Properties

Label 1638.2.a.w.1.2
Level $1638$
Weight $2$
Character 1638.1
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
Dimension $2$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{41}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.70156\) of defining polynomial
Character \(\chi\) \(=\) 1638.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} +1.00000 q^{7} -1.00000 q^{8} -3.70156 q^{10} -5.70156 q^{11} +1.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.70156 q^{17} +5.70156 q^{19} +3.70156 q^{20} +5.70156 q^{22} +1.70156 q^{23} +8.70156 q^{25} -1.00000 q^{26} +1.00000 q^{28} +3.70156 q^{29} -1.00000 q^{32} +3.70156 q^{34} +3.70156 q^{35} +4.29844 q^{37} -5.70156 q^{38} -3.70156 q^{40} +9.40312 q^{41} +9.10469 q^{43} -5.70156 q^{44} -1.70156 q^{46} +8.00000 q^{47} +1.00000 q^{49} -8.70156 q^{50} +1.00000 q^{52} +2.00000 q^{53} -21.1047 q^{55} -1.00000 q^{56} -3.70156 q^{58} -10.8062 q^{59} +7.70156 q^{61} +1.00000 q^{64} +3.70156 q^{65} -7.40312 q^{67} -3.70156 q^{68} -3.70156 q^{70} +8.00000 q^{71} -7.70156 q^{73} -4.29844 q^{74} +5.70156 q^{76} -5.70156 q^{77} -3.40312 q^{79} +3.70156 q^{80} -9.40312 q^{82} +0.596876 q^{83} -13.7016 q^{85} -9.10469 q^{86} +5.70156 q^{88} -16.8062 q^{89} +1.00000 q^{91} +1.70156 q^{92} -8.00000 q^{94} +21.1047 q^{95} +16.8062 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} + q^{5} + 2 q^{7} - 2 q^{8} - q^{10} - 5 q^{11} + 2 q^{13} - 2 q^{14} + 2 q^{16} - q^{17} + 5 q^{19} + q^{20} + 5 q^{22} - 3 q^{23} + 11 q^{25} - 2 q^{26} + 2 q^{28} + q^{29} - 2 q^{32} + q^{34} + q^{35} + 15 q^{37} - 5 q^{38} - q^{40} + 6 q^{41} - q^{43} - 5 q^{44} + 3 q^{46} + 16 q^{47} + 2 q^{49} - 11 q^{50} + 2 q^{52} + 4 q^{53} - 23 q^{55} - 2 q^{56} - q^{58} + 4 q^{59} + 9 q^{61} + 2 q^{64} + q^{65} - 2 q^{67} - q^{68} - q^{70} + 16 q^{71} - 9 q^{73} - 15 q^{74} + 5 q^{76} - 5 q^{77} + 6 q^{79} + q^{80} - 6 q^{82} + 14 q^{83} - 21 q^{85} + q^{86} + 5 q^{88} - 8 q^{89} + 2 q^{91} - 3 q^{92} - 16 q^{94} + 23 q^{95} + 8 q^{97} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 3.70156 1.65539 0.827694 0.561179i \(-0.189652\pi\)
0.827694 + 0.561179i \(0.189652\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) −3.70156 −1.17054
\(11\) −5.70156 −1.71909 −0.859543 0.511064i \(-0.829252\pi\)
−0.859543 + 0.511064i \(0.829252\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.70156 −0.897761 −0.448880 0.893592i \(-0.648177\pi\)
−0.448880 + 0.893592i \(0.648177\pi\)
\(18\) 0 0
\(19\) 5.70156 1.30803 0.654014 0.756482i \(-0.273084\pi\)
0.654014 + 0.756482i \(0.273084\pi\)
\(20\) 3.70156 0.827694
\(21\) 0 0
\(22\) 5.70156 1.21558
\(23\) 1.70156 0.354800 0.177400 0.984139i \(-0.443231\pi\)
0.177400 + 0.984139i \(0.443231\pi\)
\(24\) 0 0
\(25\) 8.70156 1.74031
\(26\) −1.00000 −0.196116
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 3.70156 0.687363 0.343681 0.939086i \(-0.388326\pi\)
0.343681 + 0.939086i \(0.388326\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 3.70156 0.634813
\(35\) 3.70156 0.625678
\(36\) 0 0
\(37\) 4.29844 0.706659 0.353329 0.935499i \(-0.385049\pi\)
0.353329 + 0.935499i \(0.385049\pi\)
\(38\) −5.70156 −0.924916
\(39\) 0 0
\(40\) −3.70156 −0.585268
\(41\) 9.40312 1.46852 0.734261 0.678868i \(-0.237529\pi\)
0.734261 + 0.678868i \(0.237529\pi\)
\(42\) 0 0
\(43\) 9.10469 1.38845 0.694226 0.719757i \(-0.255747\pi\)
0.694226 + 0.719757i \(0.255747\pi\)
\(44\) −5.70156 −0.859543
\(45\) 0 0
\(46\) −1.70156 −0.250882
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −8.70156 −1.23059
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) −21.1047 −2.84576
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −3.70156 −0.486039
\(59\) −10.8062 −1.40685 −0.703427 0.710768i \(-0.748348\pi\)
−0.703427 + 0.710768i \(0.748348\pi\)
\(60\) 0 0
\(61\) 7.70156 0.986084 0.493042 0.870006i \(-0.335885\pi\)
0.493042 + 0.870006i \(0.335885\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.70156 0.459122
\(66\) 0 0
\(67\) −7.40312 −0.904436 −0.452218 0.891908i \(-0.649367\pi\)
−0.452218 + 0.891908i \(0.649367\pi\)
\(68\) −3.70156 −0.448880
\(69\) 0 0
\(70\) −3.70156 −0.442421
\(71\) 8.00000 0.949425 0.474713 0.880141i \(-0.342552\pi\)
0.474713 + 0.880141i \(0.342552\pi\)
\(72\) 0 0
\(73\) −7.70156 −0.901400 −0.450700 0.892676i \(-0.648826\pi\)
−0.450700 + 0.892676i \(0.648826\pi\)
\(74\) −4.29844 −0.499683
