Properties

Label 8281.2.a.q.1.1
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $1$
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] = [8281,2,Mod(1,8281)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8281.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.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: \(66.1241179138\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.73205 q^{5} +3.46410 q^{6} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} -2.00000 q^{3} +1.00000 q^{4} -1.73205 q^{5} +3.46410 q^{6} +1.73205 q^{8} +1.00000 q^{9} +3.00000 q^{10} -2.00000 q^{12} +3.46410 q^{15} -5.00000 q^{16} -3.00000 q^{17} -1.73205 q^{18} -3.46410 q^{19} -1.73205 q^{20} +6.00000 q^{23} -3.46410 q^{24} -2.00000 q^{25} +4.00000 q^{27} +3.00000 q^{29} -6.00000 q^{30} +3.46410 q^{31} +5.19615 q^{32} +5.19615 q^{34} +1.00000 q^{36} -8.66025 q^{37} +6.00000 q^{38} -3.00000 q^{40} +5.19615 q^{41} -8.00000 q^{43} -1.73205 q^{45} -10.3923 q^{46} +3.46410 q^{47} +10.0000 q^{48} +3.46410 q^{50} +6.00000 q^{51} -3.00000 q^{53} -6.92820 q^{54} +6.92820 q^{57} -5.19615 q^{58} -6.92820 q^{59} +3.46410 q^{60} -1.00000 q^{61} -6.00000 q^{62} +1.00000 q^{64} +3.46410 q^{67} -3.00000 q^{68} -12.0000 q^{69} -3.46410 q^{71} +1.73205 q^{72} -1.73205 q^{73} +15.0000 q^{74} +4.00000 q^{75} -3.46410 q^{76} +4.00000 q^{79} +8.66025 q^{80} -11.0000 q^{81} -9.00000 q^{82} +13.8564 q^{83} +5.19615 q^{85} +13.8564 q^{86} -6.00000 q^{87} -6.92820 q^{89} +3.00000 q^{90} +6.00000 q^{92} -6.92820 q^{93} -6.00000 q^{94} +6.00000 q^{95} -10.3923 q^{96} +6.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{4} + 2 q^{9} + 6 q^{10} - 4 q^{12} - 10 q^{16} - 6 q^{17} + 12 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} - 12 q^{30} + 2 q^{36} + 12 q^{38} - 6 q^{40} - 16 q^{43} + 20 q^{48} + 12 q^{51} - 6 q^{53} - 2 q^{61} - 12 q^{62} + 2 q^{64} - 6 q^{68} - 24 q^{69} + 30 q^{74} + 8 q^{75} + 8 q^{79} - 22 q^{81} - 18 q^{82} - 12 q^{87} + 6 q^{90} + 12 q^{92} - 12 q^{94} + 12 q^{95}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) −2.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.73205 −0.774597 −0.387298 0.921954i \(-0.626592\pi\)
−0.387298 + 0.921954i \(0.626592\pi\)
\(6\) 3.46410 1.41421
\(7\) 0 0
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) −2.00000 −0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) −5.00000 −1.25000
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) −1.73205 −0.408248
\(19\) −3.46410 −0.794719 −0.397360 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) −1.73205 −0.387298
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) −3.46410 −0.707107
\(25\) −2.00000 −0.400000
\(26\) 0 0
\(27\) 4.00000 0.769800
\(28\) 0 0
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) −6.00000 −1.09545
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 5.19615 0.891133
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.66025 −1.42374 −0.711868 0.702313i \(-0.752151\pi\)
−0.711868 + 0.702313i \(0.752151\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 5.19615 0.811503 0.405751 0.913984i \(-0.367010\pi\)
0.405751 + 0.913984i \(0.367010\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −1.73205 −0.258199
\(46\) −10.3923 −1.53226
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 10.0000 1.44338
