(-0.296979\pi\)
0.595438 + 0.803401i \(0.296979\pi\)
\(602\) 0 0
\(603\) 9108.00i 0.615102i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 14560.0i − 0.973595i −0.873515 0.486798i \(-0.838165\pi\)
0.873515 0.486798i \(-0.161835\pi\)
\(608\) 0 0
\(609\) 1134.00 0.0754548
\(610\) 0 0
\(611\) −11904.0 −0.788190
\(612\) 0 0
\(613\) 4498.00i 0.296366i 0.988960 + 0.148183i \(0.0473425\pi\)
−0.988960 + 0.148183i \(0.952657\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 5478.00i − 0.357433i −0.983901 0.178716i \(-0.942806\pi\)
0.983901 0.178716i \(-0.0571944\pi\)
\(618\) 0 0
\(619\) −6044.00 −0.392454 −0.196227 0.980559i \(-0.562869\pi\)
−0.196227 + 0.980559i \(0.562869\pi\)
\(620\) 0 0
\(621\) 648.000 0.0418733
\(622\) 0 0
\(623\) 11466.0i 0.737360i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8208.00i 0.522801i
\(628\) 0 0
\(629\) 20292.0 1.28632
\(630\) 0 0
\(631\) −15352.0 −0.968547 −0.484274 0.874917i \(-0.660916\pi\)
−0.484274 + 0.874917i \(0.660916\pi\)
\(632\) 0 0
\(633\) − 10092.0i − 0.633682i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 3038.00i 0.188964i
\(638\) 0 0
\(639\) −7560.00 −0.468027
\(640\) 0 0
\(641\) −22398.0 −1.38014 −0.690068 0.723744i \(-0.742420\pi\)
−0.690068 + 0.723744i \(0.742420\pi\)
\(642\) 0 0
\(643\) − 3764.00i − 0.230852i −0.993316 0.115426i \(-0.963177\pi\)
0.993316 0.115426i \(-0.0368233\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 17688.0i − 1.07479i −0.843332 0.537393i \(-0.819409\pi\)
0.843332 0.537393i \(-0.180591\pi\)
\(648\) 0 0
\(649\) −14256.0 −0.862245
\(650\) 0 0
\(651\) 2352.00 0.141601
\(652\) 0 0
\(653\) − 19878.0i − 1.19125i −0.803263 0.595625i \(-0.796904\pi\)
0.803263 0.595625i \(-0.203096\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8010.00i 0.475647i
\(658\) 0 0
\(659\) 20004.0 1.18247 0.591233 0.806501i \(-0.298641\pi\)
0.591233 + 0.806501i \(0.298641\pi\)
\(660\) 0 0
\(661\) −1306.00 −0.0768495 −0.0384247 0.999261i \(-0.512234\pi\)
−0.0384247 + 0.999261i \(0.512234\pi\)
\(662\) 0 0
\(663\) 21204.0i 1.24207i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 1296.00i − 0.0752344i
\(668\) 0 0
\(669\) −14304.0 −0.826644
\(670\) 0 0
\(671\) 9144.00 0.526081
\(672\) 0 0
\(673\) 13054.0i 0.747689i 0.927491 + 0.373845i \(0.121961\pi\)
−0.927491 + 0.373845i \(0.878039\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5046.00i 0.286460i 0.989689 + 0.143230i \(0.0457489\pi\)
−0.989689 + 0.143230i \(0.954251\pi\)
\(678\) 0 0
\(679\) −7070.00 −0.399590
\(680\) 0 0
\(681\) −1260.00 −0.0709006
\(682\) 0 0
\(683\) − 12468.0i − 0.698499i −0.937030 0.349249i \(-0.886437\pi\)
0.937030 0.349249i \(-0.113563\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 5646.00i 0.313549i
\(688\) 0 0
\(689\) 24924.0 1.37813
\(690\) 0 0
\(691\) −23212.0 −1.27790 −0.638948 0.769250i \(-0.720630\pi\)
−0.638948 + 0.769250i \(0.720630\pi\)
\(692\) 0 0
\(693\) − 2268.00i − 0.124321i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 43092.0i 2.34179i
\(698\) 0 0
\(699\) 15246.0 0.824974
\(700\) 0 0
\(701\) 35958.0 1.93740 0.968698 0.248241i \(-0.0798526\pi\)
0.968698 + 0.248241i \(0.0798526\pi\)
\(702\) 0 0
\(703\) − 13528.0i − 0.725773i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 42.0000i 0.00223419i
\(708\) 0 0
\(709\) −6446.00 −0.341445 −0.170723 0.985319i \(-0.554610\pi\)