\(75\) 0 0
\(76\) 5.70156 0.654014
\(77\) −5.70156 −0.649753
\(78\) 0 0
\(79\) −3.40312 −0.382881 −0.191441 0.981504i \(-0.561316\pi\)
−0.191441 + 0.981504i \(0.561316\pi\)
\(80\) 3.70156 0.413847
\(81\) 0 0
\(82\) −9.40312 −1.03840
\(83\) 0.596876 0.0655156 0.0327578 0.999463i \(-0.489571\pi\)
0.0327578 + 0.999463i \(0.489571\pi\)
\(84\) 0 0
\(85\) −13.7016 −1.48614
\(86\) −9.10469 −0.981784
\(87\) 0 0
\(88\) 5.70156 0.607789
\(89\) −16.8062 −1.78146 −0.890729 0.454534i \(-0.849806\pi\)
−0.890729 + 0.454534i \(0.849806\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 1.70156 0.177400
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 21.1047 2.16530
\(96\) 0 0
\(97\) 16.8062 1.70642 0.853208 0.521571i \(-0.174654\pi\)
0.853208 + 0.521571i \(0.174654\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 8.70156 0.870156
\(101\) −9.40312 −0.935646 −0.467823 0.883822i \(-0.654961\pi\)
−0.467823 + 0.883822i \(0.654961\pi\)
\(102\) 0 0
\(103\) −9.70156 −0.955923 −0.477962 0.878381i \(-0.658624\pi\)
−0.477962 + 0.878381i \(0.658624\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) −7.40312 −0.715687 −0.357844 0.933782i \(-0.616488\pi\)
−0.357844 + 0.933782i \(0.616488\pi\)
\(108\) 0 0
\(109\) 15.7016 1.50394 0.751968 0.659199i \(-0.229105\pi\)
0.751968 + 0.659199i \(0.229105\pi\)
\(110\) 21.1047 2.01225
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) 20.8062 1.95729 0.978644 0.205564i \(-0.0659030\pi\)
0.978644 + 0.205564i \(0.0659030\pi\)
\(114\) 0 0
\(115\) 6.29844 0.587332
\(116\) 3.70156 0.343681
\(117\) 0 0
\(118\) 10.8062 0.994796
\(119\) −3.70156 −0.339322
\(120\) 0 0
\(121\) 21.5078 1.95526
\(122\) −7.70156 −0.697267
\(123\) 0 0
\(124\) 0 0
\(125\) 13.7016 1.22550
\(126\) 0 0
\(127\) −19.4031 −1.72175 −0.860874 0.508817i \(-0.830083\pi\)
−0.860874 + 0.508817i \(0.830083\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.70156 −0.324648
\(131\) 13.7016 1.19711 0.598556 0.801081i \(-0.295742\pi\)
0.598556 + 0.801081i \(0.295742\pi\)
\(132\) 0 0
\(133\) 5.70156 0.494388
\(134\) 7.40312 0.639533
\(135\) 0 0
\(136\) 3.70156 0.317406
\(137\) 12.2984 1.05073 0.525363 0.850878i \(-0.323929\pi\)
0.525363 + 0.850878i \(0.323929\pi\)
\(138\) 0 0
\(139\) −18.8062 −1.59513 −0.797563 0.603236i \(-0.793878\pi\)
−0.797563 + 0.603236i \(0.793878\pi\)
\(140\) 3.70156 0.312839
\(141\) 0 0
\(142\) −8.00000 −0.671345
\(143\) −5.70156 −0.476789
\(144\) 0 0
\(145\) 13.7016 1.13785
\(146\) 7.70156 0.637386
\(147\) 0 0
\(148\) 4.29844 0.353329
\(149\) −2.59688 −0.212744 −0.106372 0.994326i \(-0.533923\pi\)
−0.106372 + 0.994326i \(0.533923\pi\)
\(150\) 0 0
\(151\) 1.70156 0.138471 0.0692356 0.997600i \(-0.477944\pi\)
0.0692356 + 0.997600i \(0.477944\pi\)
\(152\) −5.70156 −0.462458
\(153\) 0 0
\(154\) 5.70156 0.459445
\(155\) 0 0
\(156\) 0 0
\(157\) 0.895314 0.0714538 0.0357269 0.999362i \(-0.488625\pi\)
0.0357269 + 0.999362i \(0.488625\pi\)
\(158\) 3.40312 0.270738
\(159\) 0 0
\(160\) −3.70156 −0.292634
\(161\) 1.70156 0.134102
\(162\) 0 0
\(163\) −7.40312 −0.579857 −0.289929 0.957048i \(-0.593632\pi\)
−0.289929 + 0.957048i \(0.593632\pi\)
\(164\) 9.40312 0.734261
\(165\) 0 0
\(166\) −0.596876 −0.0463265
\(167\) −9.70156 −0.750729 −0.375365 0.926877i \(-0.622483\pi\)
−0.375365 + 0.926877i \(0.622483\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 13.7016 1.05086
\(171\) 0 0
\(172\) 9.10469 0.694226
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 8.70156 0.657776
\(176\) −5.70156 −0.429771
\(177\) 0 0
\(178\) 16.8062 1.25968
\(179\) 10.8062 0.807697 0.403848 0.914826i \(-0.367672\pi\)
0.403848 + 0.914826i \(0.367672\pi\)
\(180\) 0 0
\(181\) −16.8062 −1.24920 −0.624599 0.780945i \(-0.714738\pi\)
−0.624599 + 0.780945i \(0.714738\pi\)
\(182\) −1.00000 −0.0741249
\(183\) 0 0
\(184\) −1.70156 −0.125441
\(185\) 15.9109 1.16980
\(186\) 0 0
\(187\) 21.1047 1.54333
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) −21.1047 −1.53109
\(191\) −6.29844 −0.455739 −0.227869 0.973692i \(-0.573176\pi\)
−0.227869 + 0.973692i \(0.573176\pi\)
\(192\) 0 0
\(193\) −1.40312 −0.100999 −0.0504995 0.998724i \(-0.516081\pi\)
−0.0504995 + 0.998724i \(0.516081\pi\)
\(194\) −16.8062 −1.20662
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −9.40312 −0.669945 −0.334972 0.942228i \(-0.608727\pi\)
−0.334972 + 0.942228i \(0.608727\pi\)
\(198\) 0 0