\(49\) 0 0
\(50\) 3.46410 0.489898
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −6.92820 −0.942809
\(55\) 0 0
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) −5.19615 −0.682288
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 3.46410 0.447214
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410 0.423207 0.211604 0.977356i \(-0.432131\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) −3.00000 −0.363803
\(69\) −12.0000 −1.44463
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 1.73205 0.204124
\(73\) −1.73205 −0.202721 −0.101361 0.994850i \(-0.532320\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 15.0000 1.74371
\(75\) 4.00000 0.461880
\(76\) −3.46410 −0.397360
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 8.66025 0.968246
\(81\) −11.0000 −1.22222
\(82\) −9.00000 −0.993884
\(83\) 13.8564 1.52094 0.760469 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) 0 0
\(85\) 5.19615 0.563602
\(86\) 13.8564 1.49417
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) 6.00000 0.625543
\(93\) −6.92820 −0.718421
\(94\) −6.00000 −0.618853
\(95\) 6.00000 0.615587
\(96\) −10.3923 −1.06066
\(97\) 6.92820 0.703452 0.351726 0.936103i \(-0.385595\pi\)
0.351726 + 0.936103i \(0.385595\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) −10.3923 −1.02899
\(103\) −10.0000 −0.985329 −0.492665 0.870219i \(-0.663977\pi\)
−0.492665 + 0.870219i \(0.663977\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 5.19615 0.504695
\(107\) 6.00000 0.580042 0.290021 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 4.00000 0.384900
\(109\) 13.8564 1.32720 0.663602 0.748086i \(-0.269027\pi\)
0.663602 + 0.748086i \(0.269027\pi\)
\(110\) 0 0
\(111\) 17.3205 1.64399
\(112\) 0 0
\(113\) −15.0000 −1.41108 −0.705541 0.708669i \(-0.749296\pi\)
−0.705541 + 0.708669i \(0.749296\pi\)
\(114\) −12.0000 −1.12390
\(115\) −10.3923 −0.969087
\(116\) 3.00000 0.278543
\(117\) 0 0
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) 6.00000 0.547723
\(121\) −11.0000 −1.00000
\(122\) 1.73205 0.156813
\(123\) −10.3923 −0.937043
\(124\) 3.46410 0.311086
\(125\) 12.1244 1.08444
\(126\) 0 0
\(127\) 2.00000 0.177471 0.0887357 0.996055i \(-0.471717\pi\)
0.0887357 + 0.996055i \(0.471717\pi\)
\(128\) −12.1244 −1.07165
\(129\) 16.0000 1.40872
\(130\) 0 0
\(131\) −18.0000 −1.57267 −0.786334 0.617802i \(-0.788023\pi\)
−0.786334 + 0.617802i \(0.788023\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −6.00000 −0.518321
\(135\) −6.92820 −0.596285
\(136\) −5.19615 −0.445566
\(137\) −15.5885 −1.33181 −0.665906 0.746036i \(-0.731955\pi\)
−0.665906 + 0.746036i \(0.731955\pi\)
\(138\) 20.7846 1.76930
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −6.92820 −0.583460
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) −5.00000 −0.416667
\(145\) −5.19615 −0.431517
\(146\) 3.00000 0.248282
\(147\) 0 0
\(148\) −8.66025 −0.711868
\(149\) 19.0526 1.56085 0.780423 0.625252i \(-0.215004\pi\)
0.780423 + 0.625252i \(0.215004\pi\)
\(150\) −6.92820 −0.565685
\(151\) 17.3205 1.40952 0.704761 0.709444i \(-0.251054\pi\)
0.704761 + 0.709444i \(0.251054\pi\)
\(152\) −6.00000 −0.486664
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 0 0
\(157\) 13.0000 1.03751 0.518756 0.854922i \(-0.326395\pi\)
0.518756 + 0.854922i \(0.326395\pi\)