−0.170723 + 0.985319i \(0.554610\pi\)
\(710\) 0 0
\(711\) 720.000 0.0379777
\(712\) 0 0
\(713\) − 2688.00i − 0.141187i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 16272.0i 0.847544i
\(718\) 0 0
\(719\) 4704.00 0.243991 0.121996 0.992531i \(-0.461071\pi\)
0.121996 + 0.992531i \(0.461071\pi\)
\(720\) 0 0
\(721\) −3304.00 −0.170662
\(722\) 0 0
\(723\) − 7770.00i − 0.399681i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 10600.0i − 0.540760i −0.962754 0.270380i \(-0.912851\pi\)
0.962754 0.270380i \(-0.0871493\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −19608.0 −0.992104
\(732\) 0 0
\(733\) − 12542.0i − 0.631991i −0.948761 0.315995i \(-0.897662\pi\)
0.948761 0.315995i \(-0.102338\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 36432.0i − 1.82088i
\(738\) 0 0
\(739\) −23324.0 −1.16101 −0.580506 0.814256i \(-0.697145\pi\)
−0.580506 + 0.814256i \(0.697145\pi\)
\(740\) 0 0
\(741\) 14136.0 0.700808
\(742\) 0 0
\(743\) 6312.00i 0.311662i 0.987784 + 0.155831i \(0.0498055\pi\)
−0.987784 + 0.155831i \(0.950194\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 972.000i − 0.0476086i
\(748\) 0 0
\(749\) 6804.00 0.331926
\(750\) 0 0
\(751\) 35840.0 1.74144 0.870719 0.491781i \(-0.163654\pi\)
0.870719 + 0.491781i \(0.163654\pi\)
\(752\) 0 0
\(753\) − 14796.0i − 0.716064i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 34594.0i − 1.66095i −0.557055 0.830476i \(-0.688069\pi\)
0.557055 0.830476i \(-0.311931\pi\)
\(758\) 0 0
\(759\) −2592.00 −0.123957
\(760\) 0 0
\(761\) 23946.0 1.14066 0.570330 0.821416i \(-0.306815\pi\)
0.570330 + 0.821416i \(0.306815\pi\)
\(762\) 0 0
\(763\) 12502.0i 0.593188i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24552.0i 1.15583i
\(768\) 0 0
\(769\) −18770.0 −0.880187 −0.440093 0.897952i \(-0.645055\pi\)
−0.440093 + 0.897952i \(0.645055\pi\)
\(770\) 0 0
\(771\) 10314.0 0.481776
\(772\) 0 0
\(773\) − 30342.0i − 1.41181i −0.708309 0.705903i \(-0.750541\pi\)
0.708309 0.705903i \(-0.249459\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 3738.00i 0.172587i
\(778\) 0 0
\(779\) 28728.0 1.32129
\(780\) 0 0
\(781\) 30240.0 1.38550
\(782\) 0 0
\(783\) 1458.00i 0.0665449i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 26188.0i − 1.18615i −0.805147 0.593076i \(-0.797913\pi\)
0.805147 0.593076i \(-0.202087\pi\)
\(788\) 0 0
\(789\) 18360.0 0.828433
\(790\) 0 0
\(791\) −16002.0 −0.719299
\(792\) 0 0
\(793\) − 15748.0i − 0.705205i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 34818.0i − 1.54745i −0.633522 0.773724i \(-0.718391\pi\)
0.633522 0.773724i \(-0.281609\pi\)
\(798\) 0 0
\(799\) 21888.0 0.969139
\(800\) 0 0
\(801\) −14742.0 −0.650291
\(802\) 0 0
\(803\) − 32040.0i − 1.40805i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 54.0000i 0.00235550i
\(808\) 0 0
\(809\) 21702.0 0.943142 0.471571 0.881828i \(-0.343687\pi\)
0.471571 + 0.881828i \(0.343687\pi\)
\(810\) 0 0
\(811\) −20356.0 −0.881376 −0.440688 0.897660i \(-0.645265\pi\)
−0.440688 + 0.897660i \(0.645265\pi\)
\(812\) 0 0
\(813\) 20688.0i 0.892448i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 13072.0i 0.559769i
\(818\) 0 0
\(819\) −3906.00 −0.166650
\(820\) 0 0
\(821\) −19890.0 −0.845513 −0.422756 0.906243i \(-0.638937\pi\)
−0.422756 + 0.906243i \(0.638937\pi\)
\(822\) 0 0
