Properties

Label 1200.4.f.f.49.1
Level $1200$
Weight $4$
Character 1200.49
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(49,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, 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("1200.49");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
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 600)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1200.49
Dual form 1200.4.f.f.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} -19.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -19.0000i q^{7} -9.00000 q^{9} -22.0000 q^{11} +1.00000i q^{13} +58.0000i q^{17} -53.0000 q^{19} -57.0000 q^{21} -58.0000i q^{23} +27.0000i q^{27} -22.0000 q^{29} +35.0000 q^{31} +66.0000i q^{33} +270.000i q^{37} +3.00000 q^{39} -468.000 q^{41} +431.000i q^{43} -230.000i q^{47} -18.0000 q^{49} +174.000 q^{51} +159.000i q^{57} +446.000 q^{59} +127.000 q^{61} +171.000i q^{63} -811.000i q^{67} -174.000 q^{69} -36.0000 q^{71} +522.000i q^{73} +418.000i q^{77} +1368.00 q^{79} +81.0000 q^{81} +1138.00i q^{83} +66.0000i q^{87} -144.000 q^{89} +19.0000 q^{91} -105.000i q^{93} +1079.00i q^{97} +198.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} - 44 q^{11} - 106 q^{19} - 114 q^{21} - 44 q^{29} + 70 q^{31} + 6 q^{39} - 936 q^{41} - 36 q^{49} + 348 q^{51} + 892 q^{59} + 254 q^{61} - 348 q^{69} - 72 q^{71} + 2736 q^{79} + 162 q^{81} - 288 q^{89} + 38 q^{91} + 396 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(7\) − 19.0000i − 1.02590i −0.858417 0.512952i \(-0.828552\pi\)
0.858417 0.512952i \(-0.171448\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −22.0000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.0213346i 0.999943 + 0.0106673i \(0.00339558\pi\)
−0.999943 + 0.0106673i \(0.996604\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 58.0000i 0.827474i 0.910396 + 0.413737i \(0.135777\pi\)
−0.910396 + 0.413737i \(0.864223\pi\)
\(18\) 0 0
\(19\) −53.0000 −0.639949 −0.319975 0.947426i \(-0.603674\pi\)
−0.319975 + 0.947426i \(0.603674\pi\)
\(20\) 0 0
\(21\) −57.0000 −0.592306
\(22\) 0 0
\(23\) − 58.0000i − 0.525819i −0.964821 0.262909i \(-0.915318\pi\)
0.964821 0.262909i \(-0.0846821\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) −22.0000 −0.140872 −0.0704362 0.997516i \(-0.522439\pi\)
−0.0704362 + 0.997516i \(0.522439\pi\)
\(30\) 0 0
\(31\) 35.0000 0.202780 0.101390 0.994847i \(-0.467671\pi\)
0.101390 + 0.994847i \(0.467671\pi\)
\(32\) 0 0
\(33\) 66.0000i 0.348155i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 270.000i 1.19967i 0.800124 + 0.599834i \(0.204767\pi\)
−0.800124 + 0.599834i \(0.795233\pi\)
\(38\) 0 0
\(39\) 3.00000 0.0123176
\(40\) 0 0
\(41\) −468.000 −1.78267 −0.891333 0.453349i \(-0.850229\pi\)
−0.891333 + 0.453349i \(0.850229\pi\)
\(42\) 0 0
\(43\) 431.000i 1.52853i 0.644901 + 0.764266i \(0.276899\pi\)
−0.644901 + 0.764266i \(0.723101\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 230.000i − 0.713807i −0.934141 0.356904i \(-0.883832\pi\)
0.934141 0.356904i \(-0.116168\pi\)
\(48\) 0 0
\(49\) −18.0000 −0.0524781
\(50\) 0 0
\(51\) 174.000 0.477743
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 159.000i 0.369475i
\(58\) 0 0
\(59\) 446.000 0.984140 0.492070 0.870556i \(-0.336240\pi\)
0.492070 + 0.870556i \(0.336240\pi\)
\(60\) 0 0
\(61\) 127.000 0.266569 0.133284 0.991078i \(-0.457448\pi\)
0.133284 + 0.991078i \(0.457448\pi\)
\(62\) 0 0
\(63\) 171.000i 0.341968i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 811.000i − 1.47880i −0.673268 0.739399i \(-0.735110\pi\)
0.673268 0.739399i \(-0.264890\pi\)
\(68\) 0 0
\(69\) −174.000 −0.303582
\(70\) 0 0
\(71\) −36.0000 −0.0601748 −0.0300874 0.999547i \(-0.509579\pi\)
−0.0300874 + 0.999547i \(0.509579\pi\)
\(72\) 0 0
\(73\) 522.000i 0.836924i 0.908234 + 0.418462i \(0.137431\pi\)
−0.908234 + 0.418462i \(0.862569\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 418.000i 0.618643i
\(78\) 0 0
\(79\) 1368.00 1.94825 0.974127 0.226002i \(-0.0725657\pi\)
0.974127 + 0.226002i \(0.0725657\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 1138.00i 1.50496i 0.658615 + 0.752480i \(0.271143\pi\)