\(158\) −6.92820 −0.551178
\(159\) 6.00000 0.475831
\(160\) −9.00000 −0.711512
\(161\) 0 0
\(162\) 19.0526 1.49691
\(163\) 20.7846 1.62798 0.813988 0.580881i \(-0.197292\pi\)
0.813988 + 0.580881i \(0.197292\pi\)
\(164\) 5.19615 0.405751
\(165\) 0 0
\(166\) −24.0000 −1.86276
\(167\) −13.8564 −1.07224 −0.536120 0.844141i \(-0.680111\pi\)
−0.536120 + 0.844141i \(0.680111\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −9.00000 −0.690268
\(171\) −3.46410 −0.264906
\(172\) −8.00000 −0.609994
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) 10.3923 0.787839
\(175\) 0 0
\(176\) 0 0
\(177\) 13.8564 1.04151
\(178\) 12.0000 0.899438
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) −1.73205 −0.129099
\(181\) 11.0000 0.817624 0.408812 0.912619i \(-0.365943\pi\)
0.408812 + 0.912619i \(0.365943\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 10.3923 0.766131
\(185\) 15.0000 1.10282
\(186\) 12.0000 0.879883
\(187\) 0 0
\(188\) 3.46410 0.252646
\(189\) 0 0
\(190\) −10.3923 −0.753937
\(191\) 18.0000 1.30243 0.651217 0.758891i \(-0.274259\pi\)
0.651217 + 0.758891i \(0.274259\pi\)
\(192\) −2.00000 −0.144338
\(193\) 5.19615 0.374027 0.187014 0.982357i \(-0.440119\pi\)
0.187014 + 0.982357i \(0.440119\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564 0.987228 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) −3.46410 −0.244949
\(201\) −6.92820 −0.488678
\(202\) 5.19615 0.365600
\(203\) 0 0
\(204\) 6.00000 0.420084
\(205\) −9.00000 −0.628587
\(206\) 17.3205 1.20678
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) −3.00000 −0.206041
\(213\) 6.92820 0.474713
\(214\) −10.3923 −0.710403
\(215\) 13.8564 0.944999
\(216\) 6.92820 0.471405
\(217\) 0 0
\(218\) −24.0000 −1.62549
\(219\) 3.46410 0.234082
\(220\) 0 0
\(221\) 0 0
\(222\) −30.0000 −2.01347
\(223\) 10.3923 0.695920 0.347960 0.937509i \(-0.386874\pi\)
0.347960 + 0.937509i \(0.386874\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 25.9808 1.72821
\(227\) 24.2487 1.60944 0.804722 0.593652i \(-0.202314\pi\)
0.804722 + 0.593652i \(0.202314\pi\)
\(228\) 6.92820 0.458831
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 18.0000 1.18688
\(231\) 0 0
\(232\) 5.19615 0.341144
\(233\) −6.00000 −0.393073 −0.196537 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) −6.92820 −0.450988
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 20.7846 1.34444 0.672222 0.740349i \(-0.265340\pi\)
0.672222 + 0.740349i \(0.265340\pi\)
\(240\) −17.3205 −1.11803
\(241\) 1.73205 0.111571 0.0557856 0.998443i \(-0.482234\pi\)
0.0557856 + 0.998443i \(0.482234\pi\)
\(242\) 19.0526 1.22474
\(243\) 10.0000 0.641500
\(244\) −1.00000 −0.0640184
\(245\) 0 0
\(246\) 18.0000 1.14764
\(247\) 0 0
\(248\) 6.00000 0.381000
\(249\) −27.7128 −1.75623
\(250\) −21.0000 −1.32816
\(251\) −18.0000 −1.13615 −0.568075 0.822977i \(-0.692312\pi\)
−0.568075 + 0.822977i \(0.692312\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3.46410 −0.217357
\(255\) −10.3923 −0.650791
\(256\) 19.0000 1.18750
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) −27.7128 −1.72532
\(259\) 0 0
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 31.1769 1.92612
\(263\) 12.0000 0.739952 0.369976 0.929041i \(-0.379366\pi\)
0.369976 + 0.929041i \(0.379366\pi\)
\(264\) 0 0
\(265\) 5.19615 0.319197
\(266\) 0 0
