Properties

Label 2205.4.a.b.1.1
Level $2205$
Weight $4$
Character 2205.1
Self dual yes
Analytic conductor $130.099$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,4,Mod(1,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

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: yes
Analytic conductor: \(130.099211563\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2205.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{2} +17.0000 q^{4} +5.00000 q^{5} -45.0000 q^{8} +O(q^{10})\) \(q-5.00000 q^{2} +17.0000 q^{4} +5.00000 q^{5} -45.0000 q^{8} -25.0000 q^{10} -12.0000 q^{11} -30.0000 q^{13} +89.0000 q^{16} -134.000 q^{17} +92.0000 q^{19} +85.0000 q^{20} +60.0000 q^{22} -112.000 q^{23} +25.0000 q^{25} +150.000 q^{26} +58.0000 q^{29} +224.000 q^{31} -85.0000 q^{32} +670.000 q^{34} -146.000 q^{37} -460.000 q^{38} -225.000 q^{40} +18.0000 q^{41} +340.000 q^{43} -204.000 q^{44} +560.000 q^{46} +208.000 q^{47} -125.000 q^{50} -510.000 q^{52} +754.000 q^{53} -60.0000 q^{55} -290.000 q^{58} +380.000 q^{59} -718.000 q^{61} -1120.00 q^{62} -287.000 q^{64} -150.000 q^{65} +412.000 q^{67} -2278.00 q^{68} +960.000 q^{71} -1066.00 q^{73} +730.000 q^{74} +1564.00 q^{76} +896.000 q^{79} +445.000 q^{80} -90.0000 q^{82} +436.000 q^{83} -670.000 q^{85} -1700.00 q^{86} +540.000 q^{88} -1038.00 q^{89} -1904.00 q^{92} -1040.00 q^{94} +460.000 q^{95} +702.000 q^{97} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.00000 −1.76777 −0.883883 0.467707i \(-0.845080\pi\)
−0.883883 + 0.467707i \(0.845080\pi\)
\(3\) 0 0
\(4\) 17.0000 2.12500
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) −45.0000 −1.98874
\(9\) 0 0
\(10\) −25.0000 −0.790569
\(11\) −12.0000 −0.328921 −0.164461 0.986384i \(-0.552588\pi\)
−0.164461 + 0.986384i \(0.552588\pi\)
\(12\) 0 0
\(13\) −30.0000 −0.640039 −0.320019 0.947411i \(-0.603689\pi\)
−0.320019 + 0.947411i \(0.603689\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) −134.000 −1.91175 −0.955876 0.293771i \(-0.905090\pi\)
−0.955876 + 0.293771i \(0.905090\pi\)
\(18\) 0 0
\(19\) 92.0000 1.11086 0.555428 0.831565i \(-0.312555\pi\)
0.555428 + 0.831565i \(0.312555\pi\)
\(20\) 85.0000 0.950329
\(21\) 0 0
\(22\) 60.0000 0.581456
\(23\) −112.000 −1.01537 −0.507687 0.861541i \(-0.669499\pi\)
−0.507687 + 0.861541i \(0.669499\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 150.000 1.13144
\(27\) 0 0
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 224.000 1.29779 0.648897 0.760877i \(-0.275231\pi\)
0.648897 + 0.760877i \(0.275231\pi\)
\(32\) −85.0000 −0.469563
\(33\) 0 0
\(34\) 670.000 3.37953
\(35\) 0 0
\(36\) 0 0
\(37\) −146.000 −0.648710 −0.324355 0.945936i \(-0.605147\pi\)
−0.324355 + 0.945936i \(0.605147\pi\)
\(38\) −460.000 −1.96373
\(39\) 0 0
\(40\) −225.000 −0.889391
\(41\) 18.0000 0.0685641 0.0342820 0.999412i \(-0.489086\pi\)
0.0342820 + 0.999412i \(0.489086\pi\)
\(42\) 0 0
\(43\) 340.000 1.20580 0.602901 0.797816i \(-0.294011\pi\)
0.602901 + 0.797816i \(0.294011\pi\)
\(44\) −204.000 −0.698958
\(45\) 0 0
\(46\) 560.000 1.79495
\(47\) 208.000 0.645530 0.322765 0.946479i \(-0.395388\pi\)
0.322765 + 0.946479i \(0.395388\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −125.000 −0.353553
\(51\) 0 0
\(52\) −510.000 −1.36008
\(53\) 754.000 1.95415 0.977074 0.212899i \(-0.0682905\pi\)
0.977074 + 0.212899i \(0.0682905\pi\)
\(54\) 0 0
\(55\) −60.0000 −0.147098
\(56\) 0 0
\(57\) 0 0
\(58\) −290.000 −0.656532
\(59\) 380.000 0.838505 0.419252 0.907870i \(-0.362292\pi\)
0.419252 + 0.907870i \(0.362292\pi\)
\(60\) 0 0
\(61\) −718.000 −1.50706 −0.753529 0.657415i \(-0.771650\pi\)
−0.753529 + 0.657415i \(0.771650\pi\)
\(62\) −1120.00 −2.29420
\(63\) 0 0
\(64\) −287.000 −0.560547
\(65\) −150.000 −0.286234
\(66\) 0 0
\(67\) 412.000 0.751251 0.375625 0.926772i \(-0.377428\pi\)
0.375625 + 0.926772i \(0.377428\pi\)
\(68\) −2278.00 −4.06247
\(69\) 0 0
\(70\) 0 0
\(71\) 960.000 1.60466 0.802331 0.596879i \(-0.203593\pi\)
0.802331 + 0.596879i \(0.203593\pi\)
\(72\) 0 0
\(73\) −1066.00 −1.70912 −0.854561 0.519352i \(-0.826174\pi\)
−0.854561 + 0.519352i \(0.826174\pi\)
\(74\) 730.000 1.14677
\(75\) 0 0
\(76\) 1564.00 2.36057
\(77\) 0 0
\(78\) 0 0
\(79\) 896.000 1.27605 0.638025 0.770016i \(-0.279752\pi\)
0.638025 + 0.770016i \(0.279752\pi\)
\(80\) 445.000 0.621906
\(81\) 0 0
\(82\) −90.0000 −0.121205