\(199\) 2.89531 0.205243 0.102622 0.994720i \(-0.467277\pi\)
0.102622 + 0.994720i \(0.467277\pi\)
\(200\) −8.70156 −0.615293
\(201\) 0 0
\(202\) 9.40312 0.661602
\(203\) 3.70156 0.259799
\(204\) 0 0
\(205\) 34.8062 2.43097
\(206\) 9.70156 0.675940
\(207\) 0 0
\(208\) 1.00000 0.0693375
\(209\) −32.5078 −2.24861
\(210\) 0 0
\(211\) −18.2984 −1.25972 −0.629858 0.776710i \(-0.716887\pi\)
−0.629858 + 0.776710i \(0.716887\pi\)
\(212\) 2.00000 0.137361
\(213\) 0 0
\(214\) 7.40312 0.506067
\(215\) 33.7016 2.29843
\(216\) 0 0
\(217\) 0 0
\(218\) −15.7016 −1.06344
\(219\) 0 0
\(220\) −21.1047 −1.42288
\(221\) −3.70156 −0.248994
\(222\) 0 0
\(223\) −3.40312 −0.227890 −0.113945 0.993487i \(-0.536349\pi\)
−0.113945 + 0.993487i \(0.536349\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −20.8062 −1.38401
\(227\) −8.59688 −0.570595 −0.285297 0.958439i \(-0.592092\pi\)
−0.285297 + 0.958439i \(0.592092\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) −6.29844 −0.415307
\(231\) 0 0
\(232\) −3.70156 −0.243019
\(233\) −20.2094 −1.32396 −0.661980 0.749521i \(-0.730284\pi\)
−0.661980 + 0.749521i \(0.730284\pi\)
\(234\) 0 0
\(235\) 29.6125 1.93171
\(236\) −10.8062 −0.703427
\(237\) 0 0
\(238\) 3.70156 0.239937
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 0.806248 0.0519350 0.0259675 0.999663i \(-0.491733\pi\)
0.0259675 + 0.999663i \(0.491733\pi\)
\(242\) −21.5078 −1.38257
\(243\) 0 0
\(244\) 7.70156 0.493042
\(245\) 3.70156 0.236484
\(246\) 0 0
\(247\) 5.70156 0.362782
\(248\) 0 0
\(249\) 0 0
\(250\) −13.7016 −0.866563
\(251\) 28.5078 1.79940 0.899699 0.436512i \(-0.143786\pi\)
0.899699 + 0.436512i \(0.143786\pi\)
\(252\) 0 0
\(253\) −9.70156 −0.609932
\(254\) 19.4031 1.21746
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 4.29844 0.267092
\(260\) 3.70156 0.229561
\(261\) 0 0
\(262\) −13.7016 −0.846485
\(263\) 1.19375 0.0736099 0.0368049 0.999322i \(-0.488282\pi\)
0.0368049 + 0.999322i \(0.488282\pi\)
\(264\) 0 0
\(265\) 7.40312 0.454770
\(266\) −5.70156 −0.349585
\(267\) 0 0
\(268\) −7.40312 −0.452218
\(269\) −20.8062 −1.26858 −0.634290 0.773095i \(-0.718707\pi\)
−0.634290 + 0.773095i \(0.718707\pi\)
\(270\) 0 0
\(271\) −26.2094 −1.59211 −0.796053 0.605227i \(-0.793082\pi\)
−0.796053 + 0.605227i \(0.793082\pi\)
\(272\) −3.70156 −0.224440
\(273\) 0 0
\(274\) −12.2984 −0.742976
\(275\) −49.6125 −2.99175
\(276\) 0 0
\(277\) −23.6125 −1.41874 −0.709369 0.704838i \(-0.751020\pi\)
−0.709369 + 0.704838i \(0.751020\pi\)
\(278\) 18.8062 1.12792
\(279\) 0 0
\(280\) −3.70156 −0.221211
\(281\) 12.8062 0.763957 0.381978 0.924171i \(-0.375243\pi\)
0.381978 + 0.924171i \(0.375243\pi\)
\(282\) 0 0
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) 8.00000 0.474713
\(285\) 0 0
\(286\) 5.70156 0.337140
\(287\) 9.40312 0.555049
\(288\) 0 0
\(289\) −3.29844 −0.194026
\(290\) −13.7016 −0.804583
\(291\) 0 0
\(292\) −7.70156 −0.450700
\(293\) −12.8062 −0.748149 −0.374075 0.927399i \(-0.622040\pi\)
−0.374075 + 0.927399i \(0.622040\pi\)
\(294\) 0 0
\(295\) −40.0000 −2.32889
\(296\) −4.29844 −0.249842
\(297\) 0 0
\(298\) 2.59688 0.150433
\(299\) 1.70156 0.0984039
\(300\) 0 0
\(301\) 9.10469 0.524785
\(302\) −1.70156 −0.0979139
\(303\) 0 0
\(304\) 5.70156 0.327007
\(305\) 28.5078 1.63235
\(306\) 0 0
\(307\) 18.8062 1.07333 0.536665 0.843796i \(-0.319684\pi\)
0.536665 + 0.843796i \(0.319684\pi\)
\(308\) −5.70156 −0.324877
\(309\) 0 0
\(310\) 0 0
\(311\) −27.4031 −1.55389 −0.776944 0.629569i \(-0.783231\pi\)
−0.776944 + 0.629569i \(0.783231\pi\)
\(312\) 0 0
\(313\) −16.2094 −0.916208 −0.458104 0.888899i \(-0.651471\pi\)
−0.458104 + 0.888899i \(0.651471\pi\)
\(314\) −0.895314 −0.0505255
\(315\) 0 0
\(316\) −3.40312 −0.191441
\(317\) 5.40312 0.303470 0.151735 0.988421i \(-0.451514\pi\)
0.151735 + 0.988421i \(0.451514\pi\)
\(318\) 0 0
\(319\) −21.1047 −1.18164
\(320\) 3.70156 0.206924
\(321\) 0 0
\(322\) −1.70156 −0.0948243
\(323\) −21.1047 −1.17430
\(324\) 0 0
\(325\) 8.70156 0.482676
\(326\) 7.40312 0.410021
\(327\) 0 0
\(328\) −9.40312 −0.519201
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −6.20937 −0.341298 −0.170649 0.985332i \(-0.554586\pi\)
−0.170649 + 0.985332i \(0.554586\pi\)
\(332\) 0.596876 0.0327578
\(333\) 0 0
\(334\) 9.70156 0.530846
\(335\) −27.4031 −1.49719
\(336\) 0 0
\(337\) 2.50781 0.136609 0.0683046 0.997665i \(-0.478241\pi\)
0.0683046 + 0.997665i \(0.478241\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 0 0
