Properties

Label 192.3.g.b.127.2
Level $192$
Weight $3$
Character 192.127
Analytic conductor $5.232$
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] = [192,3,Mod(127,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 192.g (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: \(5.23162107572\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 12)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 127.2
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 192.127
Dual form 192.3.g.b.127.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.73205i q^{3} +2.00000 q^{5} +6.92820i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+1.73205i q^{3} +2.00000 q^{5} +6.92820i q^{7} -3.00000 q^{9} +6.92820i q^{11} -2.00000 q^{13} +3.46410i q^{15} +10.0000 q^{17} +20.7846i q^{19} -12.0000 q^{21} +27.7128i q^{23} -21.0000 q^{25} -5.19615i q^{27} +26.0000 q^{29} -6.92820i q^{31} -12.0000 q^{33} +13.8564i q^{35} -26.0000 q^{37} -3.46410i q^{39} +58.0000 q^{41} -48.4974i q^{43} -6.00000 q^{45} -69.2820i q^{47} +1.00000 q^{49} +17.3205i q^{51} +74.0000 q^{53} +13.8564i q^{55} -36.0000 q^{57} -90.0666i q^{59} -26.0000 q^{61} -20.7846i q^{63} -4.00000 q^{65} +6.92820i q^{67} -48.0000 q^{69} -46.0000 q^{73} -36.3731i q^{75} -48.0000 q^{77} -117.779i q^{79} +9.00000 q^{81} +48.4974i q^{83} +20.0000 q^{85} +45.0333i q^{87} +82.0000 q^{89} -13.8564i q^{91} +12.0000 q^{93} +41.5692i q^{95} +2.00000 q^{97} -20.7846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 6 q^{9} - 4 q^{13} + 20 q^{17} - 24 q^{21} - 42 q^{25} + 52 q^{29} - 24 q^{33} - 52 q^{37} + 116 q^{41} - 12 q^{45} + 2 q^{49} + 148 q^{53} - 72 q^{57} - 52 q^{61} - 8 q^{65} - 96 q^{69} - 92 q^{73} - 96 q^{77} + 18 q^{81} + 40 q^{85} + 164 q^{89} + 24 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/192\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(133\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.73205i 0.577350i
\(4\) 0 0
\(5\) 2.00000 0.400000 0.200000 0.979796i \(-0.435906\pi\)
0.200000 + 0.979796i \(0.435906\pi\)
\(6\) 0 0
\(7\) 6.92820i 0.989743i 0.868966 + 0.494872i \(0.164785\pi\)
−0.868966 + 0.494872i \(0.835215\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 6.92820i 0.629837i 0.949119 + 0.314918i \(0.101977\pi\)
−0.949119 + 0.314918i \(0.898023\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.153846 −0.0769231 0.997037i \(-0.524510\pi\)
−0.0769231 + 0.997037i \(0.524510\pi\)
\(14\) 0 0
\(15\) 3.46410i 0.230940i
\(16\) 0 0
\(17\) 10.0000 0.588235 0.294118 0.955769i \(-0.404974\pi\)
0.294118 + 0.955769i \(0.404974\pi\)
\(18\) 0 0
\(19\) 20.7846i 1.09393i 0.837157 + 0.546963i \(0.184216\pi\)
−0.837157 + 0.546963i \(0.815784\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.571429
\(22\) 0 0
\(23\) 27.7128i 1.20490i 0.798155 + 0.602452i \(0.205810\pi\)
−0.798155 + 0.602452i \(0.794190\pi\)
\(24\) 0 0
\(25\) −21.0000 −0.840000
\(26\) 0 0
\(27\) − 5.19615i − 0.192450i
\(28\) 0 0
\(29\) 26.0000 0.896552 0.448276 0.893895i \(-0.352038\pi\)
0.448276 + 0.893895i \(0.352038\pi\)
\(30\) 0 0
\(31\) − 6.92820i − 0.223490i −0.993737 0.111745i \(-0.964356\pi\)
0.993737 0.111745i \(-0.0356441\pi\)
\(32\) 0 0
\(33\) −12.0000 −0.363636
\(34\) 0 0
\(35\) 13.8564i 0.395897i
\(36\) 0 0
\(37\) −26.0000 −0.702703 −0.351351 0.936244i \(-0.614278\pi\)
−0.351351 + 0.936244i \(0.614278\pi\)
\(38\) 0 0
\(39\) − 3.46410i − 0.0888231i
\(40\) 0 0
\(41\) 58.0000 1.41463 0.707317 0.706896i \(-0.249905\pi\)
0.707317 + 0.706896i \(0.249905\pi\)
\(42\) 0 0
\(43\) − 48.4974i − 1.12785i −0.825827 0.563924i \(-0.809291\pi\)
0.825827 0.563924i \(-0.190709\pi\)
\(44\) 0 0
\(45\) −6.00000 −0.133333
\(46\) 0 0
\(47\) − 69.2820i − 1.47409i −0.675846 0.737043i \(-0.736222\pi\)
0.675846 0.737043i \(-0.263778\pi\)
\(48\) 0 0
\(49\) 1.00000 0.0204082
\(50\) 0 0
\(51\) 17.3205i 0.339618i
\(52\) 0 0
\(53\) 74.0000 1.39623 0.698113 0.715987i \(-0.254023\pi\)
0.698113 + 0.715987i \(0.254023\pi\)
\(54\) 0 0
\(55\) 13.8564i 0.251935i
\(56\) 0 0
\(57\) −36.0000 −0.631579
\(58\) 0 0
\(59\) − 90.0666i − 1.52655i −0.646072 0.763277i \(-0.723589\pi\)
0.646072 0.763277i \(-0.276411\pi\)
\(60\) 0 0
\(61\) −26.0000 −0.426230 −0.213115 0.977027i \(-0.568361\pi\)
−0.213115 + 0.977027i \(0.568361\pi\)
\(62\) 0 0
\(63\) − 20.7846i − 0.329914i
\(64\) 0 0
\(65\) −4.00000 −0.0615385
\(66\) 0 0
\(67\) 6.92820i 0.103406i 0.998663 + 0.0517030i \(0.0164649\pi\)
−0.998663 + 0.0517030i \(0.983535\pi\)
\(68\) 0 0
\(69\) −48.0000 −0.695652
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −46.0000 −0.630137 −0.315068 0.949069i \(-0.602027\pi\)