\(83\) 436.000 0.576593 0.288296 0.957541i \(-0.406911\pi\)
0.288296 + 0.957541i \(0.406911\pi\)
\(84\) 0 0
\(85\) −670.000 −0.854961
\(86\) −1700.00 −2.13158
\(87\) 0 0
\(88\) 540.000 0.654139
\(89\) −1038.00 −1.23627 −0.618134 0.786073i \(-0.712111\pi\)
−0.618134 + 0.786073i \(0.712111\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1904.00 −2.15767
\(93\) 0 0
\(94\) −1040.00 −1.14115
\(95\) 460.000 0.496790
\(96\) 0 0
\(97\) 702.000 0.734818 0.367409 0.930060i \(-0.380245\pi\)
0.367409 + 0.930060i \(0.380245\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 425.000 0.425000
\(101\) 46.0000 0.0453185 0.0226593 0.999743i \(-0.492787\pi\)
0.0226593 + 0.999743i \(0.492787\pi\)
\(102\) 0 0
\(103\) −1880.00 −1.79847 −0.899233 0.437471i \(-0.855874\pi\)
−0.899233 + 0.437471i \(0.855874\pi\)
\(104\) 1350.00 1.27287
\(105\) 0 0
\(106\) −3770.00 −3.45448
\(107\) −732.000 −0.661356 −0.330678 0.943744i \(-0.607277\pi\)
−0.330678 + 0.943744i \(0.607277\pi\)
\(108\) 0 0
\(109\) −378.000 −0.332164 −0.166082 0.986112i \(-0.553112\pi\)
−0.166082 + 0.986112i \(0.553112\pi\)
\(110\) 300.000 0.260035
\(111\) 0 0
\(112\) 0 0
\(113\) −1458.00 −1.21378 −0.606890 0.794786i \(-0.707583\pi\)
−0.606890 + 0.794786i \(0.707583\pi\)
\(114\) 0 0
\(115\) −560.000 −0.454089
\(116\) 986.000 0.789205
\(117\) 0 0
\(118\) −1900.00 −1.48228
\(119\) 0 0
\(120\) 0 0
\(121\) −1187.00 −0.891811
\(122\) 3590.00 2.66413
\(123\) 0 0
\(124\) 3808.00 2.75781
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 608.000 0.424813 0.212407 0.977181i \(-0.431870\pi\)
0.212407 + 0.977181i \(0.431870\pi\)
\(128\) 2115.00 1.46048
\(129\) 0 0
\(130\) 750.000 0.505995
\(131\) −956.000 −0.637604 −0.318802 0.947821i \(-0.603280\pi\)
−0.318802 + 0.947821i \(0.603280\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −2060.00 −1.32804
\(135\) 0 0
\(136\) 6030.00 3.80197
\(137\) 374.000 0.233233 0.116617 0.993177i \(-0.462795\pi\)
0.116617 + 0.993177i \(0.462795\pi\)
\(138\) 0 0
\(139\) −396.000 −0.241642 −0.120821 0.992674i \(-0.538553\pi\)
−0.120821 + 0.992674i \(0.538553\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4800.00 −2.83667
\(143\) 360.000 0.210522
\(144\) 0 0
\(145\) 290.000 0.166091
\(146\) 5330.00 3.02133
\(147\) 0 0
\(148\) −2482.00 −1.37851
\(149\) 1874.00 1.03036 0.515181 0.857081i \(-0.327725\pi\)
0.515181 + 0.857081i \(0.327725\pi\)
\(150\) 0 0
\(151\) −1096.00 −0.590670 −0.295335 0.955394i \(-0.595431\pi\)
−0.295335 + 0.955394i \(0.595431\pi\)
\(152\) −4140.00 −2.20920
\(153\) 0 0
\(154\) 0 0
\(155\) 1120.00 0.580391
\(156\) 0 0
\(157\) −1918.00 −0.974988 −0.487494 0.873126i \(-0.662089\pi\)
−0.487494 + 0.873126i \(0.662089\pi\)
\(158\) −4480.00 −2.25576
\(159\) 0 0
\(160\) −425.000 −0.209995
\(161\) 0 0
\(162\) 0 0
\(163\) 2316.00 1.11290 0.556451 0.830880i \(-0.312163\pi\)
0.556451 + 0.830880i \(0.312163\pi\)
\(164\) 306.000 0.145699
\(165\) 0 0
\(166\) −2180.00 −1.01928
\(167\) −1736.00 −0.804405 −0.402203 0.915551i \(-0.631755\pi\)
−0.402203 + 0.915551i \(0.631755\pi\)
\(168\) 0 0
\(169\) −1297.00 −0.590350
\(170\) 3350.00 1.51137
\(171\) 0 0
\(172\) 5780.00 2.56233
\(173\) −2442.00 −1.07319 −0.536595 0.843840i \(-0.680290\pi\)
−0.536595 + 0.843840i \(0.680290\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1068.00 −0.457406
\(177\) 0 0
\(178\) 5190.00 2.18543
\(179\) 4092.00 1.70866 0.854331 0.519730i \(-0.173967\pi\)
0.854331 + 0.519730i \(0.173967\pi\)
\(180\) 0 0
\(181\) −1270.00 −0.521538 −0.260769 0.965401i \(-0.583976\pi\)
−0.260769 + 0.965401i \(0.583976\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5040.00 2.01931
\(185\) −730.000 −0.290112
\(186\) 0 0
\(187\) 1608.00 0.628816
\(188\) 3536.00 1.37175
\(189\) 0 0
\(190\) −2300.00 −0.878208
\(191\) −4904.00 −1.85781 −0.928903 0.370323i \(-0.879247\pi\)
−0.928903 + 0.370323i \(0.879247\pi\)
\(192\) 0 0
\(193\) 2178.00 0.812310 0.406155 0.913804i \(-0.366869\pi\)
0.406155 + 0.913804i \(0.366869\pi\)
\(194\) −3510.00 −1.29899
\(195\) 0 0
\(196\) 0 0
\(197\) 2850.00 1.03073 0.515366 0.856970i \(-0.327656\pi\)
0.515366 + 0.856970i \(0.327656\pi\)
\(198\) 0 0
\(199\) 1144.00 0.407518 0.203759 0.979021i \(-0.434684\pi\)
0.203759 + 0.979021i \(0.434684\pi\)
\(200\) −1125.00 −0.397748
\(201\) 0 0
\(202\) −230.000 −0.0801126
\(203\) 0 0
\(204\) 0 0
\(205\) 90.0000 0.0306628