\(267\) 13.8564 0.847998
\(268\) 3.46410 0.211604
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 12.0000 0.730297
\(271\) 20.7846 1.26258 0.631288 0.775549i \(-0.282527\pi\)
0.631288 + 0.775549i \(0.282527\pi\)
\(272\) 15.0000 0.909509
\(273\) 0 0
\(274\) 27.0000 1.63113
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) 7.00000 0.420589 0.210295 0.977638i \(-0.432558\pi\)
0.210295 + 0.977638i \(0.432558\pi\)
\(278\) −6.92820 −0.415526
\(279\) 3.46410 0.207390
\(280\) 0 0
\(281\) −22.5167 −1.34323 −0.671616 0.740900i \(-0.734399\pi\)
−0.671616 + 0.740900i \(0.734399\pi\)
\(282\) 12.0000 0.714590
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −3.46410 −0.205557
\(285\) −12.0000 −0.710819
\(286\) 0 0
\(287\) 0 0
\(288\) 5.19615 0.306186
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) −13.8564 −0.812277
\(292\) −1.73205 −0.101361
\(293\) −5.19615 −0.303562 −0.151781 0.988414i \(-0.548501\pi\)
−0.151781 + 0.988414i \(0.548501\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) −15.0000 −0.871857
\(297\) 0 0
\(298\) −33.0000 −1.91164
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) 0 0
\(302\) −30.0000 −1.72631
\(303\) 6.00000 0.344691
\(304\) 17.3205 0.993399
\(305\) 1.73205 0.0991769
\(306\) 5.19615 0.297044
\(307\) 17.3205 0.988534 0.494267 0.869310i \(-0.335437\pi\)
0.494267 + 0.869310i \(0.335437\pi\)
\(308\) 0 0
\(309\) 20.0000 1.13776
\(310\) 10.3923 0.590243
\(311\) −30.0000 −1.70114 −0.850572 0.525859i \(-0.823744\pi\)
−0.850572 + 0.525859i \(0.823744\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −22.5167 −1.27069
\(315\) 0 0
\(316\) 4.00000 0.225018
\(317\) 5.19615 0.291845 0.145922 0.989296i \(-0.453385\pi\)
0.145922 + 0.989296i \(0.453385\pi\)
\(318\) −10.3923 −0.582772
\(319\) 0 0
\(320\) −1.73205 −0.0968246
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 10.3923 0.578243
\(324\) −11.0000 −0.611111
\(325\) 0 0
\(326\) −36.0000 −1.99386
\(327\) −27.7128 −1.53252
\(328\) 9.00000 0.496942
\(329\) 0 0
\(330\) 0 0
\(331\) 27.7128 1.52323 0.761617 0.648027i \(-0.224406\pi\)
0.761617 + 0.648027i \(0.224406\pi\)
\(332\) 13.8564 0.760469
\(333\) −8.66025 −0.474579
\(334\) 24.0000 1.31322
\(335\) −6.00000 −0.327815
\(336\) 0 0
\(337\) 23.0000 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(338\) 0 0
\(339\) 30.0000 1.62938
\(340\) 5.19615 0.281801
\(341\) 0 0
\(342\) 6.00000 0.324443
\(343\) 0 0
\(344\) −13.8564 −0.747087
\(345\) 20.7846 1.11901
\(346\) −10.3923 −0.558694
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) −6.00000 −0.321634
\(349\) 13.8564 0.741716 0.370858 0.928689i \(-0.379064\pi\)
0.370858 + 0.928689i \(0.379064\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.9090 −1.75157 −0.875784 0.482704i \(-0.839655\pi\)
−0.875784 + 0.482704i \(0.839655\pi\)
\(354\) −24.0000 −1.27559
\(355\) 6.00000 0.318447
\(356\) −6.92820 −0.367194
\(357\) 0 0
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) −3.00000 −0.158114
\(361\) −7.00000 −0.368421
\(362\) −19.0526 −1.00138
\(363\) 22.0000 1.15470
\(364\) 0 0
\(365\) 3.00000 0.157027
\(366\) −3.46410 −0.181071
\(367\) 22.0000 1.14839 0.574195 0.818718i \(-0.305315\pi\)
0.574195 + 0.818718i \(0.305315\pi\)
\(368\) −30.0000 −1.56386
\(369\) 5.19615 0.270501
\(370\) −25.9808 −1.35068
\(371\) 0 0
\(372\) −6.92820 −0.359211