−0.658615 + 0.752480i \(0.728857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 66.0000i 0.0813327i
\(88\) 0 0
\(89\) −144.000 −0.171505 −0.0857526 0.996316i \(-0.527329\pi\)
−0.0857526 + 0.996316i \(0.527329\pi\)
\(90\) 0 0
\(91\) 19.0000 0.0218873
\(92\) 0 0
\(93\) − 105.000i − 0.117075i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1079.00i 1.12944i 0.825282 + 0.564721i \(0.191016\pi\)
−0.825282 + 0.564721i \(0.808984\pi\)
\(98\) 0 0
\(99\) 198.000 0.201008
\(100\) 0 0
\(101\) −1440.00 −1.41867 −0.709333 0.704873i \(-0.751004\pi\)
−0.709333 + 0.704873i \(0.751004\pi\)
\(102\) 0 0
\(103\) − 124.000i − 0.118622i −0.998240 0.0593111i \(-0.981110\pi\)
0.998240 0.0593111i \(-0.0188904\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 432.000i − 0.390309i −0.980773 0.195154i \(-0.937479\pi\)
0.980773 0.195154i \(-0.0625208\pi\)
\(108\) 0 0
\(109\) −701.000 −0.615997 −0.307998 0.951387i \(-0.599659\pi\)
−0.307998 + 0.951387i \(0.599659\pi\)
\(110\) 0 0
\(111\) 810.000 0.692629
\(112\) 0 0
\(113\) − 1044.00i − 0.869126i −0.900641 0.434563i \(-0.856903\pi\)
0.900641 0.434563i \(-0.143097\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 9.00000i − 0.00711154i
\(118\) 0 0
\(119\) 1102.00 0.848909
\(120\) 0 0
\(121\) −847.000 −0.636364
\(122\) 0 0
\(123\) 1404.00i 1.02922i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 504.000i 0.352148i 0.984377 + 0.176074i \(0.0563398\pi\)
−0.984377 + 0.176074i \(0.943660\pi\)
\(128\) 0 0
\(129\) 1293.00 0.882498
\(130\) 0 0
\(131\) −72.0000 −0.0480204 −0.0240102 0.999712i \(-0.507643\pi\)
−0.0240102 + 0.999712i \(0.507643\pi\)
\(132\) 0 0
\(133\) 1007.00i 0.656526i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1850.00i 1.15369i 0.816852 + 0.576847i \(0.195717\pi\)
−0.816852 + 0.576847i \(0.804283\pi\)
\(138\) 0 0
\(139\) 1836.00 1.12034 0.560171 0.828377i \(-0.310735\pi\)
0.560171 + 0.828377i \(0.310735\pi\)
\(140\) 0 0
\(141\) −690.000 −0.412117
\(142\) 0 0
\(143\) − 22.0000i − 0.0128653i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 54.0000i 0.0302983i
\(148\) 0 0
\(149\) 2398.00 1.31847 0.659234 0.751938i \(-0.270881\pi\)
0.659234 + 0.751938i \(0.270881\pi\)
\(150\) 0 0
\(151\) −1871.00 −1.00834 −0.504172 0.863604i \(-0.668202\pi\)
−0.504172 + 0.863604i \(0.668202\pi\)
\(152\) 0 0
\(153\) − 522.000i − 0.275825i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 3293.00i 1.67395i 0.547242 + 0.836975i \(0.315678\pi\)
−0.547242 + 0.836975i \(0.684322\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1102.00 −0.539440
\(162\) 0 0
\(163\) − 883.000i − 0.424306i −0.977236 0.212153i \(-0.931952\pi\)
0.977236 0.212153i \(-0.0680475\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 4104.00i 1.90166i 0.309715 + 0.950830i \(0.399766\pi\)
−0.309715 + 0.950830i \(0.600234\pi\)
\(168\) 0 0
\(169\) 2196.00 0.999545
\(170\) 0 0
\(171\) 477.000 0.213316
\(172\) 0 0
\(173\) 1706.00i 0.749739i 0.927078 + 0.374869i \(0.122312\pi\)
−0.927078 + 0.374869i \(0.877688\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1338.00i − 0.568193i
\(178\) 0 0
\(179\) 662.000 0.276426 0.138213 0.990403i \(-0.455864\pi\)
0.138213 + 0.990403i \(0.455864\pi\)
\(180\) 0 0
\(181\) 4121.00 1.69233 0.846164 0.532922i \(-0.178906\pi\)
0.846164 + 0.532922i \(0.178906\pi\)
\(182\) 0 0
\(183\) − 381.000i − 0.153903i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1276.00i − 0.498986i
\(188\) 0 0
\(189\) 513.000 0.197435
\(190\) 0 0
\(191\) 958.000 0.362924 0.181462 0.983398i \(-0.441917\pi\)
0.181462 + 0.983398i \(0.441917\pi\)
\(192\) 0 0
\(193\) 3187.00i 1.18863i 0.804233 + 0.594314i \(0.202576\pi\)
−0.804233 + 0.594314i \(0.797424\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2282.00i 0.825308i 0.910888 + 0.412654i \(0.135398\pi\)
−0.910888 + 0.412654i \(0.864602\pi\)
\(198\) 0 0
\(199\) 1043.00 0.371539 0.185770 0.982593i \(-0.440522\pi\)
0.185770 + 0.982593i \(0.440522\pi\)
\(200\) 0 0
\(201\) −2433.00 −0.853784
\(202\) 0 0
\(203\) 418.000i 0.144521i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 522.000i 0.175273i
\(208\) 0 0
\(209\) 1166.00 0.385904
\(210\) 0 0