\(206\) 9400.00 3.17927
\(207\) 0 0
\(208\) −2670.00 −0.890054
\(209\) −1104.00 −0.365384
\(210\) 0 0
\(211\) 412.000 0.134423 0.0672115 0.997739i \(-0.478590\pi\)
0.0672115 + 0.997739i \(0.478590\pi\)
\(212\) 12818.0 4.15257
\(213\) 0 0
\(214\) 3660.00 1.16912
\(215\) 1700.00 0.539251
\(216\) 0 0
\(217\) 0 0
\(218\) 1890.00 0.587188
\(219\) 0 0
\(220\) −1020.00 −0.312584
\(221\) 4020.00 1.22359
\(222\) 0 0
\(223\) 1632.00 0.490075 0.245038 0.969514i \(-0.421200\pi\)
0.245038 + 0.969514i \(0.421200\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 7290.00 2.14568
\(227\) 4084.00 1.19412 0.597059 0.802198i \(-0.296336\pi\)
0.597059 + 0.802198i \(0.296336\pi\)
\(228\) 0 0
\(229\) 3386.00 0.977088 0.488544 0.872539i \(-0.337528\pi\)
0.488544 + 0.872539i \(0.337528\pi\)
\(230\) 2800.00 0.802724
\(231\) 0 0
\(232\) −2610.00 −0.738599
\(233\) −5322.00 −1.49638 −0.748188 0.663486i \(-0.769076\pi\)
−0.748188 + 0.663486i \(0.769076\pi\)
\(234\) 0 0
\(235\) 1040.00 0.288690
\(236\) 6460.00 1.78182
\(237\) 0 0
\(238\) 0 0
\(239\) −3736.00 −1.01114 −0.505569 0.862786i \(-0.668717\pi\)
−0.505569 + 0.862786i \(0.668717\pi\)
\(240\) 0 0
\(241\) −210.000 −0.0561298 −0.0280649 0.999606i \(-0.508935\pi\)
−0.0280649 + 0.999606i \(0.508935\pi\)
\(242\) 5935.00 1.57651
\(243\) 0 0
\(244\) −12206.0 −3.20250
\(245\) 0 0
\(246\) 0 0
\(247\) −2760.00 −0.710990
\(248\) −10080.0 −2.58097
\(249\) 0 0
\(250\) −625.000 −0.158114
\(251\) −4212.00 −1.05920 −0.529600 0.848248i \(-0.677658\pi\)
−0.529600 + 0.848248i \(0.677658\pi\)
\(252\) 0 0
\(253\) 1344.00 0.333978
\(254\) −3040.00 −0.750971
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) 5130.00 1.24514 0.622569 0.782565i \(-0.286089\pi\)
0.622569 + 0.782565i \(0.286089\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2550.00 −0.608247
\(261\) 0 0
\(262\) 4780.00 1.12714
\(263\) −848.000 −0.198821 −0.0994105 0.995047i \(-0.531696\pi\)
−0.0994105 + 0.995047i \(0.531696\pi\)
\(264\) 0 0
\(265\) 3770.00 0.873922
\(266\) 0 0
\(267\) 0 0
\(268\) 7004.00 1.59641
\(269\) −1274.00 −0.288763 −0.144381 0.989522i \(-0.546119\pi\)
−0.144381 + 0.989522i \(0.546119\pi\)
\(270\) 0 0
\(271\) −864.000 −0.193669 −0.0968344 0.995301i \(-0.530872\pi\)
−0.0968344 + 0.995301i \(0.530872\pi\)
\(272\) −11926.0 −2.65853
\(273\) 0 0
\(274\) −1870.00 −0.412302
\(275\) −300.000 −0.0657843
\(276\) 0 0
\(277\) −8530.00 −1.85025 −0.925123 0.379668i \(-0.876038\pi\)
−0.925123 + 0.379668i \(0.876038\pi\)
\(278\) 1980.00 0.427167
\(279\) 0 0
\(280\) 0 0
\(281\) 5382.00 1.14257 0.571287 0.820750i \(-0.306444\pi\)
0.571287 + 0.820750i \(0.306444\pi\)
\(282\) 0 0
\(283\) −6236.00 −1.30986 −0.654932 0.755687i \(-0.727303\pi\)
−0.654932 + 0.755687i \(0.727303\pi\)
\(284\) 16320.0 3.40991
\(285\) 0 0
\(286\) −1800.00 −0.372155
\(287\) 0 0
\(288\) 0 0
\(289\) 13043.0 2.65479
\(290\) −1450.00 −0.293610
\(291\) 0 0
\(292\) −18122.0 −3.63188
\(293\) −818.000 −0.163099 −0.0815496 0.996669i \(-0.525987\pi\)
−0.0815496 + 0.996669i \(0.525987\pi\)
\(294\) 0 0
\(295\) 1900.00 0.374991
\(296\) 6570.00 1.29011
\(297\) 0 0
\(298\) −9370.00 −1.82144
\(299\) 3360.00 0.649879
\(300\) 0 0
\(301\) 0 0
\(302\) 5480.00 1.04417
\(303\) 0 0
\(304\) 8188.00 1.54478
\(305\) −3590.00 −0.673976
\(306\) 0 0
\(307\) 2268.00 0.421634 0.210817 0.977526i \(-0.432388\pi\)
0.210817 + 0.977526i \(0.432388\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5600.00 −1.02600
\(311\) 6648.00 1.21213 0.606067 0.795414i \(-0.292746\pi\)
0.606067 + 0.795414i \(0.292746\pi\)
\(312\) 0 0
\(313\) −9818.00 −1.77299 −0.886495 0.462737i \(-0.846867\pi\)
−0.886495 + 0.462737i \(0.846867\pi\)
\(314\) 9590.00 1.72355
\(315\) 0 0
\(316\) 15232.0 2.71160
\(317\) −934.000 −0.165485 −0.0827424 0.996571i \(-0.526368\pi\)
−0.0827424 + 0.996571i \(0.526368\pi\)
\(318\) 0 0
\(319\) −696.000 −0.122158
\(320\) −1435.00 −0.250684
\(321\) 0 0
\(322\) 0 0
\(323\) −12328.0 −2.12368
\(324\) 0 0
\(325\) −750.000 −0.128008
\(326\) −11580.0 −1.96735
\(327\) 0 0
\(328\) −810.000 −0.136356
\(329\) 0 0
\(330\) 0 0
\(331\) 2292.00 0.380603 0.190302 0.981726i \(-0.439053\pi\)
0.190302 + 0.981726i \(0.439053\pi\)
\(332\) 7412.00 1.22526
\(333\) 0 0
\(334\) 8680.00 1.42200
\(335\) 2060.00 0.335970
\(336\) 0 0
\(337\) −6062.00 −0.979876 −0.489938 0.871757i \(-0.662981\pi\)
−0.489938 + 0.871757i \(0.662981\pi\)