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) −24.2487 −1.25220
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) −24.2487 −1.24557 −0.622786 0.782392i \(-0.713999\pi\)
−0.622786 + 0.782392i \(0.713999\pi\)
\(380\) 6.00000 0.307794
\(381\) −4.00000 −0.204926
\(382\) −31.1769 −1.59515
\(383\) −20.7846 −1.06204 −0.531022 0.847358i \(-0.678192\pi\)
−0.531022 + 0.847358i \(0.678192\pi\)
\(384\) 24.2487 1.23744
\(385\) 0 0
\(386\) −9.00000 −0.458088
\(387\) −8.00000 −0.406663
\(388\) 6.92820 0.351726
\(389\) 9.00000 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 36.0000 1.81596
\(394\) −24.0000 −1.20910
\(395\) −6.92820 −0.348596
\(396\) 0 0
\(397\) 13.8564 0.695433 0.347717 0.937600i \(-0.386957\pi\)
0.347717 + 0.937600i \(0.386957\pi\)
\(398\) 3.46410 0.173640
\(399\) 0 0
\(400\) 10.0000 0.500000
\(401\) 1.73205 0.0864945 0.0432472 0.999064i \(-0.486230\pi\)
0.0432472 + 0.999064i \(0.486230\pi\)
\(402\) 12.0000 0.598506
\(403\) 0 0
\(404\) −3.00000 −0.149256
\(405\) 19.0526 0.946729
\(406\) 0 0
\(407\) 0 0
\(408\) 10.3923 0.514496
\(409\) 15.5885 0.770800 0.385400 0.922750i \(-0.374064\pi\)
0.385400 + 0.922750i \(0.374064\pi\)
\(410\) 15.5885 0.769859
\(411\) 31.1769 1.53784
\(412\) −10.0000 −0.492665
\(413\) 0 0
\(414\) −10.3923 −0.510754
\(415\) −24.0000 −1.17811
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −18.0000 −0.879358 −0.439679 0.898155i \(-0.644908\pi\)
−0.439679 + 0.898155i \(0.644908\pi\)
\(420\) 0 0
\(421\) −15.5885 −0.759735 −0.379867 0.925041i \(-0.624030\pi\)
−0.379867 + 0.925041i \(0.624030\pi\)
\(422\) −17.3205 −0.843149
\(423\) 3.46410 0.168430
\(424\) −5.19615 −0.252347
\(425\) 6.00000 0.291043
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) −24.0000 −1.15738
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) −20.0000 −0.962250
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) 0 0
\(435\) 10.3923 0.498273
\(436\) 13.8564 0.663602
\(437\) −20.7846 −0.994263
\(438\) −6.00000 −0.286691
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 17.3205 0.821995
\(445\) 12.0000 0.568855
\(446\) −18.0000 −0.852325
\(447\) −38.1051 −1.80231
\(448\) 0 0
\(449\) −6.92820 −0.326962 −0.163481 0.986546i \(-0.552272\pi\)
−0.163481 + 0.986546i \(0.552272\pi\)
\(450\) 3.46410 0.163299
\(451\) 0 0
\(452\) −15.0000 −0.705541
\(453\) −34.6410 −1.62758
\(454\) −42.0000 −1.97116
\(455\) 0 0
\(456\) 12.0000 0.561951
\(457\) 1.73205 0.0810219 0.0405110 0.999179i \(-0.487101\pi\)
0.0405110 + 0.999179i \(0.487101\pi\)
\(458\) 0 0
\(459\) −12.0000 −0.560112
\(460\) −10.3923 −0.484544
\(461\) 22.5167 1.04871 0.524353 0.851501i \(-0.324307\pi\)
0.524353 + 0.851501i \(0.324307\pi\)
\(462\) 0 0
\(463\) 13.8564 0.643962 0.321981 0.946746i \(-0.395651\pi\)
0.321981 + 0.946746i \(0.395651\pi\)
\(464\) −15.0000 −0.696358
\(465\) 12.0000 0.556487
\(466\) 10.3923 0.481414
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 10.3923 0.479361
\(471\) −26.0000 −1.19802
\(472\) −12.0000 −0.552345
\(473\) 0 0
\(474\) 13.8564 0.636446
\(475\) 6.92820 0.317888
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) −36.0000 −1.64660
\(479\) 24.2487 1.10795 0.553976 0.832533i \(-0.313110\pi\)
0.553976 + 0.832533i \(0.313110\pi\)
\(480\) 18.0000 0.821584
\(481\) 0 0
\(482\) −3.00000 −0.136646
\(483\) 0 0