\(211\) −4139.00 −1.35043 −0.675214 0.737621i \(-0.735949\pi\)
−0.675214 + 0.737621i \(0.735949\pi\)
\(212\) 0 0
\(213\) 108.000i 0.0347420i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 665.000i − 0.208033i
\(218\) 0 0
\(219\) 1566.00 0.483199
\(220\) 0 0
\(221\) −58.0000 −0.0176539
\(222\) 0 0
\(223\) − 413.000i − 0.124020i −0.998076 0.0620101i \(-0.980249\pi\)
0.998076 0.0620101i \(-0.0197511\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5652.00i 1.65258i 0.563242 + 0.826292i \(0.309554\pi\)
−0.563242 + 0.826292i \(0.690446\pi\)
\(228\) 0 0
\(229\) 4391.00 1.26710 0.633549 0.773703i \(-0.281597\pi\)
0.633549 + 0.773703i \(0.281597\pi\)
\(230\) 0 0
\(231\) 1254.00 0.357174
\(232\) 0 0
\(233\) − 2052.00i − 0.576957i −0.957486 0.288479i \(-0.906851\pi\)
0.957486 0.288479i \(-0.0931494\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 4104.00i − 1.12482i
\(238\) 0 0
\(239\) −4320.00 −1.16919 −0.584597 0.811324i \(-0.698748\pi\)
−0.584597 + 0.811324i \(0.698748\pi\)
\(240\) 0 0
\(241\) 4265.00 1.13997 0.569985 0.821655i \(-0.306949\pi\)
0.569985 + 0.821655i \(0.306949\pi\)
\(242\) 0 0
\(243\) − 243.000i − 0.0641500i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 53.0000i − 0.0136531i
\(248\) 0 0
\(249\) 3414.00 0.868889
\(250\) 0 0
\(251\) 2412.00 0.606550 0.303275 0.952903i \(-0.401920\pi\)
0.303275 + 0.952903i \(0.401920\pi\)
\(252\) 0 0
\(253\) 1276.00i 0.317081i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 6948.00i − 1.68640i −0.537601 0.843199i \(-0.680669\pi\)
0.537601 0.843199i \(-0.319331\pi\)
\(258\) 0 0
\(259\) 5130.00 1.23074
\(260\) 0 0
\(261\) 198.000 0.0469574
\(262\) 0 0
\(263\) 3564.00i 0.835611i 0.908537 + 0.417805i \(0.137201\pi\)
−0.908537 + 0.417805i \(0.862799\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 432.000i 0.0990186i
\(268\) 0 0
\(269\) 1534.00 0.347694 0.173847 0.984773i \(-0.444380\pi\)
0.173847 + 0.984773i \(0.444380\pi\)
\(270\) 0 0
\(271\) −7704.00 −1.72688 −0.863440 0.504451i \(-0.831695\pi\)
−0.863440 + 0.504451i \(0.831695\pi\)
\(272\) 0 0
\(273\) − 57.0000i − 0.0126366i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 4877.00i − 1.05787i −0.848662 0.528936i \(-0.822591\pi\)
0.848662 0.528936i \(-0.177409\pi\)
\(278\) 0 0
\(279\) −315.000 −0.0675934
\(280\) 0 0
\(281\) −3758.00 −0.797806 −0.398903 0.916993i \(-0.630609\pi\)
−0.398903 + 0.916993i \(0.630609\pi\)
\(282\) 0 0
\(283\) − 935.000i − 0.196396i −0.995167 0.0981978i \(-0.968692\pi\)
0.995167 0.0981978i \(-0.0313078\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8892.00i 1.82884i
\(288\) 0 0
\(289\) 1549.00 0.315286
\(290\) 0 0
\(291\) 3237.00 0.652084
\(292\) 0 0
\(293\) 6214.00i 1.23900i 0.784998 + 0.619498i \(0.212664\pi\)
−0.784998 + 0.619498i \(0.787336\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 594.000i − 0.116052i
\(298\) 0 0
\(299\) 58.0000 0.0112181
\(300\) 0 0
\(301\) 8189.00 1.56813
\(302\) 0 0
\(303\) 4320.00i 0.819068i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 3905.00i − 0.725961i −0.931797 0.362981i \(-0.881759\pi\)
0.931797 0.362981i \(-0.118241\pi\)
\(308\) 0 0
\(309\) −372.000 −0.0684865
\(310\) 0 0
\(311\) −9662.00 −1.76168 −0.880839 0.473416i \(-0.843021\pi\)
−0.880839 + 0.473416i \(0.843021\pi\)
\(312\) 0 0
\(313\) 5147.00i 0.929475i 0.885449 + 0.464737i \(0.153851\pi\)
−0.885449 + 0.464737i \(0.846149\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9216.00i 1.63288i 0.577432 + 0.816439i \(0.304055\pi\)
−0.577432 + 0.816439i \(0.695945\pi\)
\(318\) 0 0
\(319\) 484.000 0.0849492
\(320\) 0 0
\(321\) −1296.00 −0.225345
\(322\) 0 0
\(323\) − 3074.00i − 0.529542i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 2103.00i 0.355646i
\(328\) 0 0
\(329\) −4370.00 −0.732298
\(330\) 0 0
\(331\) −2196.00 −0.364662 −0.182331 0.983237i \(-0.558364\pi\)
−0.182331 + 0.983237i \(0.558364\pi\)
\(332\) 0 0
\(333\) − 2430.00i − 0.399889i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 2521.00i − 0.407500i −0.979023 0.203750i \(-0.934687\pi\)
0.979023 0.203750i \(-0.0653130\pi\)
\(338\) 0 0
\(339\) −3132.00 −0.501790
\(340\) 0 0
\(341\) −770.000 −0.122281
\(342\) 0 0