\(823\) − 4232.00i − 0.179245i −0.995976 0.0896223i \(-0.971434\pi\)
0.995976 0.0896223i \(-0.0285660\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9636.00i 0.405171i 0.979265 + 0.202586i \(0.0649344\pi\)
−0.979265 + 0.202586i \(0.935066\pi\)
\(828\) 0 0
\(829\) −35294.0 −1.47866 −0.739331 0.673342i \(-0.764858\pi\)
−0.739331 + 0.673342i \(0.764858\pi\)
\(830\) 0 0
\(831\) −18762.0 −0.783209
\(832\) 0 0
\(833\) − 5586.00i − 0.232345i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3024.00i 0.124880i
\(838\) 0 0
\(839\) 3768.00 0.155049 0.0775243 0.996990i \(-0.475298\pi\)
0.0775243 + 0.996990i \(0.475298\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 0 0
\(843\) 3582.00i 0.146347i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 245.000i − 0.00993896i
\(848\) 0 0
\(849\) −21468.0 −0.867821
\(850\) 0 0
\(851\) 4272.00 0.172083
\(852\) 0 0
\(853\) 39466.0i 1.58416i 0.610416 + 0.792081i \(0.291002\pi\)
−0.610416 + 0.792081i \(0.708998\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 34038.0i − 1.35673i −0.734726 0.678364i \(-0.762689\pi\)
0.734726 0.678364i \(-0.237311\pi\)
\(858\) 0 0
\(859\) 3364.00 0.133618 0.0668092 0.997766i \(-0.478718\pi\)
0.0668092 + 0.997766i \(0.478718\pi\)
\(860\) 0 0
\(861\) −7938.00 −0.314200
\(862\) 0 0
\(863\) − 13104.0i − 0.516878i −0.966028 0.258439i \(-0.916792\pi\)
0.966028 0.258439i \(-0.0832080\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 24249.0i − 0.949872i
\(868\) 0 0
\(869\) −2880.00 −0.112425
\(870\) 0 0
\(871\) −62744.0 −2.44087
\(872\) 0 0
\(873\) − 9090.00i − 0.352405i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 40858.0i − 1.57318i −0.617477 0.786589i \(-0.711845\pi\)
0.617477 0.786589i \(-0.288155\pi\)
\(878\) 0 0
\(879\) −11214.0 −0.430306
\(880\) 0 0
\(881\) −37374.0 −1.42924 −0.714621 0.699512i \(-0.753401\pi\)
−0.714621 + 0.699512i \(0.753401\pi\)
\(882\) 0 0
\(883\) − 9788.00i − 0.373038i −0.982451 0.186519i \(-0.940279\pi\)
0.982451 0.186519i \(-0.0597206\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 50424.0i 1.90876i 0.298591 + 0.954381i \(0.403483\pi\)
−0.298591 + 0.954381i \(0.596517\pi\)
\(888\) 0 0
\(889\) −9296.00 −0.350706
\(890\) 0 0
\(891\) 2916.00 0.109640
\(892\) 0 0
\(893\) − 14592.0i − 0.546811i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4464.00i 0.166163i
\(898\) 0 0
\(899\) 6048.00 0.224374
\(900\) 0 0
\(901\) −45828.0 −1.69451
\(902\) 0 0
\(903\) − 3612.00i − 0.133112i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 12412.0i − 0.454392i −0.973849 0.227196i \(-0.927044\pi\)
0.973849 0.227196i \(-0.0729558\pi\)
\(908\) 0 0
\(909\) −54.0000 −0.00197037
\(910\) 0 0
\(911\) 6576.00 0.239158 0.119579 0.992825i \(-0.461846\pi\)
0.119579 + 0.992825i \(0.461846\pi\)
\(912\) 0 0
\(913\) 3888.00i 0.140935i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 8484.00i − 0.305525i
\(918\) 0 0
\(919\) −8264.00 −0.296631 −0.148316 0.988940i \(-0.547385\pi\)
−0.148316 + 0.988940i \(0.547385\pi\)
\(920\) 0 0
\(921\) 2532.00 0.0905887
\(922\) 0 0
\(923\) − 52080.0i − 1.85724i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 4248.00i − 0.150510i
\(928\) 0 0
\(929\) −39426.0 −1.39238 −0.696192 0.717855i \(-0.745124\pi\)
−0.696192 + 0.717855i \(0.745124\pi\)
\(930\) 0 0
\(931\) −3724.00 −0.131095
\(932\) 0 0