−0.315068 + 0.949069i \(0.602027\pi\)
\(74\) 0 0
\(75\) − 36.3731i − 0.484974i
\(76\) 0 0
\(77\) −48.0000 −0.623377
\(78\) 0 0
\(79\) − 117.779i − 1.49088i −0.666573 0.745440i \(-0.732240\pi\)
0.666573 0.745440i \(-0.267760\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) 48.4974i 0.584306i 0.956372 + 0.292153i \(0.0943717\pi\)
−0.956372 + 0.292153i \(0.905628\pi\)
\(84\) 0 0
\(85\) 20.0000 0.235294
\(86\) 0 0
\(87\) 45.0333i 0.517624i
\(88\) 0 0
\(89\) 82.0000 0.921348 0.460674 0.887569i \(-0.347608\pi\)
0.460674 + 0.887569i \(0.347608\pi\)
\(90\) 0 0
\(91\) − 13.8564i − 0.152268i
\(92\) 0 0
\(93\) 12.0000 0.129032
\(94\) 0 0
\(95\) 41.5692i 0.437571i
\(96\) 0 0
\(97\) 2.00000 0.0206186 0.0103093 0.999947i \(-0.496718\pi\)
0.0103093 + 0.999947i \(0.496718\pi\)
\(98\) 0 0
\(99\) − 20.7846i − 0.209946i
\(100\) 0 0
\(101\) 74.0000 0.732673 0.366337 0.930482i \(-0.380612\pi\)
0.366337 + 0.930482i \(0.380612\pi\)
\(102\) 0 0
\(103\) 76.2102i 0.739905i 0.929051 + 0.369953i \(0.120626\pi\)
−0.929051 + 0.369953i \(0.879374\pi\)
\(104\) 0 0
\(105\) −24.0000 −0.228571
\(106\) 0 0
\(107\) 20.7846i 0.194249i 0.995272 + 0.0971243i \(0.0309645\pi\)
−0.995272 + 0.0971243i \(0.969036\pi\)
\(108\) 0 0
\(109\) 46.0000 0.422018 0.211009 0.977484i \(-0.432325\pi\)
0.211009 + 0.977484i \(0.432325\pi\)
\(110\) 0 0
\(111\) − 45.0333i − 0.405706i
\(112\) 0 0
\(113\) −110.000 −0.973451 −0.486726 0.873555i \(-0.661809\pi\)
−0.486726 + 0.873555i \(0.661809\pi\)
\(114\) 0 0
\(115\) 55.4256i 0.481962i
\(116\) 0 0
\(117\) 6.00000 0.0512821
\(118\) 0 0
\(119\) 69.2820i 0.582202i
\(120\) 0 0
\(121\) 73.0000 0.603306
\(122\) 0 0
\(123\) 100.459i 0.816739i
\(124\) 0 0
\(125\) −92.0000 −0.736000
\(126\) 0 0
\(127\) 145.492i 1.14561i 0.819692 + 0.572804i \(0.194144\pi\)
−0.819692 + 0.572804i \(0.805856\pi\)
\(128\) 0 0
\(129\) 84.0000 0.651163
\(130\) 0 0
\(131\) 117.779i 0.899080i 0.893260 + 0.449540i \(0.148412\pi\)
−0.893260 + 0.449540i \(0.851588\pi\)
\(132\) 0 0
\(133\) −144.000 −1.08271
\(134\) 0 0
\(135\) − 10.3923i − 0.0769800i
\(136\) 0 0
\(137\) 10.0000 0.0729927 0.0364964 0.999334i \(-0.488380\pi\)
0.0364964 + 0.999334i \(0.488380\pi\)
\(138\) 0 0
\(139\) 48.4974i 0.348902i 0.984666 + 0.174451i \(0.0558151\pi\)
−0.984666 + 0.174451i \(0.944185\pi\)
\(140\) 0 0
\(141\) 120.000 0.851064
\(142\) 0 0
\(143\) − 13.8564i − 0.0968979i
\(144\) 0 0
\(145\) 52.0000 0.358621
\(146\) 0 0
\(147\) 1.73205i 0.0117827i
\(148\) 0 0
\(149\) 2.00000 0.0134228 0.00671141 0.999977i \(-0.497864\pi\)
0.00671141 + 0.999977i \(0.497864\pi\)
\(150\) 0 0
\(151\) 90.0666i 0.596468i 0.954493 + 0.298234i \(0.0963975\pi\)
−0.954493 + 0.298234i \(0.903602\pi\)
\(152\) 0 0
\(153\) −30.0000 −0.196078
\(154\) 0 0
\(155\) − 13.8564i − 0.0893962i
\(156\) 0 0
\(157\) 214.000 1.36306 0.681529 0.731791i \(-0.261315\pi\)
0.681529 + 0.731791i \(0.261315\pi\)
\(158\) 0 0
\(159\) 128.172i 0.806112i
\(160\) 0 0
\(161\) −192.000 −1.19255
\(162\) 0 0
\(163\) 20.7846i 0.127513i 0.997965 + 0.0637565i \(0.0203081\pi\)
−0.997965 + 0.0637565i \(0.979692\pi\)
\(164\) 0 0
\(165\) −24.0000 −0.145455
\(166\) 0 0
\(167\) − 96.9948i − 0.580807i −0.956904 0.290404i \(-0.906210\pi\)
0.956904 0.290404i \(-0.0937896\pi\)
\(168\) 0 0
\(169\) −165.000 −0.976331
\(170\) 0 0
\(171\) − 62.3538i − 0.364642i
\(172\) 0 0
\(173\) −334.000 −1.93064 −0.965318 0.261077i \(-0.915922\pi\)
−0.965318 + 0.261077i \(0.915922\pi\)
\(174\) 0 0
\(175\) − 145.492i − 0.831384i
\(176\) 0 0
\(177\) 156.000 0.881356
\(178\) 0 0
\(179\) − 187.061i − 1.04504i −0.852628 0.522518i \(-0.824993\pi\)
0.852628 0.522518i \(-0.175007\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.0110497 −0.00552486 0.999985i \(-0.501759\pi\)
−0.00552486 + 0.999985i \(0.501759\pi\)
\(182\) 0 0
\(183\) − 45.0333i − 0.246084i
\(184\) 0 0
\(185\) −52.0000 −0.281081
\(186\) 0 0
\(187\) 69.2820i 0.370492i
\(188\) 0 0
\(189\) 36.0000 0.190476
\(190\) 0 0
\(191\) 221.703i 1.16075i 0.814351 + 0.580373i \(0.197093\pi\)
−0.814351 + 0.580373i \(0.802907\pi\)
\(192\) 0 0
\(193\) 290.000 1.50259 0.751295 0.659966i \(-0.229429\pi\)
0.751295 + 0.659966i \(0.229429\pi\)
\(194\) 0 0
\(195\) − 6.92820i − 0.0355292i
\(196\) 0 0
\(197\) 26.0000 0.131980 0.0659898 0.997820i \(-0.478980\pi\)
0.0659898 + 0.997820i \(0.478980\pi\)
\(198\) 0 0
\(199\) − 394.908i − 1.98446i −0.124416 0.992230i \(-0.539706\pi\)
0.124416 0.992230i \(-0.460294\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.0597015
\(202\) 0 0
\(203\) 180.133i 0.887356i
\(204\) 0 0