\(340\) −13.7016 −0.743072
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −9.10469 −0.490892
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 30.0000 1.60586 0.802932 0.596071i \(-0.203272\pi\)
0.802932 + 0.596071i \(0.203272\pi\)
\(350\) −8.70156 −0.465118
\(351\) 0 0
\(352\) 5.70156 0.303894
\(353\) 4.80625 0.255811 0.127905 0.991786i \(-0.459175\pi\)
0.127905 + 0.991786i \(0.459175\pi\)
\(354\) 0 0
\(355\) 29.6125 1.57167
\(356\) −16.8062 −0.890729
\(357\) 0 0
\(358\) −10.8062 −0.571128
\(359\) −1.19375 −0.0630038 −0.0315019 0.999504i \(-0.510029\pi\)
−0.0315019 + 0.999504i \(0.510029\pi\)
\(360\) 0 0
\(361\) 13.5078 0.710937
\(362\) 16.8062 0.883317
\(363\) 0 0
\(364\) 1.00000 0.0524142
\(365\) −28.5078 −1.49217
\(366\) 0 0
\(367\) 21.6125 1.12816 0.564082 0.825719i \(-0.309230\pi\)
0.564082 + 0.825719i \(0.309230\pi\)
\(368\) 1.70156 0.0887001
\(369\) 0 0
\(370\) −15.9109 −0.827170
\(371\) 2.00000 0.103835
\(372\) 0 0
\(373\) −6.59688 −0.341573 −0.170787 0.985308i \(-0.554631\pi\)
−0.170787 + 0.985308i \(0.554631\pi\)
\(374\) −21.1047 −1.09130
\(375\) 0 0
\(376\) −8.00000 −0.412568
\(377\) 3.70156 0.190640
\(378\) 0 0
\(379\) 33.6125 1.72656 0.863279 0.504727i \(-0.168407\pi\)
0.863279 + 0.504727i \(0.168407\pi\)
\(380\) 21.1047 1.08265
\(381\) 0 0
\(382\) 6.29844 0.322256
\(383\) 27.9109 1.42618 0.713091 0.701071i \(-0.247295\pi\)
0.713091 + 0.701071i \(0.247295\pi\)
\(384\) 0 0
\(385\) −21.1047 −1.07559
\(386\) 1.40312 0.0714171
\(387\) 0 0
\(388\) 16.8062 0.853208
\(389\) −14.0000 −0.709828 −0.354914 0.934899i \(-0.615490\pi\)
−0.354914 + 0.934899i \(0.615490\pi\)
\(390\) 0 0
\(391\) −6.29844 −0.318526
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 9.40312 0.473723
\(395\) −12.5969 −0.633818
\(396\) 0 0
\(397\) −24.8062 −1.24499 −0.622495 0.782624i \(-0.713881\pi\)
−0.622495 + 0.782624i \(0.713881\pi\)
\(398\) −2.89531 −0.145129
\(399\) 0 0
\(400\) 8.70156 0.435078
\(401\) 20.8062 1.03901 0.519507 0.854466i \(-0.326116\pi\)
0.519507 + 0.854466i \(0.326116\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −9.40312 −0.467823
\(405\) 0 0
\(406\) −3.70156 −0.183705
\(407\) −24.5078 −1.21481
\(408\) 0 0
\(409\) 11.7016 0.578605 0.289303 0.957238i \(-0.406577\pi\)
0.289303 + 0.957238i \(0.406577\pi\)
\(410\) −34.8062 −1.71896
\(411\) 0 0
\(412\) −9.70156 −0.477962
\(413\) −10.8062 −0.531741
\(414\) 0 0
\(415\) 2.20937 0.108454
\(416\) −1.00000 −0.0490290
\(417\) 0 0
\(418\) 32.5078 1.59001
\(419\) −14.8953 −0.727684 −0.363842 0.931461i \(-0.618535\pi\)
−0.363842 + 0.931461i \(0.618535\pi\)
\(420\) 0 0
\(421\) −31.6125 −1.54070 −0.770349 0.637622i \(-0.779918\pi\)
−0.770349 + 0.637622i \(0.779918\pi\)
\(422\) 18.2984 0.890754
\(423\) 0 0
\(424\) −2.00000 −0.0971286
\(425\) −32.2094 −1.56238
\(426\) 0 0
\(427\) 7.70156 0.372705
\(428\) −7.40312 −0.357844
\(429\) 0 0
\(430\) −33.7016 −1.62523
\(431\) −26.2094 −1.26246 −0.631231 0.775595i \(-0.717450\pi\)
−0.631231 + 0.775595i \(0.717450\pi\)
\(432\) 0 0
\(433\) −24.2094 −1.16343 −0.581714 0.813393i \(-0.697618\pi\)
−0.581714 + 0.813393i \(0.697618\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 15.7016 0.751968
\(437\) 9.70156 0.464089
\(438\) 0 0
\(439\) −32.5078 −1.55151 −0.775757 0.631032i \(-0.782632\pi\)
−0.775757 + 0.631032i \(0.782632\pi\)
\(440\) 21.1047 1.00613
\(441\) 0 0
\(442\) 3.70156 0.176065
\(443\) 38.2094 1.81538 0.907691 0.419639i \(-0.137843\pi\)
0.907691 + 0.419639i \(0.137843\pi\)
\(444\) 0 0
\(445\) −62.2094 −2.94901
\(446\) 3.40312 0.161143
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 27.1047 1.27915 0.639575 0.768729i \(-0.279111\pi\)
0.639575 + 0.768729i \(0.279111\pi\)
\(450\) 0 0
\(451\) −53.6125 −2.52451
\(452\) 20.8062 0.978644
\(453\) 0 0
\(454\) 8.59688 0.403471
\(455\) 3.70156 0.173532
\(456\) 0 0
\(457\) 13.4031 0.626972 0.313486 0.949593i \(-0.398503\pi\)
0.313486 + 0.949593i \(0.398503\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 6.29844 0.293666
\(461\) −37.3141 −1.73789 −0.868944 0.494910i \(-0.835201\pi\)
−0.868944 + 0.494910i \(0.835201\pi\)
\(462\) 0 0
\(463\) 23.3141 1.08350 0.541748 0.840541i \(-0.317763\pi\)
0.541748 + 0.840541i \(0.317763\pi\)
\(464\) 3.70156 0.171841
\(465\) 0 0
\(466\) 20.2094 0.936181
\(467\) 36.5078 1.68938 0.844690 0.535256i \(-0.179785\pi\)
0.844690 + 0.535256i \(0.179785\pi\)