\(343\) − 6175.00i − 0.972066i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7034.00i 1.08820i 0.839021 + 0.544099i \(0.183129\pi\)
−0.839021 + 0.544099i \(0.816871\pi\)
\(348\) 0 0
\(349\) −7362.00 −1.12917 −0.564583 0.825376i \(-0.690963\pi\)
−0.564583 + 0.825376i \(0.690963\pi\)
\(350\) 0 0
\(351\) −27.0000 −0.00410585
\(352\) 0 0
\(353\) − 9382.00i − 1.41460i −0.706914 0.707300i \(-0.749913\pi\)
0.706914 0.707300i \(-0.250087\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 3306.00i − 0.490118i
\(358\) 0 0
\(359\) −10116.0 −1.48719 −0.743596 0.668629i \(-0.766881\pi\)
−0.743596 + 0.668629i \(0.766881\pi\)
\(360\) 0 0
\(361\) −4050.00 −0.590465
\(362\) 0 0
\(363\) 2541.00i 0.367405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 3277.00i − 0.466098i −0.972465 0.233049i \(-0.925130\pi\)
0.972465 0.233049i \(-0.0748703\pi\)
\(368\) 0 0
\(369\) 4212.00 0.594222
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10891.0i 1.51184i 0.654667 + 0.755918i \(0.272809\pi\)
−0.654667 + 0.755918i \(0.727191\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 22.0000i − 0.00300546i
\(378\) 0 0
\(379\) −2591.00 −0.351163 −0.175581 0.984465i \(-0.556181\pi\)
−0.175581 + 0.984465i \(0.556181\pi\)
\(380\) 0 0
\(381\) 1512.00 0.203313
\(382\) 0 0
\(383\) 612.000i 0.0816494i 0.999166 + 0.0408247i \(0.0129985\pi\)
−0.999166 + 0.0408247i \(0.987001\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 3879.00i − 0.509511i
\(388\) 0 0
\(389\) −3708.00 −0.483298 −0.241649 0.970364i \(-0.577688\pi\)
−0.241649 + 0.970364i \(0.577688\pi\)
\(390\) 0 0
\(391\) 3364.00 0.435102
\(392\) 0 0
\(393\) 216.000i 0.0277246i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3833.00i 0.484566i 0.970206 + 0.242283i \(0.0778963\pi\)
−0.970206 + 0.242283i \(0.922104\pi\)
\(398\) 0 0
\(399\) 3021.00 0.379046
\(400\) 0 0
\(401\) −9288.00 −1.15666 −0.578330 0.815803i \(-0.696295\pi\)
−0.578330 + 0.815803i \(0.696295\pi\)
\(402\) 0 0
\(403\) 35.0000i 0.00432624i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 5940.00i − 0.723427i
\(408\) 0 0
\(409\) 755.000 0.0912771 0.0456386 0.998958i \(-0.485468\pi\)
0.0456386 + 0.998958i \(0.485468\pi\)
\(410\) 0 0
\(411\) 5550.00 0.666086
\(412\) 0 0
\(413\) − 8474.00i − 1.00963i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 5508.00i − 0.646830i
\(418\) 0 0
\(419\) −9576.00 −1.11651 −0.558256 0.829669i \(-0.688529\pi\)
−0.558256 + 0.829669i \(0.688529\pi\)
\(420\) 0 0
\(421\) 9414.00 1.08981 0.544905 0.838498i \(-0.316566\pi\)
0.544905 + 0.838498i \(0.316566\pi\)
\(422\) 0 0
\(423\) 2070.00i 0.237936i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 2413.00i − 0.273474i
\(428\) 0 0
\(429\) −66.0000 −0.00742776
\(430\) 0 0
\(431\) 2254.00 0.251906 0.125953 0.992036i \(-0.459801\pi\)
0.125953 + 0.992036i \(0.459801\pi\)
\(432\) 0 0
\(433\) − 6301.00i − 0.699323i −0.936876 0.349661i \(-0.886297\pi\)
0.936876 0.349661i \(-0.113703\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3074.00i 0.336497i
\(438\) 0 0
\(439\) −3779.00 −0.410847 −0.205423 0.978673i \(-0.565857\pi\)
−0.205423 + 0.978673i \(0.565857\pi\)
\(440\) 0 0
\(441\) 162.000 0.0174927
\(442\) 0 0
\(443\) − 7632.00i − 0.818527i −0.912416 0.409263i \(-0.865786\pi\)
0.912416 0.409263i \(-0.134214\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 7194.00i − 0.761218i
\(448\) 0 0
\(449\) −2988.00 −0.314059 −0.157029 0.987594i \(-0.550192\pi\)
−0.157029 + 0.987594i \(0.550192\pi\)
\(450\) 0 0
\(451\) 10296.0 1.07499
\(452\) 0 0
\(453\) 5613.00i 0.582167i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1370.00i 0.140232i 0.997539 + 0.0701159i \(0.0223369\pi\)
−0.997539 + 0.0701159i \(0.977663\pi\)
\(458\) 0 0
\(459\) −1566.00 −0.159248
\(460\) 0 0
\(461\) −7992.00 −0.807429 −0.403714 0.914885i \(-0.632281\pi\)
−0.403714 + 0.914885i \(0.632281\pi\)
\(462\) 0 0
\(463\) 3096.00i 0.310763i 0.987855 + 0.155382i \(0.0496607\pi\)
−0.987855 + 0.155382i \(0.950339\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 8338.00i − 0.826203i −0.910685 0.413101i \(-0.864446\pi\)
0.910685 0.413101i \(-0.135554\pi\)
\(468\) 0 0
\(469\) −15409.0 −1.51710
\(470\) 0 0