\(484\) −11.0000 −0.500000
\(485\) −12.0000 −0.544892
\(486\) −17.3205 −0.785674
\(487\) 6.92820 0.313947 0.156973 0.987603i \(-0.449826\pi\)
0.156973 + 0.987603i \(0.449826\pi\)
\(488\) −1.73205 −0.0784063
\(489\) −41.5692 −1.87983
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) −10.3923 −0.468521
\(493\) −9.00000 −0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) −17.3205 −0.777714
\(497\) 0 0
\(498\) 48.0000 2.15093
\(499\) −31.1769 −1.39567 −0.697835 0.716258i \(-0.745853\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) 12.1244 0.542218
\(501\) 27.7128 1.23812
\(502\) 31.1769 1.39149
\(503\) −36.0000 −1.60516 −0.802580 0.596544i \(-0.796540\pi\)
−0.802580 + 0.596544i \(0.796540\pi\)
\(504\) 0 0
\(505\) 5.19615 0.231226
\(506\) 0 0
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) −19.0526 −0.844490 −0.422245 0.906482i \(-0.638758\pi\)
−0.422245 + 0.906482i \(0.638758\pi\)
\(510\) 18.0000 0.797053
\(511\) 0 0
\(512\) −8.66025 −0.382733
\(513\) −13.8564 −0.611775
\(514\) −5.19615 −0.229192
\(515\) 17.3205 0.763233
\(516\) 16.0000 0.704361
\(517\) 0 0
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −9.00000 −0.394297 −0.197149 0.980374i \(-0.563168\pi\)
−0.197149 + 0.980374i \(0.563168\pi\)
\(522\) −5.19615 −0.227429
\(523\) 16.0000 0.699631 0.349816 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) −18.0000 −0.786334
\(525\) 0 0
\(526\) −20.7846 −0.906252
\(527\) −10.3923 −0.452696
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) −9.00000 −0.390935
\(531\) −6.92820 −0.300658
\(532\) 0 0
\(533\) 0 0
\(534\) −24.0000 −1.03858
\(535\) −10.3923 −0.449299
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) −10.3923 −0.448044
\(539\) 0 0
\(540\) −6.92820 −0.298142
\(541\) 29.4449 1.26593 0.632967 0.774179i \(-0.281837\pi\)
0.632967 + 0.774179i \(0.281837\pi\)
\(542\) −36.0000 −1.54633
\(543\) −22.0000 −0.944110
\(544\) −15.5885 −0.668350
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −15.5885 −0.665906
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) −10.3923 −0.442727
\(552\) −20.7846 −0.884652
\(553\) 0 0
\(554\) −12.1244 −0.515115
\(555\) −30.0000 −1.27343
\(556\) 4.00000 0.169638
\(557\) 15.5885 0.660504 0.330252 0.943893i \(-0.392866\pi\)
0.330252 + 0.943893i \(0.392866\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 39.0000 1.64512
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) −6.92820 −0.291730
\(565\) 25.9808 1.09302
\(566\) −6.92820 −0.291214
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) 42.0000 1.76073 0.880366 0.474295i \(-0.157297\pi\)
0.880366 + 0.474295i \(0.157297\pi\)
\(570\) 20.7846 0.870572
\(571\) −40.0000 −1.67395 −0.836974 0.547243i \(-0.815677\pi\)
−0.836974 + 0.547243i \(0.815677\pi\)
\(572\) 0 0
\(573\) −36.0000 −1.50392
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 1.00000 0.0416667
\(577\) −19.0526 −0.793168 −0.396584 0.917998i \(-0.629805\pi\)
−0.396584 + 0.917998i \(0.629805\pi\)
\(578\) 13.8564 0.576351
\(579\) −10.3923 −0.431889
\(580\) −5.19615 −0.215758
\(581\) 0 0
\(582\) 24.0000 0.994832
\(583\) 0 0
\(584\) −3.00000 −0.124141
\(585\) 0 0
\(586\) 9.00000 0.371787
\(587\) 20.7846 0.857873 0.428936 0.903335i \(-0.358888\pi\)
0.428936 + 0.903335i \(0.358888\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) −20.7846 −0.855689