\(338\) 6485.00 1.04360
\(339\) 0 0
\(340\) −11390.0 −1.81679
\(341\) −2688.00 −0.426872
\(342\) 0 0
\(343\) 0 0
\(344\) −15300.0 −2.39803
\(345\) 0 0
\(346\) 12210.0 1.89715
\(347\) −1484.00 −0.229583 −0.114791 0.993390i \(-0.536620\pi\)
−0.114791 + 0.993390i \(0.536620\pi\)
\(348\) 0 0
\(349\) −254.000 −0.0389579 −0.0194790 0.999810i \(-0.506201\pi\)
−0.0194790 + 0.999810i \(0.506201\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1020.00 0.154449
\(353\) −10950.0 −1.65102 −0.825509 0.564388i \(-0.809112\pi\)
−0.825509 + 0.564388i \(0.809112\pi\)
\(354\) 0 0
\(355\) 4800.00 0.717627
\(356\) −17646.0 −2.62707
\(357\) 0 0
\(358\) −20460.0 −3.02052
\(359\) −11376.0 −1.67243 −0.836215 0.548402i \(-0.815236\pi\)
−0.836215 + 0.548402i \(0.815236\pi\)
\(360\) 0 0
\(361\) 1605.00 0.233999
\(362\) 6350.00 0.921957
\(363\) 0 0
\(364\) 0 0
\(365\) −5330.00 −0.764342
\(366\) 0 0
\(367\) 1136.00 0.161577 0.0807884 0.996731i \(-0.474256\pi\)
0.0807884 + 0.996731i \(0.474256\pi\)
\(368\) −9968.00 −1.41201
\(369\) 0 0
\(370\) 3650.00 0.512850
\(371\) 0 0
\(372\) 0 0
\(373\) −8242.00 −1.14411 −0.572057 0.820214i \(-0.693854\pi\)
−0.572057 + 0.820214i \(0.693854\pi\)
\(374\) −8040.00 −1.11160
\(375\) 0 0
\(376\) −9360.00 −1.28379
\(377\) −1740.00 −0.237704
\(378\) 0 0
\(379\) 3620.00 0.490625 0.245313 0.969444i \(-0.421109\pi\)
0.245313 + 0.969444i \(0.421109\pi\)
\(380\) 7820.00 1.05568
\(381\) 0 0
\(382\) 24520.0 3.28417
\(383\) −8464.00 −1.12922 −0.564609 0.825359i \(-0.690973\pi\)
−0.564609 + 0.825359i \(0.690973\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10890.0 −1.43598
\(387\) 0 0
\(388\) 11934.0 1.56149
\(389\) −3678.00 −0.479388 −0.239694 0.970848i \(-0.577047\pi\)
−0.239694 + 0.970848i \(0.577047\pi\)
\(390\) 0 0
\(391\) 15008.0 1.94114
\(392\) 0 0
\(393\) 0 0
\(394\) −14250.0 −1.82209
\(395\) 4480.00 0.570666
\(396\) 0 0
\(397\) −12590.0 −1.59162 −0.795811 0.605545i \(-0.792955\pi\)
−0.795811 + 0.605545i \(0.792955\pi\)
\(398\) −5720.00 −0.720396
\(399\) 0 0
\(400\) 2225.00 0.278125
\(401\) −2850.00 −0.354918 −0.177459 0.984128i \(-0.556788\pi\)
−0.177459 + 0.984128i \(0.556788\pi\)
\(402\) 0 0
\(403\) −6720.00 −0.830638
\(404\) 782.000 0.0963019
\(405\) 0 0
\(406\) 0 0
\(407\) 1752.00 0.213374
\(408\) 0 0
\(409\) −1226.00 −0.148220 −0.0741098 0.997250i \(-0.523612\pi\)
−0.0741098 + 0.997250i \(0.523612\pi\)
\(410\) −450.000 −0.0542047
\(411\) 0 0
\(412\) −31960.0 −3.82174
\(413\) 0 0
\(414\) 0 0
\(415\) 2180.00 0.257860
\(416\) 2550.00 0.300539
\(417\) 0 0
\(418\) 5520.00 0.645914
\(419\) 612.000 0.0713560 0.0356780 0.999363i \(-0.488641\pi\)
0.0356780 + 0.999363i \(0.488641\pi\)
\(420\) 0 0
\(421\) 5182.00 0.599894 0.299947 0.953956i \(-0.403031\pi\)
0.299947 + 0.953956i \(0.403031\pi\)
\(422\) −2060.00 −0.237629
\(423\) 0 0
\(424\) −33930.0 −3.88629
\(425\) −3350.00 −0.382350
\(426\) 0 0
\(427\) 0 0
\(428\) −12444.0 −1.40538
\(429\) 0 0
\(430\) −8500.00 −0.953271
\(431\) 4984.00 0.557009 0.278504 0.960435i \(-0.410161\pi\)
0.278504 + 0.960435i \(0.410161\pi\)
\(432\) 0 0
\(433\) 1694.00 0.188010 0.0940051 0.995572i \(-0.470033\pi\)
0.0940051 + 0.995572i \(0.470033\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6426.00 −0.705848
\(437\) −10304.0 −1.12793
\(438\) 0 0
\(439\) −13864.0 −1.50727 −0.753636 0.657292i \(-0.771702\pi\)
−0.753636 + 0.657292i \(0.771702\pi\)
\(440\) 2700.00 0.292540
\(441\) 0 0
\(442\) −20100.0 −2.16303
\(443\) 4644.00 0.498066 0.249033 0.968495i \(-0.419887\pi\)
0.249033 + 0.968495i \(0.419887\pi\)
\(444\) 0 0
\(445\) −5190.00 −0.552875
\(446\) −8160.00 −0.866339
\(447\) 0 0
\(448\) 0 0
\(449\) 4926.00 0.517756 0.258878 0.965910i \(-0.416647\pi\)
0.258878 + 0.965910i \(0.416647\pi\)
\(450\) 0 0
\(451\) −216.000 −0.0225522
\(452\) −24786.0 −2.57928
\(453\) 0 0
\(454\) −20420.0 −2.11092
\(455\) 0 0
\(456\) 0 0
\(457\) −14694.0 −1.50406 −0.752031 0.659128i \(-0.770926\pi\)
−0.752031 + 0.659128i \(0.770926\pi\)
\(458\) −16930.0 −1.72726
\(459\) 0 0
\(460\) −9520.00 −0.964940
\(461\) 2006.00 0.202665 0.101333 0.994853i \(-0.467689\pi\)
0.101333 + 0.994853i \(0.467689\pi\)
\(462\) 0 0
\(463\) 4896.00 0.491439 0.245720 0.969341i \(-0.420976\pi\)
0.245720 + 0.969341i \(0.420976\pi\)
\(464\) 5162.00 0.516465
\(465\) 0 0
\(466\) 26610.0 2.64525