\(933\) 18936.0i 0.664455i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 4678.00i − 0.163099i −0.996669 0.0815494i \(-0.974013\pi\)
0.996669 0.0815494i \(-0.0259868\pi\)
\(938\) 0 0
\(939\) 24846.0 0.863492
\(940\) 0 0
\(941\) −17346.0 −0.600918 −0.300459 0.953795i \(-0.597140\pi\)
−0.300459 + 0.953795i \(0.597140\pi\)
\(942\) 0 0
\(943\) 9072.00i 0.313282i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19452.0i 0.667482i 0.942665 + 0.333741i \(0.108311\pi\)
−0.942665 + 0.333741i \(0.891689\pi\)
\(948\) 0 0
\(949\) −55180.0 −1.88748
\(950\) 0 0
\(951\) −27954.0 −0.953176
\(952\) 0 0
\(953\) − 4458.00i − 0.151531i −0.997126 0.0757654i \(-0.975860\pi\)
0.997126 0.0757654i \(-0.0241400\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 5832.00i − 0.196992i
\(958\) 0 0
\(959\) 8778.00 0.295575
\(960\) 0 0
\(961\) −17247.0 −0.578933
\(962\) 0 0
\(963\) 8748.00i 0.292731i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 52520.0i 1.74657i 0.487213 + 0.873283i \(0.338013\pi\)
−0.487213 + 0.873283i \(0.661987\pi\)
\(968\) 0 0
\(969\) −25992.0 −0.861696
\(970\) 0 0
\(971\) −10404.0 −0.343852 −0.171926 0.985110i \(-0.554999\pi\)
−0.171926 + 0.985110i \(0.554999\pi\)
\(972\) 0 0
\(973\) 2380.00i 0.0784165i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7566.00i − 0.247756i −0.992297 0.123878i \(-0.960467\pi\)
0.992297 0.123878i \(-0.0395332\pi\)
\(978\) 0 0
\(979\) 58968.0 1.92505
\(980\) 0 0
\(981\) −16074.0 −0.523143
\(982\) 0 0
\(983\) − 44376.0i − 1.43985i −0.694051 0.719926i \(-0.744176\pi\)
0.694051 0.719926i \(-0.255824\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 4032.00i 0.130030i
\(988\) 0 0
\(989\) −4128.00 −0.132723
\(990\) 0 0
\(991\) −27328.0 −0.875986 −0.437993 0.898978i \(-0.644311\pi\)
−0.437993 + 0.898978i \(0.644311\pi\)
\(992\) 0 0
\(993\) 4956.00i 0.158383i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2774.00i 0.0881178i 0.999029 + 0.0440589i \(0.0140289\pi\)
−0.999029 + 0.0440589i \(0.985971\pi\)
\(998\) 0 0
\(999\) −4806.00 −0.152207
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 2100.4.k.j.1849.2 2
5.2 odd 4 2100.4.a.l.1.1 1
5.3 odd 4 84.4.a.a.1.1 1
5.4 even 2 inner 2100.4.k.j.1849.1 2
15.8 even 4 252.4.a.b.1.1 1
20.3 even 4 336.4.a.k.1.1 1
35.3 even 12 588.4.i.c.373.1 2
35.13 even 4 588.4.a.d.1.1 1
35.18 odd 12 588.4.i.f.373.1 2
35.23 odd 12 588.4.i.f.361.1 2
35.33 even 12 588.4.i.c.361.1 2
40.3 even 4 1344.4.a.d.1.1 1
40.13 odd 4 1344.4.a.q.1.1 1
60.23 odd 4 1008.4.a.h.1.1 1
105.23 even 12 1764.4.k.l.361.1 2
105.38 odd 12 1764.4.k.f.1549.1 2
105.53 even 12 1764.4.k.l.1549.1 2
105.68 odd 12 1764.4.k.f.361.1 2
105.83 odd 4 1764.4.a.j.1.1 1
140.83 odd 4 2352.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.a.a.1.1 1 5.3 odd 4
252.4.a.b.1.1 1 15.8 even 4
336.4.a.k.1.1 1 20.3 even 4
588.4.a.d.1.1 1 35.13 even 4
588.4.i.c.361.1 2 35.33 even 12
588.4.i.c.373.1 2 35.3 even 12
588.4.i.f.361.1 2 35.23 odd 12
588.4.i.f.373.1 2 35.18 odd 12
1008.4.a.h.1.1 1 60.23 odd 4
1344.4.a.d.1.1 1 40.3 even 4
1344.4.a.q.1.1 1 40.13 odd 4
1764.4.a.j.1.1 1 105.83 odd 4
1764.4.k.f.361.1 2 105.68 odd 12
1764.4.k.f.1549.1 2 105.38 odd 12
1764.4.k.l.361.1 2 105.23 even 12
1764.4.k.l.1549.1 2 105.53 even 12
2100.4.a.l.1.1 1 5.2 odd 4
2100.4.k.j.1849.1 2 5.4 even 2 inner
2100.4.k.j.1849.2 2 1.1 even 1 trivial
2352.4.a.d.1.1 1 140.83 odd 4