\(205\) 116.000 0.565854
\(206\) 0 0
\(207\) − 83.1384i − 0.401635i
\(208\) 0 0
\(209\) −144.000 −0.688995
\(210\) 0 0
\(211\) − 242.487i − 1.14923i −0.818425 0.574614i \(-0.805152\pi\)
0.818425 0.574614i \(-0.194848\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 96.9948i − 0.451139i
\(216\) 0 0
\(217\) 48.0000 0.221198
\(218\) 0 0
\(219\) − 79.6743i − 0.363810i
\(220\) 0 0
\(221\) −20.0000 −0.0904977
\(222\) 0 0
\(223\) 339.482i 1.52234i 0.648552 + 0.761170i \(0.275375\pi\)
−0.648552 + 0.761170i \(0.724625\pi\)
\(224\) 0 0
\(225\) 63.0000 0.280000
\(226\) 0 0
\(227\) − 284.056i − 1.25135i −0.780084 0.625675i \(-0.784824\pi\)
0.780084 0.625675i \(-0.215176\pi\)
\(228\) 0 0
\(229\) 142.000 0.620087 0.310044 0.950722i \(-0.399656\pi\)
0.310044 + 0.950722i \(0.399656\pi\)
\(230\) 0 0
\(231\) − 83.1384i − 0.359907i
\(232\) 0 0
\(233\) 82.0000 0.351931 0.175966 0.984396i \(-0.443695\pi\)
0.175966 + 0.984396i \(0.443695\pi\)
\(234\) 0 0
\(235\) − 138.564i − 0.589634i
\(236\) 0 0
\(237\) 204.000 0.860759
\(238\) 0 0
\(239\) − 387.979i − 1.62334i −0.584113 0.811672i \(-0.698558\pi\)
0.584113 0.811672i \(-0.301442\pi\)
\(240\) 0 0
\(241\) −46.0000 −0.190871 −0.0954357 0.995436i \(-0.530424\pi\)
−0.0954357 + 0.995436i \(0.530424\pi\)
\(242\) 0 0
\(243\) 15.5885i 0.0641500i
\(244\) 0 0
\(245\) 2.00000 0.00816327
\(246\) 0 0
\(247\) − 41.5692i − 0.168296i
\(248\) 0 0
\(249\) −84.0000 −0.337349
\(250\) 0 0
\(251\) 145.492i 0.579650i 0.957080 + 0.289825i \(0.0935972\pi\)
−0.957080 + 0.289825i \(0.906403\pi\)
\(252\) 0 0
\(253\) −192.000 −0.758893
\(254\) 0 0
\(255\) 34.6410i 0.135847i
\(256\) 0 0
\(257\) −254.000 −0.988327 −0.494163 0.869369i \(-0.664526\pi\)
−0.494163 + 0.869369i \(0.664526\pi\)
\(258\) 0 0
\(259\) − 180.133i − 0.695495i
\(260\) 0 0
\(261\) −78.0000 −0.298851
\(262\) 0 0
\(263\) − 152.420i − 0.579546i −0.957095 0.289773i \(-0.906420\pi\)
0.957095 0.289773i \(-0.0935797\pi\)
\(264\) 0 0
\(265\) 148.000 0.558491
\(266\) 0 0
\(267\) 142.028i 0.531941i
\(268\) 0 0
\(269\) −262.000 −0.973978 −0.486989 0.873408i \(-0.661905\pi\)
−0.486989 + 0.873408i \(0.661905\pi\)
\(270\) 0 0
\(271\) − 20.7846i − 0.0766960i −0.999264 0.0383480i \(-0.987790\pi\)
0.999264 0.0383480i \(-0.0122095\pi\)
\(272\) 0 0
\(273\) 24.0000 0.0879121
\(274\) 0 0
\(275\) − 145.492i − 0.529063i
\(276\) 0 0
\(277\) −290.000 −1.04693 −0.523466 0.852047i \(-0.675361\pi\)
−0.523466 + 0.852047i \(0.675361\pi\)
\(278\) 0 0
\(279\) 20.7846i 0.0744968i
\(280\) 0 0
\(281\) 226.000 0.804270 0.402135 0.915580i \(-0.368268\pi\)
0.402135 + 0.915580i \(0.368268\pi\)
\(282\) 0 0
\(283\) 297.913i 1.05270i 0.850269 + 0.526348i \(0.176439\pi\)
−0.850269 + 0.526348i \(0.823561\pi\)
\(284\) 0 0
\(285\) −72.0000 −0.252632
\(286\) 0 0
\(287\) 401.836i 1.40012i
\(288\) 0 0
\(289\) −189.000 −0.653979
\(290\) 0 0
\(291\) 3.46410i 0.0119041i
\(292\) 0 0
\(293\) 362.000 1.23549 0.617747 0.786377i \(-0.288045\pi\)
0.617747 + 0.786377i \(0.288045\pi\)
\(294\) 0 0
\(295\) − 180.133i − 0.610621i
\(296\) 0 0
\(297\) 36.0000 0.121212
\(298\) 0 0
\(299\) − 55.4256i − 0.185370i
\(300\) 0 0
\(301\) 336.000 1.11628
\(302\) 0 0
\(303\) 128.172i 0.423009i
\(304\) 0 0
\(305\) −52.0000 −0.170492
\(306\) 0 0
\(307\) 145.492i 0.473916i 0.971520 + 0.236958i \(0.0761504\pi\)
−0.971520 + 0.236958i \(0.923850\pi\)
\(308\) 0 0
\(309\) −132.000 −0.427184
\(310\) 0 0
\(311\) 235.559i 0.757424i 0.925515 + 0.378712i \(0.123633\pi\)
−0.925515 + 0.378712i \(0.876367\pi\)
\(312\) 0 0
\(313\) −478.000 −1.52716 −0.763578 0.645715i \(-0.776559\pi\)
−0.763578 + 0.645715i \(0.776559\pi\)
\(314\) 0 0
\(315\) − 41.5692i − 0.131966i
\(316\) 0 0
\(317\) 170.000 0.536278 0.268139 0.963380i \(-0.413591\pi\)
0.268139 + 0.963380i \(0.413591\pi\)
\(318\) 0 0
\(319\) 180.133i 0.564681i
\(320\) 0 0
\(321\) −36.0000 −0.112150
\(322\) 0 0
\(323\) 207.846i 0.643486i
\(324\) 0 0
\(325\) 42.0000 0.129231
\(326\) 0 0
\(327\) 79.6743i 0.243652i
\(328\) 0 0
\(329\) 480.000 1.45897
\(330\) 0 0
\(331\) 408.764i 1.23494i 0.786596 + 0.617468i \(0.211842\pi\)
−0.786596 + 0.617468i \(0.788158\pi\)
\(332\) 0 0
\(333\) 78.0000 0.234234
\(334\) 0 0
\(335\) 13.8564i 0.0413624i
\(336\) 0 0
\(337\) 338.000 1.00297 0.501484 0.865167i \(-0.332788\pi\)
0.501484 + 0.865167i \(0.332788\pi\)
\(338\) 0 0
\(339\) − 190.526i − 0.562022i
\(340\) 0 0
\(341\) 48.0000 0.140762
\(342\) 0 0
\(343\) 346.410i 1.00994i
\(344\) 0 0
\(345\) −96.0000 −0.278261
\(346\) 0 0
\(347\) 200.918i 0.579014i 0.957176 + 0.289507i \(0.0934914\pi\)
−0.957176 + 0.289507i \(0.906509\pi\)