\(467\) 2660.00 0.263576 0.131788 0.991278i \(-0.457928\pi\)
0.131788 + 0.991278i \(0.457928\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −5200.00 −0.510336
\(471\) 0 0
\(472\) −17100.0 −1.66757
\(473\) −4080.00 −0.396614
\(474\) 0 0
\(475\) 2300.00 0.222171
\(476\) 0 0
\(477\) 0 0
\(478\) 18680.0 1.78745
\(479\) −5600.00 −0.534176 −0.267088 0.963672i \(-0.586062\pi\)
−0.267088 + 0.963672i \(0.586062\pi\)
\(480\) 0 0
\(481\) 4380.00 0.415199
\(482\) 1050.00 0.0992245
\(483\) 0 0
\(484\) −20179.0 −1.89510
\(485\) 3510.00 0.328620
\(486\) 0 0
\(487\) −6424.00 −0.597740 −0.298870 0.954294i \(-0.596610\pi\)
−0.298870 + 0.954294i \(0.596610\pi\)
\(488\) 32310.0 2.99714
\(489\) 0 0
\(490\) 0 0
\(491\) 18900.0 1.73716 0.868579 0.495550i \(-0.165033\pi\)
0.868579 + 0.495550i \(0.165033\pi\)
\(492\) 0 0
\(493\) −7772.00 −0.710007
\(494\) 13800.0 1.25687
\(495\) 0 0
\(496\) 19936.0 1.80474
\(497\) 0 0
\(498\) 0 0
\(499\) −15364.0 −1.37833 −0.689165 0.724604i \(-0.742023\pi\)
−0.689165 + 0.724604i \(0.742023\pi\)
\(500\) 2125.00 0.190066
\(501\) 0 0
\(502\) 21060.0 1.87242
\(503\) 2216.00 0.196435 0.0982173 0.995165i \(-0.468686\pi\)
0.0982173 + 0.995165i \(0.468686\pi\)
\(504\) 0 0
\(505\) 230.000 0.0202671
\(506\) −6720.00 −0.590396
\(507\) 0 0
\(508\) 10336.0 0.902728
\(509\) −3754.00 −0.326902 −0.163451 0.986551i \(-0.552263\pi\)
−0.163451 + 0.986551i \(0.552263\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 24475.0 2.11260
\(513\) 0 0
\(514\) −25650.0 −2.20111
\(515\) −9400.00 −0.804298
\(516\) 0 0
\(517\) −2496.00 −0.212329
\(518\) 0 0
\(519\) 0 0
\(520\) 6750.00 0.569244
\(521\) −4702.00 −0.395390 −0.197695 0.980264i \(-0.563346\pi\)
−0.197695 + 0.980264i \(0.563346\pi\)
\(522\) 0 0
\(523\) 22660.0 1.89456 0.947278 0.320413i \(-0.103822\pi\)
0.947278 + 0.320413i \(0.103822\pi\)
\(524\) −16252.0 −1.35491
\(525\) 0 0
\(526\) 4240.00 0.351469
\(527\) −30016.0 −2.48106
\(528\) 0 0
\(529\) 377.000 0.0309855
\(530\) −18850.0 −1.54489
\(531\) 0 0
\(532\) 0 0
\(533\) −540.000 −0.0438837
\(534\) 0 0
\(535\) −3660.00 −0.295767
\(536\) −18540.0 −1.49404
\(537\) 0 0
\(538\) 6370.00 0.510465
\(539\) 0 0
\(540\) 0 0
\(541\) −8634.00 −0.686145 −0.343073 0.939309i \(-0.611468\pi\)
−0.343073 + 0.939309i \(0.611468\pi\)
\(542\) 4320.00 0.342361
\(543\) 0 0
\(544\) 11390.0 0.897688
\(545\) −1890.00 −0.148548
\(546\) 0 0
\(547\) −19284.0 −1.50736 −0.753679 0.657243i \(-0.771722\pi\)
−0.753679 + 0.657243i \(0.771722\pi\)
\(548\) 6358.00 0.495621
\(549\) 0 0
\(550\) 1500.00 0.116291
\(551\) 5336.00 0.412561
\(552\) 0 0
\(553\) 0 0
\(554\) 42650.0 3.27080
\(555\) 0 0
\(556\) −6732.00 −0.513490
\(557\) 19658.0 1.49540 0.747699 0.664038i \(-0.231159\pi\)
0.747699 + 0.664038i \(0.231159\pi\)
\(558\) 0 0
\(559\) −10200.0 −0.771760
\(560\) 0 0
\(561\) 0 0
\(562\) −26910.0 −2.01980
\(563\) −25612.0 −1.91726 −0.958630 0.284656i \(-0.908121\pi\)
−0.958630 + 0.284656i \(0.908121\pi\)
\(564\) 0 0
\(565\) −7290.00 −0.542819
\(566\) 31180.0 2.31554
\(567\) 0 0
\(568\) −43200.0 −3.19125
\(569\) −7002.00 −0.515886 −0.257943 0.966160i \(-0.583045\pi\)
−0.257943 + 0.966160i \(0.583045\pi\)
\(570\) 0 0
\(571\) −4524.00 −0.331565 −0.165782 0.986162i \(-0.553015\pi\)
−0.165782 + 0.986162i \(0.553015\pi\)
\(572\) 6120.00 0.447360
\(573\) 0 0
\(574\) 0 0
\(575\) −2800.00 −0.203075
\(576\) 0 0
\(577\) 6014.00 0.433910 0.216955 0.976182i \(-0.430388\pi\)
0.216955 + 0.976182i \(0.430388\pi\)
\(578\) −65215.0 −4.69306
\(579\) 0 0
\(580\) 4930.00 0.352943
\(581\) 0 0
\(582\) 0 0
\(583\) −9048.00 −0.642761
\(584\) 47970.0 3.39899
\(585\) 0 0
\(586\) 4090.00 0.288321
\(587\) −11748.0 −0.826051 −0.413025 0.910719i \(-0.635528\pi\)
−0.413025 + 0.910719i \(0.635528\pi\)
\(588\) 0 0
\(589\) 20608.0 1.44166
\(590\) −9500.00 −0.662896
\(591\) 0 0
\(592\) −12994.0 −0.902112
\(593\) −9462.00 −0.655241 −0.327620 0.944809i \(-0.606247\pi\)
−0.327620 + 0.944809i \(0.606247\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 31858.0 2.18952
\(597\) 0 0
\(598\) −16800.0 −1.14883
\(599\) −2320.00 −0.158251 −0.0791257 0.996865i \(-0.525213\pi\)
−0.0791257 + 0.996865i \(0.525213\pi\)
\(600\) 0 0
\(601\) −4650.00 −0.315603 −0.157802 0.987471i \(-0.550441\pi\)
−0.157802 + 0.987471i \(0.550441\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −18632.0 −1.25517
\(605\) −5935.00 −0.398830
\(606\) 0 0