\(468\) 0 0
\(469\) −7.40312 −0.341845
\(470\) −29.6125 −1.36592
\(471\) 0 0
\(472\) 10.8062 0.497398
\(473\) −51.9109 −2.38687
\(474\) 0 0
\(475\) 49.6125 2.27638
\(476\) −3.70156 −0.169661
\(477\) 0 0
\(478\) 16.0000 0.731823
\(479\) 24.5078 1.11979 0.559895 0.828563i \(-0.310841\pi\)
0.559895 + 0.828563i \(0.310841\pi\)
\(480\) 0 0
\(481\) 4.29844 0.195992
\(482\) −0.806248 −0.0367236
\(483\) 0 0
\(484\) 21.5078 0.977628
\(485\) 62.2094 2.82478
\(486\) 0 0
\(487\) −8.00000 −0.362515 −0.181257 0.983436i \(-0.558017\pi\)
−0.181257 + 0.983436i \(0.558017\pi\)
\(488\) −7.70156 −0.348633
\(489\) 0 0
\(490\) −3.70156 −0.167220
\(491\) −39.4031 −1.77824 −0.889119 0.457676i \(-0.848682\pi\)
−0.889119 + 0.457676i \(0.848682\pi\)
\(492\) 0 0
\(493\) −13.7016 −0.617087
\(494\) −5.70156 −0.256525
\(495\) 0 0
\(496\) 0 0
\(497\) 8.00000 0.358849
\(498\) 0 0
\(499\) −26.8062 −1.20001 −0.600006 0.799995i \(-0.704835\pi\)
−0.600006 + 0.799995i \(0.704835\pi\)
\(500\) 13.7016 0.612752
\(501\) 0 0
\(502\) −28.5078 −1.27237
\(503\) 20.5969 0.918369 0.459185 0.888341i \(-0.348142\pi\)
0.459185 + 0.888341i \(0.348142\pi\)
\(504\) 0 0
\(505\) −34.8062 −1.54886
\(506\) 9.70156 0.431287
\(507\) 0 0
\(508\) −19.4031 −0.860874
\(509\) 5.91093 0.261998 0.130999 0.991383i \(-0.458182\pi\)
0.130999 + 0.991383i \(0.458182\pi\)
\(510\) 0 0
\(511\) −7.70156 −0.340697
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −6.00000 −0.264649
\(515\) −35.9109 −1.58242
\(516\) 0 0
\(517\) −45.6125 −2.00604
\(518\) −4.29844 −0.188863
\(519\) 0 0
\(520\) −3.70156 −0.162324
\(521\) 7.70156 0.337412 0.168706 0.985666i \(-0.446041\pi\)
0.168706 + 0.985666i \(0.446041\pi\)
\(522\) 0 0
\(523\) 17.6125 0.770141 0.385070 0.922887i \(-0.374177\pi\)
0.385070 + 0.922887i \(0.374177\pi\)
\(524\) 13.7016 0.598556
\(525\) 0 0
\(526\) −1.19375 −0.0520500
\(527\) 0 0
\(528\) 0 0
\(529\) −20.1047 −0.874117
\(530\) −7.40312 −0.321571
\(531\) 0 0
\(532\) 5.70156 0.247194
\(533\) 9.40312 0.407295
\(534\) 0 0
\(535\) −27.4031 −1.18474
\(536\) 7.40312 0.319766
\(537\) 0 0
\(538\) 20.8062 0.897021
\(539\) −5.70156 −0.245584
\(540\) 0 0
\(541\) −0.298438 −0.0128308 −0.00641542 0.999979i \(-0.502042\pi\)
−0.00641542 + 0.999979i \(0.502042\pi\)
\(542\) 26.2094 1.12579
\(543\) 0 0
\(544\) 3.70156 0.158703
\(545\) 58.1203 2.48960
\(546\) 0 0
\(547\) −41.6125 −1.77922 −0.889611 0.456719i \(-0.849024\pi\)
−0.889611 + 0.456719i \(0.849024\pi\)
\(548\) 12.2984 0.525363
\(549\) 0 0
\(550\) 49.6125 2.11548
\(551\) 21.1047 0.899090
\(552\) 0 0
\(553\) −3.40312 −0.144716
\(554\) 23.6125 1.00320
\(555\) 0 0
\(556\) −18.8062 −0.797563
\(557\) 37.4031 1.58482 0.792411 0.609988i \(-0.208826\pi\)
0.792411 + 0.609988i \(0.208826\pi\)
\(558\) 0 0
\(559\) 9.10469 0.385087
\(560\) 3.70156 0.156420
\(561\) 0 0
\(562\) −12.8062 −0.540199
\(563\) −2.29844 −0.0968676 −0.0484338 0.998826i \(-0.515423\pi\)
−0.0484338 + 0.998826i \(0.515423\pi\)
\(564\) 0 0
\(565\) 77.0156 3.24007
\(566\) −4.00000 −0.168133
\(567\) 0 0
\(568\) −8.00000 −0.335673
\(569\) −26.0000 −1.08998 −0.544988 0.838444i \(-0.683466\pi\)
−0.544988 + 0.838444i \(0.683466\pi\)
\(570\) 0 0
\(571\) −26.8062 −1.12181 −0.560903 0.827881i \(-0.689546\pi\)
−0.560903 + 0.827881i \(0.689546\pi\)
\(572\) −5.70156 −0.238394
\(573\) 0 0
\(574\) −9.40312 −0.392479
\(575\) 14.8062 0.617463
\(576\) 0 0
\(577\) 16.8062 0.699653 0.349827 0.936814i \(-0.386240\pi\)
0.349827 + 0.936814i \(0.386240\pi\)
\(578\) 3.29844 0.137197
\(579\) 0 0
\(580\) 13.7016 0.568926
\(581\) 0.596876 0.0247626
\(582\) 0 0
\(583\) −11.4031 −0.472269
\(584\) 7.70156 0.318693
\(585\) 0 0
\(586\) 12.8062 0.529021
\(587\) 29.0156 1.19760 0.598801 0.800898i \(-0.295644\pi\)
0.598801 + 0.800898i \(0.295644\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 40.0000 1.64677
\(591\) 0 0
\(592\) 4.29844 0.176665
\(593\) −8.80625 −0.361629 −0.180815 0.983517i \(-0.557873\pi\)
−0.180815 + 0.983517i \(0.557873\pi\)
\(594\) 0 0
\(595\) −13.7016 −0.561709
\(596\) −2.59688 −0.106372
\(597\) 0 0
\(598\) −1.70156 −0.0695820
\(599\) −17.7016 −0.723266 −0.361633 0.932320i \(-0.617781\pi\)
−0.361633 + 0.932320i \(0.617781\pi\)
\(600\) 0 0
\(601\) −2.59688 −0.105929 −0.0529644 0.998596i \(-0.516867\pi\)
−0.0529644 + 0.998596i \(0.516867\pi\)
\(602\) −9.10469 −0.371079
\(603\) 0 0
\(604\) 1.70156 0.0692356
\(605\) 79.6125 3.23671
\(606\) 0 0