\(348\) 0 0
\(349\) −506.000 −1.44986 −0.724928 0.688824i \(-0.758127\pi\)
−0.724928 + 0.688824i \(0.758127\pi\)
\(350\) 0 0
\(351\) 10.3923i 0.0296077i
\(352\) 0 0
\(353\) 178.000 0.504249 0.252125 0.967695i \(-0.418871\pi\)
0.252125 + 0.967695i \(0.418871\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −120.000 −0.336134
\(358\) 0 0
\(359\) − 166.277i − 0.463167i −0.972815 0.231583i \(-0.925609\pi\)
0.972815 0.231583i \(-0.0743906\pi\)
\(360\) 0 0
\(361\) −71.0000 −0.196676
\(362\) 0 0
\(363\) 126.440i 0.348319i
\(364\) 0 0
\(365\) −92.0000 −0.252055
\(366\) 0 0
\(367\) − 200.918i − 0.547460i −0.961807 0.273730i \(-0.911742\pi\)
0.961807 0.273730i \(-0.0882575\pi\)
\(368\) 0 0
\(369\) −174.000 −0.471545
\(370\) 0 0
\(371\) 512.687i 1.38191i
\(372\) 0 0
\(373\) 310.000 0.831099 0.415550 0.909571i \(-0.363589\pi\)
0.415550 + 0.909571i \(0.363589\pi\)
\(374\) 0 0
\(375\) − 159.349i − 0.424930i
\(376\) 0 0
\(377\) −52.0000 −0.137931
\(378\) 0 0
\(379\) − 436.477i − 1.15165i −0.817572 0.575827i \(-0.804680\pi\)
0.817572 0.575827i \(-0.195320\pi\)
\(380\) 0 0
\(381\) −252.000 −0.661417
\(382\) 0 0
\(383\) 609.682i 1.59186i 0.605390 + 0.795929i \(0.293017\pi\)
−0.605390 + 0.795929i \(0.706983\pi\)
\(384\) 0 0
\(385\) −96.0000 −0.249351
\(386\) 0 0
\(387\) 145.492i 0.375949i
\(388\) 0 0
\(389\) 578.000 1.48586 0.742931 0.669368i \(-0.233435\pi\)
0.742931 + 0.669368i \(0.233435\pi\)
\(390\) 0 0
\(391\) 277.128i 0.708768i
\(392\) 0 0
\(393\) −204.000 −0.519084
\(394\) 0 0
\(395\) − 235.559i − 0.596352i
\(396\) 0 0
\(397\) −26.0000 −0.0654912 −0.0327456 0.999464i \(-0.510425\pi\)
−0.0327456 + 0.999464i \(0.510425\pi\)
\(398\) 0 0
\(399\) − 249.415i − 0.625101i
\(400\) 0 0
\(401\) 250.000 0.623441 0.311721 0.950174i \(-0.399095\pi\)
0.311721 + 0.950174i \(0.399095\pi\)
\(402\) 0 0
\(403\) 13.8564i 0.0343831i
\(404\) 0 0
\(405\) 18.0000 0.0444444
\(406\) 0 0
\(407\) − 180.133i − 0.442588i
\(408\) 0 0
\(409\) 290.000 0.709046 0.354523 0.935047i \(-0.384643\pi\)
0.354523 + 0.935047i \(0.384643\pi\)
\(410\) 0 0
\(411\) 17.3205i 0.0421424i
\(412\) 0 0
\(413\) 624.000 1.51090
\(414\) 0 0
\(415\) 96.9948i 0.233723i
\(416\) 0 0
\(417\) −84.0000 −0.201439
\(418\) 0 0
\(419\) − 339.482i − 0.810219i −0.914268 0.405110i \(-0.867233\pi\)
0.914268 0.405110i \(-0.132767\pi\)
\(420\) 0 0
\(421\) −674.000 −1.60095 −0.800475 0.599366i \(-0.795419\pi\)
−0.800475 + 0.599366i \(0.795419\pi\)
\(422\) 0 0
\(423\) 207.846i 0.491362i
\(424\) 0 0
\(425\) −210.000 −0.494118
\(426\) 0 0
\(427\) − 180.133i − 0.421858i
\(428\) 0 0
\(429\) 24.0000 0.0559441
\(430\) 0 0
\(431\) − 540.400i − 1.25383i −0.779088 0.626914i \(-0.784318\pi\)
0.779088 0.626914i \(-0.215682\pi\)
\(432\) 0 0
\(433\) −334.000 −0.771363 −0.385681 0.922632i \(-0.626034\pi\)
−0.385681 + 0.922632i \(0.626034\pi\)
\(434\) 0 0
\(435\) 90.0666i 0.207050i
\(436\) 0 0
\(437\) −576.000 −1.31808
\(438\) 0 0
\(439\) − 117.779i − 0.268290i −0.990962 0.134145i \(-0.957171\pi\)
0.990962 0.134145i \(-0.0428288\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.00680272
\(442\) 0 0
\(443\) − 76.2102i − 0.172032i −0.996294 0.0860161i \(-0.972586\pi\)
0.996294 0.0860161i \(-0.0274136\pi\)
\(444\) 0 0
\(445\) 164.000 0.368539
\(446\) 0 0
\(447\) 3.46410i 0.00774967i
\(448\) 0 0
\(449\) 394.000 0.877506 0.438753 0.898608i \(-0.355420\pi\)
0.438753 + 0.898608i \(0.355420\pi\)
\(450\) 0 0
\(451\) 401.836i 0.890988i
\(452\) 0 0
\(453\) −156.000 −0.344371
\(454\) 0 0
\(455\) − 27.7128i − 0.0609073i
\(456\) 0 0
\(457\) −478.000 −1.04595 −0.522976 0.852347i \(-0.675178\pi\)
−0.522976 + 0.852347i \(0.675178\pi\)
\(458\) 0 0
\(459\) − 51.9615i − 0.113206i
\(460\) 0 0
\(461\) −142.000 −0.308026 −0.154013 0.988069i \(-0.549220\pi\)
−0.154013 + 0.988069i \(0.549220\pi\)
\(462\) 0 0
\(463\) − 630.466i − 1.36170i −0.732423 0.680849i \(-0.761611\pi\)
0.732423 0.680849i \(-0.238389\pi\)
\(464\) 0 0
\(465\) 24.0000 0.0516129
\(466\) 0 0
\(467\) 20.7846i 0.0445067i 0.999752 + 0.0222533i \(0.00708404\pi\)
−0.999752 + 0.0222533i \(0.992916\pi\)
\(468\) 0 0
\(469\) −48.0000 −0.102345
\(470\) 0 0
\(471\) 370.659i 0.786962i
\(472\) 0 0
\(473\) 336.000 0.710359
\(474\) 0 0
\(475\) − 436.477i − 0.918899i
\(476\) 0 0
\(477\) −222.000 −0.465409
\(478\) 0 0
\(479\) − 734.390i − 1.53317i −0.642141 0.766586i \(-0.721954\pi\)
0.642141 0.766586i \(-0.278046\pi\)
\(480\) 0 0
\(481\) 52.0000 0.108108
\(482\) 0 0
\(483\) − 332.554i − 0.688517i
\(484\) 0 0
\(485\) 4.00000 0.00824742
\(486\) 0 0
\(487\) − 103.923i − 0.213394i −0.994292 0.106697i \(-0.965972\pi\)