\(607\) 14656.0 0.980014 0.490007 0.871718i \(-0.336994\pi\)
0.490007 + 0.871718i \(0.336994\pi\)
\(608\) −7820.00 −0.521617
\(609\) 0 0
\(610\) 17950.0 1.19143
\(611\) −6240.00 −0.413164
\(612\) 0 0
\(613\) 29166.0 1.92170 0.960851 0.277065i \(-0.0893616\pi\)
0.960851 + 0.277065i \(0.0893616\pi\)
\(614\) −11340.0 −0.745350
\(615\) 0 0
\(616\) 0 0
\(617\) −28554.0 −1.86311 −0.931557 0.363597i \(-0.881549\pi\)
−0.931557 + 0.363597i \(0.881549\pi\)
\(618\) 0 0
\(619\) 3876.00 0.251679 0.125840 0.992051i \(-0.459837\pi\)
0.125840 + 0.992051i \(0.459837\pi\)
\(620\) 19040.0 1.23333
\(621\) 0 0
\(622\) −33240.0 −2.14277
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 49090.0 3.13423
\(627\) 0 0
\(628\) −32606.0 −2.07185
\(629\) 19564.0 1.24017
\(630\) 0 0
\(631\) 2904.00 0.183211 0.0916057 0.995795i \(-0.470800\pi\)
0.0916057 + 0.995795i \(0.470800\pi\)
\(632\) −40320.0 −2.53773
\(633\) 0 0
\(634\) 4670.00 0.292538
\(635\) 3040.00 0.189982
\(636\) 0 0
\(637\) 0 0
\(638\) 3480.00 0.215948
\(639\) 0 0
\(640\) 10575.0 0.653146
\(641\) −9330.00 −0.574903 −0.287452 0.957795i \(-0.592808\pi\)
−0.287452 + 0.957795i \(0.592808\pi\)
\(642\) 0 0
\(643\) 18332.0 1.12433 0.562164 0.827025i \(-0.309969\pi\)
0.562164 + 0.827025i \(0.309969\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 61640.0 3.75417
\(647\) −2088.00 −0.126874 −0.0634372 0.997986i \(-0.520206\pi\)
−0.0634372 + 0.997986i \(0.520206\pi\)
\(648\) 0 0
\(649\) −4560.00 −0.275802
\(650\) 3750.00 0.226288
\(651\) 0 0
\(652\) 39372.0 2.36492
\(653\) −22.0000 −0.00131842 −0.000659209 1.00000i \(-0.500210\pi\)
−0.000659209 1.00000i \(0.500210\pi\)
\(654\) 0 0
\(655\) −4780.00 −0.285145
\(656\) 1602.00 0.0953469
\(657\) 0 0
\(658\) 0 0
\(659\) −16260.0 −0.961153 −0.480576 0.876953i \(-0.659573\pi\)
−0.480576 + 0.876953i \(0.659573\pi\)
\(660\) 0 0
\(661\) 23818.0 1.40153 0.700766 0.713391i \(-0.252842\pi\)
0.700766 + 0.713391i \(0.252842\pi\)
\(662\) −11460.0 −0.672818
\(663\) 0 0
\(664\) −19620.0 −1.14669
\(665\) 0 0
\(666\) 0 0
\(667\) −6496.00 −0.377101
\(668\) −29512.0 −1.70936
\(669\) 0 0
\(670\) −10300.0 −0.593916
\(671\) 8616.00 0.495703
\(672\) 0 0
\(673\) 31106.0 1.78165 0.890823 0.454350i \(-0.150128\pi\)
0.890823 + 0.454350i \(0.150128\pi\)
\(674\) 30310.0 1.73219
\(675\) 0 0
\(676\) −22049.0 −1.25449
\(677\) −1090.00 −0.0618790 −0.0309395 0.999521i \(-0.509850\pi\)
−0.0309395 + 0.999521i \(0.509850\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 30150.0 1.70029
\(681\) 0 0
\(682\) 13440.0 0.754610
\(683\) 12372.0 0.693121 0.346560 0.938028i \(-0.387350\pi\)
0.346560 + 0.938028i \(0.387350\pi\)
\(684\) 0 0
\(685\) 1870.00 0.104305
\(686\) 0 0
\(687\) 0 0
\(688\) 30260.0 1.67682
\(689\) −22620.0 −1.25073
\(690\) 0 0
\(691\) −3252.00 −0.179033 −0.0895166 0.995985i \(-0.528532\pi\)
−0.0895166 + 0.995985i \(0.528532\pi\)
\(692\) −41514.0 −2.28053
\(693\) 0 0
\(694\) 7420.00 0.405849
\(695\) −1980.00 −0.108066
\(696\) 0 0
\(697\) −2412.00 −0.131077
\(698\) 1270.00 0.0688685
\(699\) 0 0
\(700\) 0 0
\(701\) 5434.00 0.292781 0.146390 0.989227i \(-0.453234\pi\)
0.146390 + 0.989227i \(0.453234\pi\)
\(702\) 0 0
\(703\) −13432.0 −0.720622
\(704\) 3444.00 0.184376
\(705\) 0 0
\(706\) 54750.0 2.91862
\(707\) 0 0
\(708\) 0 0
\(709\) −5330.00 −0.282331 −0.141165 0.989986i \(-0.545085\pi\)
−0.141165 + 0.989986i \(0.545085\pi\)
\(710\) −24000.0 −1.26860
\(711\) 0 0
\(712\) 46710.0 2.45861
\(713\) −25088.0 −1.31775
\(714\) 0 0
\(715\) 1800.00 0.0941485
\(716\) 69564.0 3.63091
\(717\) 0 0
\(718\) 56880.0 2.95647
\(719\) −7520.00 −0.390054 −0.195027 0.980798i \(-0.562479\pi\)
−0.195027 + 0.980798i \(0.562479\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −8025.00 −0.413656
\(723\) 0 0
\(724\) −21590.0 −1.10827
\(725\) 1450.00 0.0742781
\(726\) 0 0
\(727\) −19336.0 −0.986427 −0.493214 0.869908i \(-0.664178\pi\)
−0.493214 + 0.869908i \(0.664178\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 26650.0 1.35118
\(731\) −45560.0 −2.30519
\(732\) 0 0
\(733\) 22498.0 1.13367 0.566837 0.823830i \(-0.308167\pi\)
0.566837 + 0.823830i \(0.308167\pi\)
\(734\) −5680.00 −0.285630
\(735\) 0 0
\(736\) 9520.00 0.476782
\(737\) −4944.00 −0.247103
\(738\) 0 0
\(739\) −18292.0 −0.910531 −0.455265 0.890356i \(-0.650456\pi\)
−0.455265 + 0.890356i \(0.650456\pi\)
\(740\) −12410.0 −0.616487