\(471\) 9879.00 0.966455
\(472\) 0 0
\(473\) − 9482.00i − 0.921740i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4918.00 0.469121 0.234561 0.972101i \(-0.424635\pi\)
0.234561 + 0.972101i \(0.424635\pi\)
\(480\) 0 0
\(481\) −270.000 −0.0255945
\(482\) 0 0
\(483\) 3306.00i 0.311446i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 19637.0i − 1.82718i −0.406635 0.913591i \(-0.633298\pi\)
0.406635 0.913591i \(-0.366702\pi\)
\(488\) 0 0
\(489\) −2649.00 −0.244973
\(490\) 0 0
\(491\) −3096.00 −0.284563 −0.142282 0.989826i \(-0.545444\pi\)
−0.142282 + 0.989826i \(0.545444\pi\)
\(492\) 0 0
\(493\) − 1276.00i − 0.116568i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 684.000i 0.0617336i
\(498\) 0 0
\(499\) 6875.00 0.616768 0.308384 0.951262i \(-0.400212\pi\)
0.308384 + 0.951262i \(0.400212\pi\)
\(500\) 0 0
\(501\) 12312.0 1.09792
\(502\) 0 0
\(503\) − 11268.0i − 0.998838i −0.866361 0.499419i \(-0.833547\pi\)
0.866361 0.499419i \(-0.166453\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 6588.00i − 0.577087i
\(508\) 0 0
\(509\) −16078.0 −1.40009 −0.700044 0.714100i \(-0.746836\pi\)
−0.700044 + 0.714100i \(0.746836\pi\)
\(510\) 0 0
\(511\) 9918.00 0.858604
\(512\) 0 0
\(513\) − 1431.00i − 0.123158i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 5060.00i 0.430442i
\(518\) 0 0
\(519\) 5118.00 0.432862
\(520\) 0 0
\(521\) −17366.0 −1.46030 −0.730152 0.683285i \(-0.760551\pi\)
−0.730152 + 0.683285i \(0.760551\pi\)
\(522\) 0 0
\(523\) 4913.00i 0.410766i 0.978682 + 0.205383i \(0.0658440\pi\)
−0.978682 + 0.205383i \(0.934156\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2030.00i 0.167795i
\(528\) 0 0
\(529\) 8803.00 0.723514
\(530\) 0 0
\(531\) −4014.00 −0.328047
\(532\) 0 0
\(533\) − 468.000i − 0.0380325i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 1986.00i − 0.159594i
\(538\) 0 0
\(539\) 396.000 0.0316455
\(540\) 0 0
\(541\) 17605.0 1.39907 0.699536 0.714597i \(-0.253390\pi\)
0.699536 + 0.714597i \(0.253390\pi\)
\(542\) 0 0
\(543\) − 12363.0i − 0.977067i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 14560.0i − 1.13810i −0.822303 0.569050i \(-0.807311\pi\)
0.822303 0.569050i \(-0.192689\pi\)
\(548\) 0 0
\(549\) −1143.00 −0.0888562
\(550\) 0 0
\(551\) 1166.00 0.0901511
\(552\) 0 0
\(553\) − 25992.0i − 1.99872i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20614.0i 1.56812i 0.620685 + 0.784060i \(0.286855\pi\)
−0.620685 + 0.784060i \(0.713145\pi\)
\(558\) 0 0
\(559\) −431.000 −0.0326107
\(560\) 0 0
\(561\) −3828.00 −0.288090
\(562\) 0 0
\(563\) − 3442.00i − 0.257661i −0.991667 0.128830i \(-0.958878\pi\)
0.991667 0.128830i \(-0.0411223\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 1539.00i − 0.113989i
\(568\) 0 0
\(569\) −22082.0 −1.62693 −0.813467 0.581611i \(-0.802423\pi\)
−0.813467 + 0.581611i \(0.802423\pi\)
\(570\) 0 0
\(571\) −451.000 −0.0330539 −0.0165269 0.999863i \(-0.505261\pi\)
−0.0165269 + 0.999863i \(0.505261\pi\)
\(572\) 0 0
\(573\) − 2874.00i − 0.209534i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 20987.0i − 1.51421i −0.653292 0.757106i \(-0.726613\pi\)
0.653292 0.757106i \(-0.273387\pi\)
\(578\) 0 0
\(579\) 9561.00 0.686255
\(580\) 0 0
\(581\) 21622.0 1.54394
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3888.00i 0.273381i 0.990614 + 0.136691i \(0.0436467\pi\)
−0.990614 + 0.136691i \(0.956353\pi\)
\(588\) 0 0
\(589\) −1855.00 −0.129769
\(590\) 0 0
\(591\) 6846.00 0.476492
\(592\) 0 0
\(593\) − 11268.0i − 0.780306i −0.920750 0.390153i \(-0.872422\pi\)
0.920750 0.390153i \(-0.127578\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 3129.00i − 0.214508i
\(598\) 0 0
\(599\) −3996.00 −0.272575 −0.136287 0.990669i \(-0.543517\pi\)
−0.136287 + 0.990669i \(0.543517\pi\)
\(600\) 0 0
\(601\) −24965.0 −1.69442 −0.847208 0.531262i \(-0.821718\pi\)
−0.847208 + 0.531262i \(0.821718\pi\)
\(602\) 0 0
\(603\) 7299.00i 0.492932i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 4176.00i 0.279240i 0.990205 + 0.139620i \(0.0445881\pi\)
−0.990205 + 0.139620i \(0.955412\pi\)
\(608\) 0 0
\(609\) 1254.00 0.0834395
\(610\) 0 0
\(611\) 230.000 0.0152288
\(612\) 0 0