0.994292 0.106697i \(-0.0340275\pi\)
\(488\) 0 0
\(489\) −36.0000 −0.0736196
\(490\) 0 0
\(491\) − 921.451i − 1.87668i −0.345711 0.938341i \(-0.612362\pi\)
0.345711 0.938341i \(-0.387638\pi\)
\(492\) 0 0
\(493\) 260.000 0.527383
\(494\) 0 0
\(495\) − 41.5692i − 0.0839782i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 76.2102i − 0.152726i −0.997080 0.0763630i \(-0.975669\pi\)
0.997080 0.0763630i \(-0.0243308\pi\)
\(500\) 0 0
\(501\) 168.000 0.335329
\(502\) 0 0
\(503\) 581.969i 1.15700i 0.815684 + 0.578498i \(0.196361\pi\)
−0.815684 + 0.578498i \(0.803639\pi\)
\(504\) 0 0
\(505\) 148.000 0.293069
\(506\) 0 0
\(507\) − 285.788i − 0.563685i
\(508\) 0 0
\(509\) 842.000 1.65422 0.827112 0.562037i \(-0.189982\pi\)
0.827112 + 0.562037i \(0.189982\pi\)
\(510\) 0 0
\(511\) − 318.697i − 0.623674i
\(512\) 0 0
\(513\) 108.000 0.210526
\(514\) 0 0
\(515\) 152.420i 0.295962i
\(516\) 0 0
\(517\) 480.000 0.928433
\(518\) 0 0
\(519\) − 578.505i − 1.11465i
\(520\) 0 0
\(521\) −326.000 −0.625720 −0.312860 0.949799i \(-0.601287\pi\)
−0.312860 + 0.949799i \(0.601287\pi\)
\(522\) 0 0
\(523\) 311.769i 0.596117i 0.954548 + 0.298058i \(0.0963390\pi\)
−0.954548 + 0.298058i \(0.903661\pi\)
\(524\) 0 0
\(525\) 252.000 0.480000
\(526\) 0 0
\(527\) − 69.2820i − 0.131465i
\(528\) 0 0
\(529\) −239.000 −0.451796
\(530\) 0 0
\(531\) 270.200i 0.508851i
\(532\) 0 0
\(533\) −116.000 −0.217636
\(534\) 0 0
\(535\) 41.5692i 0.0776995i
\(536\) 0 0
\(537\) 324.000 0.603352
\(538\) 0 0
\(539\) 6.92820i 0.0128538i
\(540\) 0 0
\(541\) −530.000 −0.979667 −0.489834 0.871816i \(-0.662942\pi\)
−0.489834 + 0.871816i \(0.662942\pi\)
\(542\) 0 0
\(543\) − 3.46410i − 0.00637956i
\(544\) 0 0
\(545\) 92.0000 0.168807
\(546\) 0 0
\(547\) − 339.482i − 0.620625i −0.950635 0.310313i \(-0.899566\pi\)
0.950635 0.310313i \(-0.100434\pi\)
\(548\) 0 0
\(549\) 78.0000 0.142077
\(550\) 0 0
\(551\) 540.400i 0.980762i
\(552\) 0 0
\(553\) 816.000 1.47559
\(554\) 0 0
\(555\) − 90.0666i − 0.162282i
\(556\) 0 0
\(557\) −766.000 −1.37522 −0.687612 0.726078i \(-0.741341\pi\)
−0.687612 + 0.726078i \(0.741341\pi\)
\(558\) 0 0
\(559\) 96.9948i 0.173515i
\(560\) 0 0
\(561\) −120.000 −0.213904
\(562\) 0 0
\(563\) 491.902i 0.873717i 0.899530 + 0.436858i \(0.143909\pi\)
−0.899530 + 0.436858i \(0.856091\pi\)
\(564\) 0 0
\(565\) −220.000 −0.389381
\(566\) 0 0
\(567\) 62.3538i 0.109971i
\(568\) 0 0
\(569\) −422.000 −0.741652 −0.370826 0.928702i \(-0.620925\pi\)
−0.370826 + 0.928702i \(0.620925\pi\)
\(570\) 0 0
\(571\) − 284.056i − 0.497472i −0.968571 0.248736i \(-0.919985\pi\)
0.968571 0.248736i \(-0.0800151\pi\)
\(572\) 0 0
\(573\) −384.000 −0.670157
\(574\) 0 0
\(575\) − 581.969i − 1.01212i
\(576\) 0 0
\(577\) −46.0000 −0.0797227 −0.0398614 0.999205i \(-0.512692\pi\)
−0.0398614 + 0.999205i \(0.512692\pi\)
\(578\) 0 0
\(579\) 502.295i 0.867521i
\(580\) 0 0
\(581\) −336.000 −0.578313
\(582\) 0 0
\(583\) 512.687i 0.879395i
\(584\) 0 0
\(585\) 12.0000 0.0205128
\(586\) 0 0
\(587\) 630.466i 1.07405i 0.843567 + 0.537024i \(0.180452\pi\)
−0.843567 + 0.537024i \(0.819548\pi\)
\(588\) 0 0
\(589\) 144.000 0.244482
\(590\) 0 0
\(591\) 45.0333i 0.0761985i
\(592\) 0 0
\(593\) 82.0000 0.138280 0.0691400 0.997607i \(-0.477974\pi\)
0.0691400 + 0.997607i \(0.477974\pi\)
\(594\) 0 0
\(595\) 138.564i 0.232881i
\(596\) 0 0
\(597\) 684.000 1.14573
\(598\) 0 0
\(599\) − 55.4256i − 0.0925303i −0.998929 0.0462651i \(-0.985268\pi\)
0.998929 0.0462651i \(-0.0147319\pi\)
\(600\) 0 0
\(601\) −334.000 −0.555740 −0.277870 0.960619i \(-0.589629\pi\)
−0.277870 + 0.960619i \(0.589629\pi\)
\(602\) 0 0
\(603\) − 20.7846i − 0.0344687i
\(604\) 0 0
\(605\) 146.000 0.241322
\(606\) 0 0
\(607\) 367.195i 0.604934i 0.953160 + 0.302467i \(0.0978102\pi\)
−0.953160 + 0.302467i \(0.902190\pi\)
\(608\) 0 0
\(609\) −312.000 −0.512315
\(610\) 0 0
\(611\) 138.564i 0.226782i
\(612\) 0 0
\(613\) 214.000 0.349103 0.174551 0.984648i \(-0.444152\pi\)
0.174551 + 0.984648i \(0.444152\pi\)
\(614\) 0 0
\(615\) 200.918i 0.326696i
\(616\) 0 0
\(617\) −1118.00 −1.81199 −0.905997 0.423285i \(-0.860877\pi\)
−0.905997 + 0.423285i \(0.860877\pi\)
\(618\) 0 0
\(619\) − 672.036i − 1.08568i −0.839836 0.542840i \(-0.817349\pi\)
0.839836 0.542840i \(-0.182651\pi\)
\(620\) 0 0
\(621\) 144.000 0.231884
\(622\) 0 0
\(623\) 568.113i 0.911898i
\(624\) 0 0
\(625\) 341.000 0.545600
\(626\) 0 0
\(627\) − 249.415i − 0.397792i
\(628\) 0 0
\(629\) −260.000 −0.413355
\(630\) 0 0
\(631\) 145.492i 0.230574i 0.993332 + 0.115287i \(0.0367788\pi\)
−0.993332 + 0.115287i \(0.963221\pi\)
\(632\) 0 0