\(741\) 0 0
\(742\) 0 0
\(743\) −17904.0 −0.884030 −0.442015 0.897008i \(-0.645736\pi\)
−0.442015 + 0.897008i \(0.645736\pi\)
\(744\) 0 0
\(745\) 9370.00 0.460792
\(746\) 41210.0 2.02253
\(747\) 0 0
\(748\) 27336.0 1.33623
\(749\) 0 0
\(750\) 0 0
\(751\) 5408.00 0.262771 0.131385 0.991331i \(-0.458057\pi\)
0.131385 + 0.991331i \(0.458057\pi\)
\(752\) 18512.0 0.897690
\(753\) 0 0
\(754\) 8700.00 0.420206
\(755\) −5480.00 −0.264156
\(756\) 0 0
\(757\) 8318.00 0.399370 0.199685 0.979860i \(-0.436008\pi\)
0.199685 + 0.979860i \(0.436008\pi\)
\(758\) −18100.0 −0.867311
\(759\) 0 0
\(760\) −20700.0 −0.987984
\(761\) 6690.00 0.318676 0.159338 0.987224i \(-0.449064\pi\)
0.159338 + 0.987224i \(0.449064\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −83368.0 −3.94784
\(765\) 0 0
\(766\) 42320.0 1.99619
\(767\) −11400.0 −0.536676
\(768\) 0 0
\(769\) −9266.00 −0.434513 −0.217257 0.976115i \(-0.569711\pi\)
−0.217257 + 0.976115i \(0.569711\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 37026.0 1.72616
\(773\) 9678.00 0.450315 0.225157 0.974322i \(-0.427710\pi\)
0.225157 + 0.974322i \(0.427710\pi\)
\(774\) 0 0
\(775\) 5600.00 0.259559
\(776\) −31590.0 −1.46136
\(777\) 0 0
\(778\) 18390.0 0.847447
\(779\) 1656.00 0.0761648
\(780\) 0 0
\(781\) −11520.0 −0.527808
\(782\) −75040.0 −3.43149
\(783\) 0 0
\(784\) 0 0
\(785\) −9590.00 −0.436028
\(786\) 0 0
\(787\) 6860.00 0.310715 0.155357 0.987858i \(-0.450347\pi\)
0.155357 + 0.987858i \(0.450347\pi\)
\(788\) 48450.0 2.19030
\(789\) 0 0
\(790\) −22400.0 −1.00881
\(791\) 0 0
\(792\) 0 0
\(793\) 21540.0 0.964575
\(794\) 62950.0 2.81362
\(795\) 0 0
\(796\) 19448.0 0.865975
\(797\) 10950.0 0.486661 0.243331 0.969943i \(-0.421760\pi\)
0.243331 + 0.969943i \(0.421760\pi\)
\(798\) 0 0
\(799\) −27872.0 −1.23409
\(800\) −2125.00 −0.0939126
\(801\) 0 0
\(802\) 14250.0 0.627413
\(803\) 12792.0 0.562167
\(804\) 0 0
\(805\) 0 0
\(806\) 33600.0 1.46837
\(807\) 0 0
\(808\) −2070.00 −0.0901267
\(809\) −26010.0 −1.13036 −0.565181 0.824967i \(-0.691194\pi\)
−0.565181 + 0.824967i \(0.691194\pi\)
\(810\) 0 0
\(811\) 14628.0 0.633364 0.316682 0.948532i \(-0.397431\pi\)
0.316682 + 0.948532i \(0.397431\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −8760.00 −0.377196
\(815\) 11580.0 0.497705
\(816\) 0 0
\(817\) 31280.0 1.33947
\(818\) 6130.00 0.262018
\(819\) 0 0
\(820\) 1530.00 0.0651584
\(821\) −8718.00 −0.370597 −0.185299 0.982682i \(-0.559325\pi\)
−0.185299 + 0.982682i \(0.559325\pi\)
\(822\) 0 0
\(823\) −7432.00 −0.314779 −0.157390 0.987537i \(-0.550308\pi\)
−0.157390 + 0.987537i \(0.550308\pi\)
\(824\) 84600.0 3.57668
\(825\) 0 0
\(826\) 0 0
\(827\) −17388.0 −0.731125 −0.365562 0.930787i \(-0.619123\pi\)
−0.365562 + 0.930787i \(0.619123\pi\)
\(828\) 0 0
\(829\) −7902.00 −0.331059 −0.165529 0.986205i \(-0.552933\pi\)
−0.165529 + 0.986205i \(0.552933\pi\)
\(830\) −10900.0 −0.455837
\(831\) 0 0
\(832\) 8610.00 0.358772
\(833\) 0 0
\(834\) 0 0
\(835\) −8680.00 −0.359741
\(836\) −18768.0 −0.776441
\(837\) 0 0
\(838\) −3060.00 −0.126141
\(839\) −31848.0 −1.31051 −0.655253 0.755409i \(-0.727438\pi\)
−0.655253 + 0.755409i \(0.727438\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) −25910.0 −1.06047
\(843\) 0 0
\(844\) 7004.00 0.285649
\(845\) −6485.00 −0.264013
\(846\) 0 0
\(847\) 0 0
\(848\) 67106.0 2.71749
\(849\) 0 0
\(850\) 16750.0 0.675906
\(851\) 16352.0 0.658683
\(852\) 0 0
\(853\) −30150.0 −1.21022 −0.605109 0.796142i \(-0.706871\pi\)
−0.605109 + 0.796142i \(0.706871\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 32940.0 1.31526
\(857\) −4350.00 −0.173388 −0.0866938 0.996235i \(-0.527630\pi\)
−0.0866938 + 0.996235i \(0.527630\pi\)
\(858\) 0 0
\(859\) 30676.0 1.21845 0.609227 0.792996i \(-0.291480\pi\)
0.609227 + 0.792996i \(0.291480\pi\)
\(860\) 28900.0 1.14591
\(861\) 0 0
\(862\) −24920.0 −0.984662
\(863\) 23688.0 0.934356 0.467178 0.884163i \(-0.345271\pi\)
0.467178 + 0.884163i \(0.345271\pi\)
\(864\) 0 0
\(865\) −12210.0 −0.479945
\(866\) −8470.00 −0.332358
\(867\) 0 0
\(868\) 0 0
\(869\) −10752.0 −0.419720
\(870\) 0 0
\(871\) −12360.0 −0.480830
\(872\) 17010.0 0.660586
\(873\) 0 0
\(874\) 51520.0 1.99392
\(875\) 0 0
\(876\) 0 0
\(877\) 31910.0 1.22865 0.614324 0.789054i \(-0.289429\pi\)
0.614324 + 0.789054i \(0.289429\pi\)
\(878\) 69320.0 2.66451
\(879\) 0 0