\(613\) − 9558.00i − 0.629762i −0.949131 0.314881i \(-0.898035\pi\)
0.949131 0.314881i \(-0.101965\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9000.00i 0.587239i 0.955922 + 0.293619i \(0.0948598\pi\)
−0.955922 + 0.293619i \(0.905140\pi\)
\(618\) 0 0
\(619\) −15625.0 −1.01457 −0.507287 0.861777i \(-0.669352\pi\)
−0.507287 + 0.861777i \(0.669352\pi\)
\(620\) 0 0
\(621\) 1566.00 0.101194
\(622\) 0 0
\(623\) 2736.00i 0.175948i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3498.00i − 0.222802i
\(628\) 0 0
\(629\) −15660.0 −0.992695
\(630\) 0 0
\(631\) 31175.0 1.96681 0.983405 0.181424i \(-0.0580705\pi\)
0.983405 + 0.181424i \(0.0580705\pi\)
\(632\) 0 0
\(633\) 12417.0i 0.779671i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 18.0000i − 0.00111960i
\(638\) 0 0
\(639\) 324.000 0.0200583
\(640\) 0 0
\(641\) −6732.00 −0.414817 −0.207409 0.978254i \(-0.566503\pi\)
−0.207409 + 0.978254i \(0.566503\pi\)
\(642\) 0 0
\(643\) 6228.00i 0.381973i 0.981593 + 0.190986i \(0.0611686\pi\)
−0.981593 + 0.190986i \(0.938831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 396.000i − 0.0240624i −0.999928 0.0120312i \(-0.996170\pi\)
0.999928 0.0120312i \(-0.00382974\pi\)
\(648\) 0 0
\(649\) −9812.00 −0.593459
\(650\) 0 0
\(651\) −1995.00 −0.120108
\(652\) 0 0
\(653\) − 4702.00i − 0.281782i −0.990025 0.140891i \(-0.955003\pi\)
0.990025 0.140891i \(-0.0449967\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 4698.00i − 0.278975i
\(658\) 0 0
\(659\) −22140.0 −1.30873 −0.654364 0.756180i \(-0.727064\pi\)
−0.654364 + 0.756180i \(0.727064\pi\)
\(660\) 0 0
\(661\) −13518.0 −0.795445 −0.397723 0.917506i \(-0.630199\pi\)
−0.397723 + 0.917506i \(0.630199\pi\)
\(662\) 0 0
\(663\) 174.000i 0.0101925i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1276.00i 0.0740733i
\(668\) 0 0
\(669\) −1239.00 −0.0716032
\(670\) 0 0
\(671\) −2794.00 −0.160747
\(672\) 0 0
\(673\) − 11250.0i − 0.644362i −0.946678 0.322181i \(-0.895584\pi\)
0.946678 0.322181i \(-0.104416\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15170.0i 0.861197i 0.902544 + 0.430599i \(0.141698\pi\)
−0.902544 + 0.430599i \(0.858302\pi\)
\(678\) 0 0
\(679\) 20501.0 1.15870
\(680\) 0 0
\(681\) 16956.0 0.954119
\(682\) 0 0
\(683\) − 22680.0i − 1.27061i −0.772262 0.635305i \(-0.780875\pi\)
0.772262 0.635305i \(-0.219125\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 13173.0i − 0.731559i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −128.000 −0.00704682 −0.00352341 0.999994i \(-0.501122\pi\)
−0.00352341 + 0.999994i \(0.501122\pi\)
\(692\) 0 0
\(693\) − 3762.00i − 0.206214i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 27144.0i − 1.47511i
\(698\) 0 0
\(699\) −6156.00 −0.333106
\(700\) 0 0
\(701\) 25682.0 1.38373 0.691866 0.722026i \(-0.256789\pi\)
0.691866 + 0.722026i \(0.256789\pi\)
\(702\) 0 0
\(703\) − 14310.0i − 0.767727i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 27360.0i 1.45542i
\(708\) 0 0
\(709\) 4951.00 0.262255 0.131127 0.991366i \(-0.458140\pi\)
0.131127 + 0.991366i \(0.458140\pi\)
\(710\) 0 0
\(711\) −12312.0 −0.649418
\(712\) 0 0
\(713\) − 2030.00i − 0.106626i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 12960.0i 0.675035i
\(718\) 0 0
\(719\) −22190.0 −1.15097 −0.575485 0.817812i \(-0.695187\pi\)
−0.575485 + 0.817812i \(0.695187\pi\)
\(720\) 0 0
\(721\) −2356.00 −0.121695
\(722\) 0 0
\(723\) − 12795.0i − 0.658162i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 7685.00i 0.392051i 0.980599 + 0.196025i \(0.0628035\pi\)
−0.980599 + 0.196025i \(0.937197\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −24998.0 −1.26482
\(732\) 0 0
\(733\) 29574.0i 1.49023i 0.666934 + 0.745116i \(0.267606\pi\)
−0.666934 + 0.745116i \(0.732394\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 17842.0i 0.891748i
\(738\) 0 0
\(739\) −32580.0 −1.62175 −0.810876 0.585218i \(-0.801009\pi\)
−0.810876 + 0.585218i \(0.801009\pi\)
\(740\) 0 0
\(741\) −159.000 −0.00788261
\(742\) 0 0
\(743\) 3060.00i 0.151091i 0.997142 + 0.0755454i \(0.0240698\pi\)
−0.997142 + 0.0755454i \(0.975930\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 10242.0i − 0.501654i
\(748\) 0 0