\(633\) 420.000 0.663507
\(634\) 0 0
\(635\) 290.985i 0.458243i
\(636\) 0 0
\(637\) −2.00000 −0.00313972
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 10.0000 0.0156006 0.00780031 0.999970i \(-0.497517\pi\)
0.00780031 + 0.999970i \(0.497517\pi\)
\(642\) 0 0
\(643\) 1212.44i 1.88559i 0.333370 + 0.942796i \(0.391814\pi\)
−0.333370 + 0.942796i \(0.608186\pi\)
\(644\) 0 0
\(645\) 168.000 0.260465
\(646\) 0 0
\(647\) 332.554i 0.513993i 0.966412 + 0.256997i \(0.0827330\pi\)
−0.966412 + 0.256997i \(0.917267\pi\)
\(648\) 0 0
\(649\) 624.000 0.961479
\(650\) 0 0
\(651\) 83.1384i 0.127709i
\(652\) 0 0
\(653\) −670.000 −1.02603 −0.513017 0.858379i \(-0.671472\pi\)
−0.513017 + 0.858379i \(0.671472\pi\)
\(654\) 0 0
\(655\) 235.559i 0.359632i
\(656\) 0 0
\(657\) 138.000 0.210046
\(658\) 0 0
\(659\) − 824.456i − 1.25107i −0.780195 0.625536i \(-0.784880\pi\)
0.780195 0.625536i \(-0.215120\pi\)
\(660\) 0 0
\(661\) 1222.00 1.84871 0.924357 0.381529i \(-0.124602\pi\)
0.924357 + 0.381529i \(0.124602\pi\)
\(662\) 0 0
\(663\) − 34.6410i − 0.0522489i
\(664\) 0 0
\(665\) −288.000 −0.433083
\(666\) 0 0
\(667\) 720.533i 1.08026i
\(668\) 0 0
\(669\) −588.000 −0.878924
\(670\) 0 0
\(671\) − 180.133i − 0.268455i
\(672\) 0 0
\(673\) −334.000 −0.496285 −0.248143 0.968724i \(-0.579820\pi\)
−0.248143 + 0.968724i \(0.579820\pi\)
\(674\) 0 0
\(675\) 109.119i 0.161658i
\(676\) 0 0
\(677\) −1006.00 −1.48597 −0.742984 0.669309i \(-0.766590\pi\)
−0.742984 + 0.669309i \(0.766590\pi\)
\(678\) 0 0
\(679\) 13.8564i 0.0204071i
\(680\) 0 0
\(681\) 492.000 0.722467
\(682\) 0 0
\(683\) − 187.061i − 0.273882i −0.990579 0.136941i \(-0.956273\pi\)
0.990579 0.136941i \(-0.0437271\pi\)
\(684\) 0 0
\(685\) 20.0000 0.0291971
\(686\) 0 0
\(687\) 245.951i 0.358008i
\(688\) 0 0
\(689\) −148.000 −0.214804
\(690\) 0 0
\(691\) 990.733i 1.43377i 0.697193 + 0.716884i \(0.254432\pi\)
−0.697193 + 0.716884i \(0.745568\pi\)
\(692\) 0 0
\(693\) 144.000 0.207792
\(694\) 0 0
\(695\) 96.9948i 0.139561i
\(696\) 0 0
\(697\) 580.000 0.832138
\(698\) 0 0
\(699\) 142.028i 0.203188i
\(700\) 0 0
\(701\) 1034.00 1.47504 0.737518 0.675328i \(-0.235998\pi\)
0.737518 + 0.675328i \(0.235998\pi\)
\(702\) 0 0
\(703\) − 540.400i − 0.768705i
\(704\) 0 0
\(705\) 240.000 0.340426
\(706\) 0 0
\(707\) 512.687i 0.725158i
\(708\) 0 0
\(709\) −530.000 −0.747532 −0.373766 0.927523i \(-0.621934\pi\)
−0.373766 + 0.927523i \(0.621934\pi\)
\(710\) 0 0
\(711\) 353.338i 0.496960i
\(712\) 0 0
\(713\) 192.000 0.269285
\(714\) 0 0
\(715\) − 27.7128i − 0.0387592i
\(716\) 0 0
\(717\) 672.000 0.937238
\(718\) 0 0
\(719\) 706.677i 0.982861i 0.870917 + 0.491430i \(0.163526\pi\)
−0.870917 + 0.491430i \(0.836474\pi\)
\(720\) 0 0
\(721\) −528.000 −0.732316
\(722\) 0 0
\(723\) − 79.6743i − 0.110200i
\(724\) 0 0
\(725\) −546.000 −0.753103
\(726\) 0 0
\(727\) − 242.487i − 0.333545i −0.985995 0.166772i \(-0.946665\pi\)
0.985995 0.166772i \(-0.0533345\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 484.974i − 0.663439i
\(732\) 0 0
\(733\) −194.000 −0.264666 −0.132333 0.991205i \(-0.542247\pi\)
−0.132333 + 0.991205i \(0.542247\pi\)
\(734\) 0 0
\(735\) 3.46410i 0.00471306i
\(736\) 0 0
\(737\) −48.0000 −0.0651289
\(738\) 0 0
\(739\) − 1351.00i − 1.82815i −0.405550 0.914073i \(-0.632920\pi\)
0.405550 0.914073i \(-0.367080\pi\)
\(740\) 0 0
\(741\) 72.0000 0.0971660
\(742\) 0 0
\(743\) − 678.964i − 0.913814i −0.889514 0.456907i \(-0.848957\pi\)
0.889514 0.456907i \(-0.151043\pi\)
\(744\) 0 0
\(745\) 4.00000 0.00536913
\(746\) 0 0
\(747\) − 145.492i − 0.194769i
\(748\) 0 0
\(749\) −144.000 −0.192256
\(750\) 0 0
\(751\) 658.179i 0.876404i 0.898877 + 0.438202i \(0.144385\pi\)
−0.898877 + 0.438202i \(0.855615\pi\)
\(752\) 0 0
\(753\) −252.000 −0.334661
\(754\) 0 0
\(755\) 180.133i 0.238587i
\(756\) 0 0
\(757\) 1006.00 1.32893 0.664465 0.747319i \(-0.268659\pi\)
0.664465 + 0.747319i \(0.268659\pi\)
\(758\) 0 0
\(759\) − 332.554i − 0.438147i
\(760\) 0 0
\(761\) −758.000 −0.996058 −0.498029 0.867160i \(-0.665943\pi\)
−0.498029 + 0.867160i \(0.665943\pi\)
\(762\) 0 0
\(763\) 318.697i 0.417690i
\(764\) 0 0
\(765\) −60.0000 −0.0784314
\(766\) 0 0
\(767\) 180.133i 0.234854i
\(768\) 0 0
\(769\) 2.00000 0.00260078 0.00130039 0.999999i \(-0.499586\pi\)
0.00130039 + 0.999999i \(0.499586\pi\)
\(770\) 0 0
\(771\) − 439.941i − 0.570611i
\(772\) 0 0
\(773\) −262.000 −0.338939 −0.169470 0.985535i \(-0.554205\pi\)
−0.169470 + 0.985535i \(0.554205\pi\)
\(774\) 0 0
\(775\) 145.492i 0.187732i
\(776\) 0 0
\(777\) 312.000 0.401544
\(778\) 0 0