\(880\) −5340.00 −0.204558
\(881\) 50250.0 1.92164 0.960820 0.277172i \(-0.0893971\pi\)
0.960820 + 0.277172i \(0.0893971\pi\)
\(882\) 0 0
\(883\) 5980.00 0.227908 0.113954 0.993486i \(-0.463648\pi\)
0.113954 + 0.993486i \(0.463648\pi\)
\(884\) 68340.0 2.60014
\(885\) 0 0
\(886\) −23220.0 −0.880464
\(887\) −24568.0 −0.930003 −0.465002 0.885310i \(-0.653946\pi\)
−0.465002 + 0.885310i \(0.653946\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 25950.0 0.977355
\(891\) 0 0
\(892\) 27744.0 1.04141
\(893\) 19136.0 0.717091
\(894\) 0 0
\(895\) 20460.0 0.764137
\(896\) 0 0
\(897\) 0 0
\(898\) −24630.0 −0.915271
\(899\) 12992.0 0.481988
\(900\) 0 0
\(901\) −101036. −3.73585
\(902\) 1080.00 0.0398670
\(903\) 0 0
\(904\) 65610.0 2.41389
\(905\) −6350.00 −0.233239
\(906\) 0 0
\(907\) 13252.0 0.485144 0.242572 0.970133i \(-0.422009\pi\)
0.242572 + 0.970133i \(0.422009\pi\)
\(908\) 69428.0 2.53750
\(909\) 0 0
\(910\) 0 0
\(911\) 6744.00 0.245267 0.122634 0.992452i \(-0.460866\pi\)
0.122634 + 0.992452i \(0.460866\pi\)
\(912\) 0 0
\(913\) −5232.00 −0.189654
\(914\) 73470.0 2.65883
\(915\) 0 0
\(916\) 57562.0 2.07631
\(917\) 0 0
\(918\) 0 0
\(919\) −45336.0 −1.62731 −0.813654 0.581349i \(-0.802525\pi\)
−0.813654 + 0.581349i \(0.802525\pi\)
\(920\) 25200.0 0.903065
\(921\) 0 0
\(922\) −10030.0 −0.358265
\(923\) −28800.0 −1.02705
\(924\) 0 0
\(925\) −3650.00 −0.129742
\(926\) −24480.0 −0.868750
\(927\) 0 0
\(928\) −4930.00 −0.174391
\(929\) 30074.0 1.06211 0.531053 0.847339i \(-0.321797\pi\)
0.531053 + 0.847339i \(0.321797\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −90474.0 −3.17980
\(933\) 0 0
\(934\) −13300.0 −0.465941
\(935\) 8040.00 0.281215
\(936\) 0 0
\(937\) −21754.0 −0.758455 −0.379227 0.925303i \(-0.623810\pi\)
−0.379227 + 0.925303i \(0.623810\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 17680.0 0.613466
\(941\) 14550.0 0.504056 0.252028 0.967720i \(-0.418903\pi\)
0.252028 + 0.967720i \(0.418903\pi\)
\(942\) 0 0
\(943\) −2016.00 −0.0696182
\(944\) 33820.0 1.16605
\(945\) 0 0
\(946\) 20400.0 0.701122
\(947\) −46660.0 −1.60110 −0.800552 0.599263i \(-0.795460\pi\)
−0.800552 + 0.599263i \(0.795460\pi\)
\(948\) 0 0
\(949\) 31980.0 1.09390
\(950\) −11500.0 −0.392747
\(951\) 0 0
\(952\) 0 0
\(953\) −20810.0 −0.707347 −0.353674 0.935369i \(-0.615068\pi\)
−0.353674 + 0.935369i \(0.615068\pi\)
\(954\) 0 0
\(955\) −24520.0 −0.830836
\(956\) −63512.0 −2.14867
\(957\) 0 0
\(958\) 28000.0 0.944300
\(959\) 0 0
\(960\) 0 0
\(961\) 20385.0 0.684267
\(962\) −21900.0 −0.733975
\(963\) 0 0
\(964\) −3570.00 −0.119276
\(965\) 10890.0 0.363276
\(966\) 0 0
\(967\) 2776.00 0.0923166 0.0461583 0.998934i \(-0.485302\pi\)
0.0461583 + 0.998934i \(0.485302\pi\)
\(968\) 53415.0 1.77358
\(969\) 0 0
\(970\) −17550.0 −0.580924
\(971\) 27292.0 0.902000 0.451000 0.892524i \(-0.351067\pi\)
0.451000 + 0.892524i \(0.351067\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 32120.0 1.05666
\(975\) 0 0
\(976\) −63902.0 −2.09575
\(977\) 62.0000 0.00203025 0.00101513 0.999999i \(-0.499677\pi\)
0.00101513 + 0.999999i \(0.499677\pi\)
\(978\) 0 0
\(979\) 12456.0 0.406635
\(980\) 0 0
\(981\) 0 0
\(982\) −94500.0 −3.07089
\(983\) 37912.0 1.23012 0.615058 0.788481i \(-0.289132\pi\)
0.615058 + 0.788481i \(0.289132\pi\)
\(984\) 0 0
\(985\) 14250.0 0.460957
\(986\) 38860.0 1.25513
\(987\) 0 0
\(988\) −46920.0 −1.51085
\(989\) −38080.0 −1.22434
\(990\) 0 0
\(991\) 10656.0 0.341573 0.170787 0.985308i \(-0.445369\pi\)
0.170787 + 0.985308i \(0.445369\pi\)
\(992\) −19040.0 −0.609396
\(993\) 0 0
\(994\) 0 0
\(995\) 5720.00 0.182247
\(996\) 0 0
\(997\) 29434.0 0.934989 0.467495 0.883996i \(-0.345157\pi\)
0.467495 + 0.883996i \(0.345157\pi\)
\(998\) 76820.0 2.43657
\(999\) 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 2205.4.a.b.1.1 1
3.2 odd 2 735.4.a.j.1.1 1
7.6 odd 2 315.4.a.a.1.1 1
21.20 even 2 105.4.a.b.1.1 1
35.34 odd 2 1575.4.a.l.1.1 1
84.83 odd 2 1680.4.a.u.1.1 1
105.62 odd 4 525.4.d.a.274.2 2
105.83 odd 4 525.4.d.a.274.1 2
105.104 even 2 525.4.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.b.1.1 1 21.20 even 2
315.4.a.a.1.1 1 7.6 odd 2
525.4.a.a.1.1 1 105.104 even 2
525.4.d.a.274.1 2 105.83 odd 4
525.4.d.a.274.2 2 105.62 odd 4
735.4.a.j.1.1 1 3.2 odd 2
1575.4.a.l.1.1 1 35.34 odd 2
1680.4.a.u.1.1 1 84.83 odd 2
2205.4.a.b.1.1 1 1.1 even 1 trivial