\(749\) −8208.00 −0.400419
\(750\) 0 0
\(751\) 7992.00 0.388325 0.194163 0.980969i \(-0.437801\pi\)
0.194163 + 0.980969i \(0.437801\pi\)
\(752\) 0 0
\(753\) − 7236.00i − 0.350192i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 22841.0i − 1.09666i −0.836263 0.548329i \(-0.815264\pi\)
0.836263 0.548329i \(-0.184736\pi\)
\(758\) 0 0
\(759\) 3828.00 0.183067
\(760\) 0 0
\(761\) −17172.0 −0.817982 −0.408991 0.912538i \(-0.634119\pi\)
−0.408991 + 0.912538i \(0.634119\pi\)
\(762\) 0 0
\(763\) 13319.0i 0.631953i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 446.000i 0.0209963i
\(768\) 0 0
\(769\) 30869.0 1.44755 0.723774 0.690037i \(-0.242406\pi\)
0.723774 + 0.690037i \(0.242406\pi\)
\(770\) 0 0
\(771\) −20844.0 −0.973642
\(772\) 0 0
\(773\) 34884.0i 1.62314i 0.584252 + 0.811572i \(0.301388\pi\)
−0.584252 + 0.811572i \(0.698612\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 15390.0i − 0.710570i
\(778\) 0 0
\(779\) 24804.0 1.14082
\(780\) 0 0
\(781\) 792.000 0.0362868
\(782\) 0 0
\(783\) − 594.000i − 0.0271109i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 5039.00i 0.228235i 0.993467 + 0.114118i \(0.0364040\pi\)
−0.993467 + 0.114118i \(0.963596\pi\)
\(788\) 0 0
\(789\) 10692.0 0.482440
\(790\) 0 0
\(791\) −19836.0 −0.891640
\(792\) 0 0
\(793\) 127.000i 0.00568714i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 540.000i − 0.0239997i −0.999928 0.0119999i \(-0.996180\pi\)
0.999928 0.0119999i \(-0.00381977\pi\)
\(798\) 0 0
\(799\) 13340.0 0.590657
\(800\) 0 0
\(801\) 1296.00 0.0571684
\(802\) 0 0
\(803\) − 11484.0i − 0.504684i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 4602.00i − 0.200741i
\(808\) 0 0
\(809\) 41328.0 1.79606 0.898032 0.439931i \(-0.144997\pi\)
0.898032 + 0.439931i \(0.144997\pi\)
\(810\) 0 0
\(811\) 12853.0 0.556510 0.278255 0.960507i \(-0.410244\pi\)
0.278255 + 0.960507i \(0.410244\pi\)
\(812\) 0 0
\(813\) 23112.0i 0.997015i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 22843.0i − 0.978183i
\(818\) 0 0
\(819\) −171.000 −0.00729576
\(820\) 0 0
\(821\) 29470.0 1.25275 0.626376 0.779521i \(-0.284537\pi\)
0.626376 + 0.779521i \(0.284537\pi\)
\(822\) 0 0
\(823\) 24407.0i 1.03375i 0.856062 + 0.516874i \(0.172904\pi\)
−0.856062 + 0.516874i \(0.827096\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15048.0i 0.632733i 0.948637 + 0.316367i \(0.102463\pi\)
−0.948637 + 0.316367i \(0.897537\pi\)
\(828\) 0 0
\(829\) 28406.0 1.19009 0.595043 0.803694i \(-0.297135\pi\)
0.595043 + 0.803694i \(0.297135\pi\)
\(830\) 0 0
\(831\) −14631.0 −0.610763
\(832\) 0 0
\(833\) − 1044.00i − 0.0434243i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 945.000i 0.0390251i
\(838\) 0 0
\(839\) −26914.0 −1.10748 −0.553739 0.832690i \(-0.686800\pi\)
−0.553739 + 0.832690i \(0.686800\pi\)
\(840\) 0 0
\(841\) −23905.0 −0.980155
\(842\) 0 0
\(843\) 11274.0i 0.460614i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 16093.0i 0.652848i
\(848\) 0 0
\(849\) −2805.00 −0.113389
\(850\) 0 0
\(851\) 15660.0 0.630808
\(852\) 0 0
\(853\) 21275.0i 0.853977i 0.904257 + 0.426988i \(0.140425\pi\)
−0.904257 + 0.426988i \(0.859575\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 39132.0i − 1.55977i −0.625922 0.779885i \(-0.715277\pi\)
0.625922 0.779885i \(-0.284723\pi\)
\(858\) 0 0
\(859\) −448.000 −0.0177946 −0.00889730 0.999960i \(-0.502832\pi\)
−0.00889730 + 0.999960i \(0.502832\pi\)
\(860\) 0 0
\(861\) 26676.0 1.05588
\(862\) 0 0
\(863\) 26856.0i 1.05932i 0.848212 + 0.529658i \(0.177680\pi\)
−0.848212 + 0.529658i \(0.822320\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 4647.00i − 0.182030i
\(868\) 0 0
\(869\) −30096.0 −1.17484
\(870\) 0 0
\(871\) 811.000 0.0315496
\(872\) 0 0
\(873\) − 9711.00i − 0.376481i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3653.00i 0.140653i 0.997524 + 0.0703267i \(0.0224042\pi\)
−0.997524 + 0.0703267i \(0.977596\pi\)
\(878\) 0 0
\(879\) 18642.0 0.715335
\(880\) 0 0
\(881\) −6552.00 −0.250559 −0.125280 0.992121i \(-0.539983\pi\)
−0.125280 + 0.992121i \(0.539983\pi\)
\(882\) 0 0
\(883\) − 4481.00i − 0.170779i −0.996348 0.0853894i \(-0.972787\pi\)