\(779\) 1205.51i 1.54751i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) − 135.100i − 0.172541i
\(784\) 0 0
\(785\) 428.000 0.545223
\(786\) 0 0
\(787\) − 1447.99i − 1.83989i −0.392046 0.919946i \(-0.628233\pi\)
0.392046 0.919946i \(-0.371767\pi\)
\(788\) 0 0
\(789\) 264.000 0.334601
\(790\) 0 0
\(791\) − 762.102i − 0.963467i
\(792\) 0 0
\(793\) 52.0000 0.0655738
\(794\) 0 0
\(795\) 256.344i 0.322445i
\(796\) 0 0
\(797\) 866.000 1.08657 0.543287 0.839547i \(-0.317179\pi\)
0.543287 + 0.839547i \(0.317179\pi\)
\(798\) 0 0
\(799\) − 692.820i − 0.867109i
\(800\) 0 0
\(801\) −246.000 −0.307116
\(802\) 0 0
\(803\) − 318.697i − 0.396883i
\(804\) 0 0
\(805\) −384.000 −0.477019
\(806\) 0 0
\(807\) − 453.797i − 0.562326i
\(808\) 0 0
\(809\) 10.0000 0.0123609 0.00618047 0.999981i \(-0.498033\pi\)
0.00618047 + 0.999981i \(0.498033\pi\)
\(810\) 0 0
\(811\) − 436.477i − 0.538196i −0.963113 0.269098i \(-0.913274\pi\)
0.963113 0.269098i \(-0.0867255\pi\)
\(812\) 0 0
\(813\) 36.0000 0.0442804
\(814\) 0 0
\(815\) 41.5692i 0.0510052i
\(816\) 0 0
\(817\) 1008.00 1.23378
\(818\) 0 0
\(819\) 41.5692i 0.0507561i
\(820\) 0 0
\(821\) −838.000 −1.02071 −0.510353 0.859965i \(-0.670485\pi\)
−0.510353 + 0.859965i \(0.670485\pi\)
\(822\) 0 0
\(823\) − 879.882i − 1.06912i −0.845132 0.534558i \(-0.820478\pi\)
0.845132 0.534558i \(-0.179522\pi\)
\(824\) 0 0
\(825\) 252.000 0.305455
\(826\) 0 0
\(827\) − 727.461i − 0.879639i −0.898086 0.439819i \(-0.855042\pi\)
0.898086 0.439819i \(-0.144958\pi\)
\(828\) 0 0
\(829\) −1298.00 −1.56574 −0.782871 0.622184i \(-0.786246\pi\)
−0.782871 + 0.622184i \(0.786246\pi\)
\(830\) 0 0
\(831\) − 502.295i − 0.604446i
\(832\) 0 0
\(833\) 10.0000 0.0120048
\(834\) 0 0
\(835\) − 193.990i − 0.232323i
\(836\) 0 0
\(837\) −36.0000 −0.0430108
\(838\) 0 0
\(839\) − 193.990i − 0.231215i −0.993295 0.115608i \(-0.963118\pi\)
0.993295 0.115608i \(-0.0368815\pi\)
\(840\) 0 0
\(841\) −165.000 −0.196195
\(842\) 0 0
\(843\) 391.443i 0.464346i
\(844\) 0 0
\(845\) −330.000 −0.390533
\(846\) 0 0
\(847\) 505.759i 0.597118i
\(848\) 0 0
\(849\) −516.000 −0.607774
\(850\) 0 0
\(851\) − 720.533i − 0.846690i
\(852\) 0 0
\(853\) −506.000 −0.593200 −0.296600 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(854\) 0 0
\(855\) − 124.708i − 0.145857i
\(856\) 0 0
\(857\) −998.000 −1.16453 −0.582264 0.813000i \(-0.697833\pi\)
−0.582264 + 0.813000i \(0.697833\pi\)
\(858\) 0 0
\(859\) 505.759i 0.588776i 0.955686 + 0.294388i \(0.0951158\pi\)
−0.955686 + 0.294388i \(0.904884\pi\)
\(860\) 0 0
\(861\) −696.000 −0.808362
\(862\) 0 0
\(863\) − 166.277i − 0.192673i −0.995349 0.0963365i \(-0.969287\pi\)
0.995349 0.0963365i \(-0.0307125\pi\)
\(864\) 0 0
\(865\) −668.000 −0.772254
\(866\) 0 0
\(867\) − 327.358i − 0.377575i
\(868\) 0 0
\(869\) 816.000 0.939010
\(870\) 0 0
\(871\) − 13.8564i − 0.0159086i
\(872\) 0 0
\(873\) −6.00000 −0.00687285
\(874\) 0 0
\(875\) − 637.395i − 0.728451i
\(876\) 0 0
\(877\) 646.000 0.736602 0.368301 0.929707i \(-0.379940\pi\)
0.368301 + 0.929707i \(0.379940\pi\)
\(878\) 0 0
\(879\) 627.002i 0.713313i
\(880\) 0 0
\(881\) 898.000 1.01930 0.509648 0.860383i \(-0.329776\pi\)
0.509648 + 0.860383i \(0.329776\pi\)
\(882\) 0 0
\(883\) − 727.461i − 0.823852i −0.911217 0.411926i \(-0.864856\pi\)
0.911217 0.411926i \(-0.135144\pi\)
\(884\) 0 0
\(885\) 312.000 0.352542
\(886\) 0 0
\(887\) − 845.241i − 0.952921i −0.879196 0.476460i \(-0.841920\pi\)
0.879196 0.476460i \(-0.158080\pi\)
\(888\) 0 0
\(889\) −1008.00 −1.13386
\(890\) 0 0
\(891\) 62.3538i 0.0699819i
\(892\) 0 0
\(893\) 1440.00 1.61254
\(894\) 0 0
\(895\) − 374.123i − 0.418014i
\(896\) 0 0
\(897\) 96.0000 0.107023
\(898\) 0 0
\(899\) − 180.133i − 0.200371i
\(900\) 0 0
\(901\) 740.000 0.821310
\(902\) 0 0
\(903\) 581.969i 0.644484i
\(904\) 0 0
\(905\) −4.00000 −0.00441989
\(906\) 0 0
\(907\) 1364.86i 1.50480i 0.658705 + 0.752401i \(0.271105\pi\)
−0.658705 + 0.752401i \(0.728895\pi\)
\(908\) 0 0
\(909\) −222.000 −0.244224
\(910\) 0 0
\(911\) 387.979i 0.425883i 0.977065 + 0.212941i \(0.0683044\pi\)
−0.977065 + 0.212941i \(0.931696\pi\)
\(912\) 0 0
\(913\) −336.000 −0.368018
\(914\) 0 0
\(915\) − 90.0666i − 0.0984335i
\(916\) 0 0
\(917\) −816.000 −0.889858
\(918\) 0 0
\(919\) − 602.754i − 0.655880i −0.944699 0.327940i \(-0.893646\pi\)
0.944699 0.327940i \(-0.106354\pi\)
\(920\) 0 0
\(921\) −252.000 −0.273616
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 546.000 0.590270
\(926\) 0 0
\(927\) − 228.631i − 0.246635i
\(928\) 0 0
\(929\) 1594.00 1.71582 0.857912 0.513797i \(-0.171762\pi\)
0.857912 + 0.513797i \(0.171762\pi\)