\(607\) −24.5078 −0.994741 −0.497371 0.867538i \(-0.665701\pi\)
−0.497371 + 0.867538i \(0.665701\pi\)
\(608\) −5.70156 −0.231229
\(609\) 0 0
\(610\) −28.5078 −1.15425
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −5.91093 −0.238740 −0.119370 0.992850i \(-0.538088\pi\)
−0.119370 + 0.992850i \(0.538088\pi\)
\(614\) −18.8062 −0.758958
\(615\) 0 0
\(616\) 5.70156 0.229722
\(617\) −13.9109 −0.560033 −0.280017 0.959995i \(-0.590340\pi\)
−0.280017 + 0.959995i \(0.590340\pi\)
\(618\) 0 0
\(619\) 13.7016 0.550712 0.275356 0.961342i \(-0.411204\pi\)
0.275356 + 0.961342i \(0.411204\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 27.4031 1.09877
\(623\) −16.8062 −0.673328
\(624\) 0 0
\(625\) 7.20937 0.288375
\(626\) 16.2094 0.647857
\(627\) 0 0
\(628\) 0.895314 0.0357269
\(629\) −15.9109 −0.634411
\(630\) 0 0
\(631\) 24.5078 0.975641 0.487820 0.872944i \(-0.337792\pi\)
0.487820 + 0.872944i \(0.337792\pi\)
\(632\) 3.40312 0.135369
\(633\) 0 0
\(634\) −5.40312 −0.214585
\(635\) −71.8219 −2.85016
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) 21.1047 0.835543
\(639\) 0 0
\(640\) −3.70156 −0.146317
\(641\) 23.1938 0.916098 0.458049 0.888927i \(-0.348548\pi\)
0.458049 + 0.888927i \(0.348548\pi\)
\(642\) 0 0
\(643\) −11.3141 −0.446183 −0.223091 0.974798i \(-0.571615\pi\)
−0.223091 + 0.974798i \(0.571615\pi\)
\(644\) 1.70156 0.0670509
\(645\) 0 0
\(646\) 21.1047 0.830353
\(647\) 8.00000 0.314512 0.157256 0.987558i \(-0.449735\pi\)
0.157256 + 0.987558i \(0.449735\pi\)
\(648\) 0 0
\(649\) 61.6125 2.41850
\(650\) −8.70156 −0.341303
\(651\) 0 0
\(652\) −7.40312 −0.289929
\(653\) −22.5078 −0.880799 −0.440399 0.897802i \(-0.645163\pi\)
−0.440399 + 0.897802i \(0.645163\pi\)
\(654\) 0 0
\(655\) 50.7172 1.98168
\(656\) 9.40312 0.367130
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) −29.0156 −1.13029 −0.565144 0.824992i \(-0.691179\pi\)
−0.565144 + 0.824992i \(0.691179\pi\)
\(660\) 0 0
\(661\) 26.4187 1.02757 0.513785 0.857919i \(-0.328243\pi\)
0.513785 + 0.857919i \(0.328243\pi\)
\(662\) 6.20937 0.241334
\(663\) 0 0
\(664\) −0.596876 −0.0231633
\(665\) 21.1047 0.818405
\(666\) 0 0
\(667\) 6.29844 0.243876
\(668\) −9.70156 −0.375365
\(669\) 0 0
\(670\) 27.4031 1.05868
\(671\) −43.9109 −1.69516
\(672\) 0 0
\(673\) −1.91093 −0.0736611 −0.0368306 0.999322i \(-0.511726\pi\)
−0.0368306 + 0.999322i \(0.511726\pi\)
\(674\) −2.50781 −0.0965973
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) −6.00000 −0.230599 −0.115299 0.993331i \(-0.536783\pi\)
−0.115299 + 0.993331i \(0.536783\pi\)
\(678\) 0 0
\(679\) 16.8062 0.644965
\(680\) 13.7016 0.525431
\(681\) 0 0
\(682\) 0 0
\(683\) 20.5078 0.784710 0.392355 0.919814i \(-0.371661\pi\)
0.392355 + 0.919814i \(0.371661\pi\)
\(684\) 0 0
\(685\) 45.5234 1.73936
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 9.10469 0.347113
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −1.61250 −0.0613423 −0.0306711 0.999530i \(-0.509764\pi\)
−0.0306711 + 0.999530i \(0.509764\pi\)
\(692\) 18.0000 0.684257
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) −69.6125 −2.64055
\(696\) 0 0
\(697\) −34.8062 −1.31838
\(698\) −30.0000 −1.13552
\(699\) 0 0
\(700\) 8.70156 0.328888
\(701\) −28.8062 −1.08800 −0.543998 0.839086i \(-0.683090\pi\)
−0.543998 + 0.839086i \(0.683090\pi\)
\(702\) 0 0
\(703\) 24.5078 0.924330
\(704\) −5.70156 −0.214886
\(705\) 0 0
\(706\) −4.80625 −0.180886
\(707\) −9.40312 −0.353641
\(708\) 0 0
\(709\) 0.387503 0.0145530 0.00727649 0.999974i \(-0.497684\pi\)
0.00727649 + 0.999974i \(0.497684\pi\)
\(710\) −29.6125 −1.11134
\(711\) 0 0
\(712\) 16.8062 0.629841
\(713\) 0 0
\(714\) 0 0
\(715\) −21.1047 −0.789271
\(716\) 10.8062 0.403848
\(717\) 0 0
\(718\) 1.19375 0.0445504
\(719\) −33.0156 −1.23127 −0.615637 0.788030i \(-0.711101\pi\)
−0.615637 + 0.788030i \(0.711101\pi\)
\(720\) 0 0
\(721\) −9.70156 −0.361305
\(722\) −13.5078 −0.502709
\(723\) 0 0
\(724\) −16.8062 −0.624599
\(725\) 32.2094 1.19623
\(726\) 0 0
\(727\) −2.89531 −0.107381 −0.0536906 0.998558i \(-0.517098\pi\)
−0.0536906 + 0.998558i \(0.517098\pi\)
\(728\) −1.00000 −0.0370625
\(729\) 0 0
\(730\) 28.5078 1.05512
\(731\) −33.7016 −1.24650
\(732\) 0 0
\(733\) 33.4031 1.23377 0.616886 0.787052i \(-0.288394\pi\)
0.616886 + 0.787052i \(0.288394\pi\)
\(734\) −21.6125 −0.797732
\(735\) 0 0
\(736\) −1.70156 −0.0627204
\(737\) 42.2094 1.55480
\(738\) 0 0