0.996348 0.0853894i \(-0.0272134\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 14666.0i − 0.555170i −0.960701 0.277585i \(-0.910466\pi\)
0.960701 0.277585i \(-0.0895341\pi\)
\(888\) 0 0
\(889\) 9576.00 0.361270
\(890\) 0 0
\(891\) −1782.00 −0.0670025
\(892\) 0 0
\(893\) 12190.0i 0.456800i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 174.000i − 0.00647680i
\(898\) 0 0
\(899\) −770.000 −0.0285661
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) − 24567.0i − 0.905358i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 51804.0i 1.89650i 0.317528 + 0.948249i \(0.397147\pi\)
−0.317528 + 0.948249i \(0.602853\pi\)
\(908\) 0 0
\(909\) 12960.0 0.472889
\(910\) 0 0
\(911\) −31198.0 −1.13462 −0.567308 0.823505i \(-0.692015\pi\)
−0.567308 + 0.823505i \(0.692015\pi\)
\(912\) 0 0
\(913\) − 25036.0i − 0.907525i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1368.00i 0.0492643i
\(918\) 0 0
\(919\) −27001.0 −0.969185 −0.484592 0.874740i \(-0.661032\pi\)
−0.484592 + 0.874740i \(0.661032\pi\)
\(920\) 0 0
\(921\) −11715.0 −0.419134
\(922\) 0 0
\(923\) − 36.0000i − 0.00128381i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1116.00i 0.0395407i
\(928\) 0 0
\(929\) 22694.0 0.801470 0.400735 0.916194i \(-0.368755\pi\)
0.400735 + 0.916194i \(0.368755\pi\)
\(930\) 0 0
\(931\) 954.000 0.0335833
\(932\) 0 0
\(933\) 28986.0i 1.01711i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 29503.0i − 1.02862i −0.857603 0.514312i \(-0.828047\pi\)
0.857603 0.514312i \(-0.171953\pi\)
\(938\) 0 0
\(939\) 15441.0 0.536633
\(940\) 0 0
\(941\) 14566.0 0.504610 0.252305 0.967648i \(-0.418811\pi\)
0.252305 + 0.967648i \(0.418811\pi\)
\(942\) 0 0
\(943\) 27144.0i 0.937360i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42322.0i 1.45225i 0.687563 + 0.726125i \(0.258681\pi\)
−0.687563 + 0.726125i \(0.741319\pi\)
\(948\) 0 0
\(949\) −522.000 −0.0178555
\(950\) 0 0
\(951\) 27648.0 0.942742
\(952\) 0 0
\(953\) 43416.0i 1.47574i 0.674942 + 0.737871i \(0.264169\pi\)
−0.674942 + 0.737871i \(0.735831\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 1452.00i − 0.0490454i
\(958\) 0 0
\(959\) 35150.0 1.18358
\(960\) 0 0
\(961\) −28566.0 −0.958880
\(962\) 0 0
\(963\) 3888.00i 0.130103i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 21528.0i 0.715919i 0.933737 + 0.357960i \(0.116527\pi\)
−0.933737 + 0.357960i \(0.883473\pi\)
\(968\) 0 0
\(969\) −9222.00 −0.305731
\(970\) 0 0
\(971\) −27050.0 −0.894002 −0.447001 0.894533i \(-0.647508\pi\)
−0.447001 + 0.894533i \(0.647508\pi\)
\(972\) 0 0
\(973\) − 34884.0i − 1.14936i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24934.0i 0.816489i 0.912873 + 0.408244i \(0.133859\pi\)
−0.912873 + 0.408244i \(0.866141\pi\)
\(978\) 0 0
\(979\) 3168.00 0.103422
\(980\) 0 0
\(981\) 6309.00 0.205332
\(982\) 0 0
\(983\) − 8388.00i − 0.272162i −0.990698 0.136081i \(-0.956549\pi\)
0.990698 0.136081i \(-0.0434508\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 13110.0i 0.422792i
\(988\) 0 0
\(989\) 24998.0 0.803731
\(990\) 0 0
\(991\) −29033.0 −0.930639 −0.465320 0.885143i \(-0.654061\pi\)
−0.465320 + 0.885143i \(0.654061\pi\)
\(992\) 0 0
\(993\) 6588.00i 0.210538i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 25326.0i 0.804496i 0.915531 + 0.402248i \(0.131771\pi\)
−0.915531 + 0.402248i \(0.868229\pi\)
\(998\) 0 0
\(999\) −7290.00 −0.230876
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 1200.4.f.f.49.1 2
4.3 odd 2 600.4.f.h.49.2 2
5.2 odd 4 1200.4.a.n.1.1 1
5.3 odd 4 1200.4.a.x.1.1 1
5.4 even 2 inner 1200.4.f.f.49.2 2
12.11 even 2 1800.4.f.g.649.2 2
20.3 even 4 600.4.a.g.1.1 1
20.7 even 4 600.4.a.j.1.1 yes 1
20.19 odd 2 600.4.f.h.49.1 2
60.23 odd 4 1800.4.a.bf.1.1 1
60.47 odd 4 1800.4.a.g.1.1 1
60.59 even 2 1800.4.f.g.649.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
600.4.a.g.1.1 1 20.3 even 4
600.4.a.j.1.1 yes 1 20.7 even 4
600.4.f.h.49.1 2 20.19 odd 2
600.4.f.h.49.2 2 4.3 odd 2
1200.4.a.n.1.1 1 5.2 odd 4
1200.4.a.x.1.1 1 5.3 odd 4
1200.4.f.f.49.1 2 1.1 even 1 trivial
1200.4.f.f.49.2 2 5.4 even 2 inner
1800.4.a.g.1.1 1 60.47 odd 4
1800.4.a.bf.1.1 1 60.23 odd 4
1800.4.f.g.649.1 2 60.59 even 2
1800.4.f.g.649.2 2 12.11 even 2