\(930\) 0 0
\(931\) 20.7846i 0.0223250i
\(932\) 0 0
\(933\) −408.000 −0.437299
\(934\) 0 0
\(935\) 138.564i 0.148197i
\(936\) 0 0
\(937\) 674.000 0.719317 0.359658 0.933084i \(-0.382893\pi\)
0.359658 + 0.933084i \(0.382893\pi\)
\(938\) 0 0
\(939\) − 827.920i − 0.881704i
\(940\) 0 0
\(941\) −430.000 −0.456961 −0.228480 0.973549i \(-0.573376\pi\)
−0.228480 + 0.973549i \(0.573376\pi\)
\(942\) 0 0
\(943\) 1607.34i 1.70450i
\(944\) 0 0
\(945\) 72.0000 0.0761905
\(946\) 0 0
\(947\) − 76.2102i − 0.0804754i −0.999190 0.0402377i \(-0.987188\pi\)
0.999190 0.0402377i \(-0.0128115\pi\)
\(948\) 0 0
\(949\) 92.0000 0.0969442
\(950\) 0 0
\(951\) 294.449i 0.309620i
\(952\) 0 0
\(953\) 730.000 0.766002 0.383001 0.923748i \(-0.374891\pi\)
0.383001 + 0.923748i \(0.374891\pi\)
\(954\) 0 0
\(955\) 443.405i 0.464298i
\(956\) 0 0
\(957\) −312.000 −0.326019
\(958\) 0 0
\(959\) 69.2820i 0.0722440i
\(960\) 0 0
\(961\) 913.000 0.950052
\(962\) 0 0
\(963\) − 62.3538i − 0.0647496i
\(964\) 0 0
\(965\) 580.000 0.601036
\(966\) 0 0
\(967\) − 921.451i − 0.952897i −0.879202 0.476448i \(-0.841924\pi\)
0.879202 0.476448i \(-0.158076\pi\)
\(968\) 0 0
\(969\) −360.000 −0.371517
\(970\) 0 0
\(971\) 1475.71i 1.51978i 0.650051 + 0.759890i \(0.274747\pi\)
−0.650051 + 0.759890i \(0.725253\pi\)
\(972\) 0 0
\(973\) −336.000 −0.345324
\(974\) 0 0
\(975\) 72.7461i 0.0746114i
\(976\) 0 0
\(977\) 346.000 0.354145 0.177073 0.984198i \(-0.443337\pi\)
0.177073 + 0.984198i \(0.443337\pi\)
\(978\) 0 0
\(979\) 568.113i 0.580299i
\(980\) 0 0
\(981\) −138.000 −0.140673
\(982\) 0 0
\(983\) − 734.390i − 0.747090i −0.927612 0.373545i \(-0.878142\pi\)
0.927612 0.373545i \(-0.121858\pi\)
\(984\) 0 0
\(985\) 52.0000 0.0527919
\(986\) 0 0
\(987\) 831.384i 0.842335i
\(988\) 0 0
\(989\) 1344.00 1.35895
\(990\) 0 0
\(991\) − 976.877i − 0.985748i −0.870101 0.492874i \(-0.835946\pi\)
0.870101 0.492874i \(-0.164054\pi\)
\(992\) 0 0
\(993\) −708.000 −0.712991
\(994\) 0 0
\(995\) − 789.815i − 0.793784i
\(996\) 0 0
\(997\) −458.000 −0.459378 −0.229689 0.973264i \(-0.573771\pi\)
−0.229689 + 0.973264i \(0.573771\pi\)
\(998\) 0 0
\(999\) 135.100i 0.135235i
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 192.3.g.b.127.2 2
3.2 odd 2 576.3.g.e.127.2 2
4.3 odd 2 inner 192.3.g.b.127.1 2
8.3 odd 2 12.3.d.a.7.1 2
8.5 even 2 12.3.d.a.7.2 yes 2
12.11 even 2 576.3.g.e.127.1 2
16.3 odd 4 768.3.b.c.127.3 4
16.5 even 4 768.3.b.c.127.4 4
16.11 odd 4 768.3.b.c.127.2 4
16.13 even 4 768.3.b.c.127.1 4
24.5 odd 2 36.3.d.c.19.1 2
24.11 even 2 36.3.d.c.19.2 2
40.3 even 4 300.3.f.a.199.2 4
40.13 odd 4 300.3.f.a.199.4 4
40.19 odd 2 300.3.c.b.151.2 2
40.27 even 4 300.3.f.a.199.3 4
40.29 even 2 300.3.c.b.151.1 2
40.37 odd 4 300.3.f.a.199.1 4
48.5 odd 4 2304.3.b.l.127.1 4
48.11 even 4 2304.3.b.l.127.2 4
48.29 odd 4 2304.3.b.l.127.3 4
48.35 even 4 2304.3.b.l.127.4 4
56.13 odd 2 588.3.g.b.295.2 2
56.27 even 2 588.3.g.b.295.1 2
72.5 odd 6 324.3.f.g.55.1 2
72.11 even 6 324.3.f.g.271.1 2
72.13 even 6 324.3.f.d.55.1 2
72.29 odd 6 324.3.f.a.271.1 2
72.43 odd 6 324.3.f.d.271.1 2
72.59 even 6 324.3.f.a.55.1 2
72.61 even 6 324.3.f.j.271.1 2
72.67 odd 6 324.3.f.j.55.1 2
120.29 odd 2 900.3.c.e.451.2 2
120.53 even 4 900.3.f.c.199.1 4
120.59 even 2 900.3.c.e.451.1 2
120.77 even 4 900.3.f.c.199.4 4
120.83 odd 4 900.3.f.c.199.3 4
120.107 odd 4 900.3.f.c.199.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
12.3.d.a.7.1 2 8.3 odd 2
12.3.d.a.7.2 yes 2 8.5 even 2
36.3.d.c.19.1 2 24.5 odd 2
36.3.d.c.19.2 2 24.11 even 2
192.3.g.b.127.1 2 4.3 odd 2 inner
192.3.g.b.127.2 2 1.1 even 1 trivial
300.3.c.b.151.1 2 40.29 even 2
300.3.c.b.151.2 2 40.19 odd 2
300.3.f.a.199.1 4 40.37 odd 4
300.3.f.a.199.2 4 40.3 even 4
300.3.f.a.199.3 4 40.27 even 4
300.3.f.a.199.4 4 40.13 odd 4
324.3.f.a.55.1 2 72.59 even 6
324.3.f.a.271.1 2 72.29 odd 6
324.3.f.d.55.1 2 72.13 even 6
324.3.f.d.271.1 2 72.43 odd 6
324.3.f.g.55.1 2 72.5 odd 6
324.3.f.g.271.1 2 72.11 even 6
324.3.f.j.55.1 2 72.67 odd 6
324.3.f.j.271.1 2 72.61 even 6
576.3.g.e.127.1 2 12.11 even 2
576.3.g.e.127.2 2 3.2 odd 2
588.3.g.b.295.1 2 56.27 even 2
588.3.g.b.295.2 2 56.13 odd 2
768.3.b.c.127.1 4 16.13 even 4
768.3.b.c.127.2 4 16.11 odd 4
768.3.b.c.127.3 4 16.3 odd 4
768.3.b.c.127.4 4 16.5 even 4
900.3.c.e.451.1 2 120.59 even 2
900.3.c.e.451.2 2 120.29 odd 2
900.3.f.c.199.1 4 120.53 even 4
900.3.f.c.199.2 4 120.107 odd 4
900.3.f.c.199.3 4 120.83 odd 4
900.3.f.c.199.4 4 120.77 even 4
2304.3.b.l.127.1 4 48.5 odd 4
2304.3.b.l.127.2 4 48.11 even 4
2304.3.b.l.127.3 4 48.29 odd 4
2304.3.b.l.127.4 4 48.35 even 4