\(739\) 1.79063 0.0658693 0.0329346 0.999458i \(-0.489515\pi\)
0.0329346 + 0.999458i \(0.489515\pi\)
\(740\) 15.9109 0.584898
\(741\) 0 0
\(742\) −2.00000 −0.0734223
\(743\) 41.0156 1.50472 0.752359 0.658754i \(-0.228916\pi\)
0.752359 + 0.658754i \(0.228916\pi\)
\(744\) 0 0
\(745\) −9.61250 −0.352175
\(746\) 6.59688 0.241529
\(747\) 0 0
\(748\) 21.1047 0.771664
\(749\) −7.40312 −0.270504
\(750\) 0 0
\(751\) 9.19375 0.335485 0.167742 0.985831i \(-0.446352\pi\)
0.167742 + 0.985831i \(0.446352\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −3.70156 −0.134803
\(755\) 6.29844 0.229224
\(756\) 0 0
\(757\) −46.4187 −1.68712 −0.843559 0.537037i \(-0.819544\pi\)
−0.843559 + 0.537037i \(0.819544\pi\)
\(758\) −33.6125 −1.22086
\(759\) 0 0
\(760\) −21.1047 −0.765547
\(761\) −45.4031 −1.64586 −0.822931 0.568141i \(-0.807663\pi\)
−0.822931 + 0.568141i \(0.807663\pi\)
\(762\) 0 0
\(763\) 15.7016 0.568435
\(764\) −6.29844 −0.227869
\(765\) 0 0
\(766\) −27.9109 −1.00846
\(767\) −10.8062 −0.390191
\(768\) 0 0
\(769\) 6.08907 0.219577 0.109789 0.993955i \(-0.464983\pi\)
0.109789 + 0.993955i \(0.464983\pi\)
\(770\) 21.1047 0.760560
\(771\) 0 0
\(772\) −1.40312 −0.0504995
\(773\) −29.3141 −1.05435 −0.527177 0.849756i \(-0.676749\pi\)
−0.527177 + 0.849756i \(0.676749\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.8062 −0.603309
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) 53.6125 1.92087
\(780\) 0 0
\(781\) −45.6125 −1.63214
\(782\) 6.29844 0.225232
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 3.31406 0.118284
\(786\) 0 0
\(787\) −23.9109 −0.852333 −0.426166 0.904645i \(-0.640136\pi\)
−0.426166 + 0.904645i \(0.640136\pi\)
\(788\) −9.40312 −0.334972
\(789\) 0 0
\(790\) 12.5969 0.448177
\(791\) 20.8062 0.739785
\(792\) 0 0
\(793\) 7.70156 0.273490
\(794\) 24.8062 0.880341
\(795\) 0 0
\(796\) 2.89531 0.102622
\(797\) 53.4031 1.89164 0.945818 0.324698i \(-0.105263\pi\)
0.945818 + 0.324698i \(0.105263\pi\)
\(798\) 0 0
\(799\) −29.6125 −1.04761
\(800\) −8.70156 −0.307647
\(801\) 0 0
\(802\) −20.8062 −0.734694
\(803\) 43.9109 1.54958
\(804\) 0 0
\(805\) 6.29844 0.221991
\(806\) 0 0
\(807\) 0 0
\(808\) 9.40312 0.330801
\(809\) −39.6125 −1.39270 −0.696351 0.717702i \(-0.745194\pi\)
−0.696351 + 0.717702i \(0.745194\pi\)
\(810\) 0 0
\(811\) −41.1047 −1.44338 −0.721690 0.692216i \(-0.756635\pi\)
−0.721690 + 0.692216i \(0.756635\pi\)
\(812\) 3.70156 0.129899
\(813\) 0 0
\(814\) 24.5078 0.858998
\(815\) −27.4031 −0.959890
\(816\) 0 0
\(817\) 51.9109 1.81613
\(818\) −11.7016 −0.409136
\(819\) 0 0
\(820\) 34.8062 1.21549
\(821\) −41.4031 −1.44498 −0.722489 0.691382i \(-0.757002\pi\)
−0.722489 + 0.691382i \(0.757002\pi\)
\(822\) 0 0
\(823\) 30.8062 1.07384 0.536919 0.843634i \(-0.319588\pi\)
0.536919 + 0.843634i \(0.319588\pi\)
\(824\) 9.70156 0.337970
\(825\) 0 0
\(826\) 10.8062 0.375997
\(827\) −8.08907 −0.281284 −0.140642 0.990060i \(-0.544917\pi\)
−0.140642 + 0.990060i \(0.544917\pi\)
\(828\) 0 0
\(829\) 21.3141 0.740268 0.370134 0.928978i \(-0.379312\pi\)
0.370134 + 0.928978i \(0.379312\pi\)
\(830\) −2.20937 −0.0766884
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) −3.70156 −0.128252
\(834\) 0 0
\(835\) −35.9109 −1.24275
\(836\) −32.5078 −1.12431
\(837\) 0 0
\(838\) 14.8953 0.514550
\(839\) −9.19375 −0.317404 −0.158702 0.987327i \(-0.550731\pi\)
−0.158702 + 0.987327i \(0.550731\pi\)
\(840\) 0 0
\(841\) −15.2984 −0.527532
\(842\) 31.6125 1.08944
\(843\) 0 0
\(844\) −18.2984 −0.629858
\(845\) 3.70156 0.127338
\(846\) 0 0
\(847\) 21.5078 0.739017
\(848\) 2.00000 0.0686803
\(849\) 0 0
\(850\) 32.2094 1.10477
\(851\) 7.31406 0.250723
\(852\) 0 0
\(853\) 16.2094 0.554998 0.277499 0.960726i \(-0.410494\pi\)
0.277499 + 0.960726i \(0.410494\pi\)
\(854\) −7.70156 −0.263542
\(855\) 0 0
\(856\) 7.40312 0.253034
\(857\) −8.80625 −0.300816 −0.150408 0.988624i \(-0.548059\pi\)
−0.150408 + 0.988624i \(0.548059\pi\)
\(858\) 0 0
\(859\) 13.1938 0.450165 0.225082 0.974340i \(-0.427735\pi\)
0.225082 + 0.974340i \(0.427735\pi\)
\(860\) 33.7016 1.14921
\(861\) 0 0
\(862\) 26.2094 0.892695
\(863\) 35.4031 1.20514 0.602568 0.798067i \(-0.294144\pi\)
0.602568 + 0.798067i \(0.294144\pi\)
\(864\) 0 0
\(865\) 66.6281 2.26542
\(866\) 24.2094 0.822668
\(867\) 0 0
\(868\) 0 0
\(869\) 19.4031 0.658206
\(870\) 0 0
\(871\) −7.40312 −0.250845
\(872\) −15.7016 −0.531722
\(873\) 0 0
\(874\) −9.70156 −0.328160
\(875\) 13.7016 0.463197
\(876\) 0 0
\(877\) 51.6125 1.74283 0.871415 0.490546i \(-0.163203\pi\)
0.871415 + 0.490546i \(0.163203\pi\)
\(878\) 32.5078 1.09709
\(879\) 0 0
\(880\) −21.1047 −0.711439
\(881\) −49.3141 −1.66143 −0.830716 0.556696i \(-0.812069\pi\)
−0.830716 + 0.556696i \(0.812069\pi\)
\(882\) 0 0
\(883\) 33.1047 1.11406 0.557031 0.830492i \(-0.311941\pi\)
0.557031 + 0.830492i \(0.311941\pi\)
\(884\) −3.70156 −0.124497
\(885\) 0 0
\(886\) −38.2094 −1.28367
\(887\) 8.00000 0.268614 0.134307 0.990940i \(-0.457119\pi\)
0.134307 + 0.990940i \(0.457119\pi\)
\(888\) 0 0
\(889\) −19.4031 −0.650760
\(890\) 62.2094 2.08526
\(891\) 0 0
\(892\) −3.40312 −0.113945
\(893\) 45.6125 1.52636
\(894\) 0 0
\(895\) 40.0000 1.33705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −27.1047 −0.904495
\(899\) 0 0
\(900\) 0 0
\(901\) −7.40312 −0.246634
\(902\) 53.6125 1.78510
\(903\) 0 0
\(904\) −20.8062 −0.692006
\(905\) −62.2094 −2.06791
\(906\) 0 0
\(907\) 25.6125 0.850449 0.425225 0.905088i \(-0.360195\pi\)
0.425225 + 0.905088i \(0.360195\pi\)
\(908\) −8.59688 −0.285297
\(909\) 0 0
\(910\) −3.70156 −0.122706
\(911\) 3.91093 0.129575 0.0647875 0.997899i \(-0.479363\pi\)
0.0647875 + 0.997899i \(0.479363\pi\)
\(912\) 0 0
\(913\) −3.40312 −0.112627
\(914\) −13.4031 −0.443336
\(915\) 0 0
\(916\) 6.00000 0.198246
\(917\) 13.7016 0.452465
\(918\) 0 0
\(919\) −27.4031 −0.903946 −0.451973 0.892032i \(-0.649280\pi\)
−0.451973 + 0.892032i \(0.649280\pi\)
\(920\) −6.29844 −0.207653
\(921\) 0 0
\(922\) 37.3141 1.22887
\(923\) 8.00000 0.263323
\(924\) 0 0
\(925\) 37.4031 1.22981
\(926\) −23.3141 −0.766148
\(927\) 0 0
\(928\) −3.70156 −0.121510
\(929\) 31.0156 1.01759 0.508795 0.860888i \(-0.330091\pi\)
0.508795 + 0.860888i \(0.330091\pi\)
\(930\) 0 0
\(931\) 5.70156 0.186861
\(932\) −20.2094 −0.661980
\(933\) 0 0
\(934\) −36.5078 −1.19457
\(935\) 78.1203 2.55481
\(936\) 0 0
\(937\) 3.19375 0.104335 0.0521677 0.998638i \(-0.483387\pi\)
0.0521677 + 0.998638i \(0.483387\pi\)
\(938\) 7.40312 0.241721
\(939\) 0 0
\(940\) 29.6125 0.965853
\(941\) 50.0000 1.62995 0.814977 0.579494i \(-0.196750\pi\)
0.814977 + 0.579494i \(0.196750\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) −10.8062 −0.351713
\(945\) 0 0
\(946\) 51.9109 1.68777
\(947\) −36.5078 −1.18634 −0.593172 0.805076i \(-0.702125\pi\)
−0.593172 + 0.805076i \(0.702125\pi\)
\(948\) 0 0
\(949\) −7.70156 −0.250003
\(950\) −49.6125 −1.60964
\(951\) 0 0
\(952\) 3.70156 0.119968
\(953\) −49.8219 −1.61389 −0.806944 0.590628i \(-0.798880\pi\)
−0.806944 + 0.590628i \(0.798880\pi\)
\(954\) 0 0
\(955\) −23.3141 −0.754425
\(956\) −16.0000 −0.517477
\(957\) 0 0
\(958\) −24.5078 −0.791811
\(959\) 12.2984 0.397137
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) −4.29844 −0.138587
\(963\) 0 0
\(964\) 0.806248 0.0259675
\(965\) −5.19375 −0.167193
\(966\) 0 0
\(967\) 18.7172 0.601904 0.300952 0.953639i \(-0.402696\pi\)
0.300952 + 0.953639i \(0.402696\pi\)
\(968\) −21.5078 −0.691287
\(969\) 0 0
\(970\) −62.2094 −1.99742
\(971\) −29.1938 −0.936872 −0.468436 0.883497i \(-0.655182\pi\)
−0.468436 + 0.883497i \(0.655182\pi\)
\(972\) 0 0
\(973\) −18.8062 −0.602901
\(974\) 8.00000 0.256337
\(975\) 0 0
\(976\) 7.70156 0.246521
\(977\) −28.7172 −0.918744 −0.459372 0.888244i \(-0.651926\pi\)
−0.459372 + 0.888244i \(0.651926\pi\)
\(978\) 0 0
\(979\) 95.8219 3.06248
\(980\) 3.70156 0.118242
\(981\) 0 0
\(982\) 39.4031 1.25740
\(983\) 18.8953 0.602667 0.301333 0.953519i \(-0.402568\pi\)
0.301333 + 0.953519i \(0.402568\pi\)
\(984\) 0 0
\(985\) −34.8062 −1.10902
\(986\) 13.7016 0.436347
\(987\) 0 0
\(988\) 5.70156 0.181391
\(989\) 15.4922 0.492623
\(990\) 0 0
\(991\) −51.4031 −1.63287 −0.816437 0.577434i \(-0.804054\pi\)
−0.816437 + 0.577434i \(0.804054\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 10.7172 0.339758
\(996\) 0 0
\(997\) 12.8062 0.405578 0.202789 0.979222i \(-0.434999\pi\)
0.202789 + 0.979222i \(0.434999\pi\)
\(998\) 26.8062 0.848537
\(999\) 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 1638.2.a.w.1.2 2
3.2 odd 2 546.2.a.i.1.1 2
12.11 even 2 4368.2.a.bg.1.1 2
21.20 even 2 3822.2.a.bt.1.2 2
39.38 odd 2 7098.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.a.i.1.1 2 3.2 odd 2
1638.2.a.w.1.2 2 1.1 even 1 trivial
3822.2.a.bt.1.2 2 21.20 even 2
4368.2.a.bg.1.1 2 12.11 even 2
7098.2.a.bh.1.2 2 39.38 odd 2