\(591\) −27.7128 −1.13995
\(592\) 43.3013 1.77967
\(593\) 25.9808 1.06690 0.533451 0.845831i \(-0.320895\pi\)
0.533451 + 0.845831i \(0.320895\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 19.0526 0.780423
\(597\) 4.00000 0.163709
\(598\) 0 0
\(599\) 30.0000 1.22577 0.612883 0.790173i \(-0.290010\pi\)
0.612883 + 0.790173i \(0.290010\pi\)
\(600\) 6.92820 0.282843
\(601\) −25.0000 −1.01977 −0.509886 0.860242i \(-0.670312\pi\)
−0.509886 + 0.860242i \(0.670312\pi\)
\(602\) 0 0
\(603\) 3.46410 0.141069
\(604\) 17.3205 0.704761
\(605\) 19.0526 0.774597
\(606\) −10.3923 −0.422159
\(607\) 34.0000 1.38002 0.690009 0.723801i \(-0.257607\pi\)
0.690009 + 0.723801i \(0.257607\pi\)
\(608\) −18.0000 −0.729996
\(609\) 0 0
\(610\) −3.00000 −0.121466
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) −12.1244 −0.489698 −0.244849 0.969561i \(-0.578738\pi\)
−0.244849 + 0.969561i \(0.578738\pi\)
\(614\) −30.0000 −1.21070
\(615\) 18.0000 0.725830
\(616\) 0 0
\(617\) 22.5167 0.906487 0.453243 0.891387i \(-0.350267\pi\)
0.453243 + 0.891387i \(0.350267\pi\)
\(618\) −34.6410 −1.39347
\(619\) 20.7846 0.835404 0.417702 0.908584i \(-0.362836\pi\)
0.417702 + 0.908584i \(0.362836\pi\)
\(620\) −6.00000 −0.240966
\(621\) 24.0000 0.963087
\(622\) 51.9615 2.08347
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 17.3205 0.692267
\(627\) 0 0
\(628\) 13.0000 0.518756
\(629\) 25.9808 1.03592
\(630\) 0 0
\(631\) −48.4974 −1.93065 −0.965326 0.261048i \(-0.915932\pi\)
−0.965326 + 0.261048i \(0.915932\pi\)
\(632\) 6.92820 0.275589
\(633\) −20.0000 −0.794929
\(634\) −9.00000 −0.357436
\(635\) −3.46410 −0.137469
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) 0 0
\(639\) −3.46410 −0.137038
\(640\) 21.0000 0.830098
\(641\) −33.0000 −1.30342 −0.651711 0.758468i \(-0.725948\pi\)
−0.651711 + 0.758468i \(0.725948\pi\)
\(642\) 20.7846 0.820303
\(643\) 13.8564 0.546443 0.273222 0.961951i \(-0.411911\pi\)
0.273222 + 0.961951i \(0.411911\pi\)
\(644\) 0 0
\(645\) −27.7128 −1.09119
\(646\) −18.0000 −0.708201
\(647\) 18.0000 0.707653 0.353827 0.935311i \(-0.384880\pi\)
0.353827 + 0.935311i \(0.384880\pi\)
\(648\) −19.0526 −0.748455
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 20.7846 0.813988
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 48.0000 1.87695
\(655\) 31.1769 1.21818
\(656\) −25.9808 −1.01438
\(657\) −1.73205 −0.0675737
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) −46.7654 −1.81896 −0.909481 0.415745i \(-0.863521\pi\)
−0.909481 + 0.415745i \(0.863521\pi\)
\(662\) −48.0000 −1.86557
\(663\) 0 0
\(664\) 24.0000 0.931381
\(665\) 0 0
\(666\) 15.0000 0.581238
\(667\) 18.0000 0.696963
\(668\) −13.8564 −0.536120
\(669\) −20.7846 −0.803579
\(670\) 10.3923 0.401490
\(671\) 0 0
\(672\) 0 0
\(673\) 19.0000 0.732396 0.366198 0.930537i \(-0.380659\pi\)
0.366198 + 0.930537i \(0.380659\pi\)
\(674\) −39.8372 −1.53447
\(675\) −8.00000 −0.307920
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −51.9615 −1.99557
\(679\) 0 0
\(680\) 9.00000 0.345134
\(681\) −48.4974 −1.85843
\(682\) 0 0
\(683\) 24.2487 0.927851 0.463926 0.885874i \(-0.346441\pi\)
0.463926 + 0.885874i \(0.346441\pi\)
\(684\) −3.46410 −0.132453
\(685\) 27.0000 1.03162
\(686\) 0 0
\(687\) 0 0
\(688\) 40.0000 1.52499
\(689\) 0 0
\(690\) −36.0000 −1.37050
\(691\) −13.8564 −0.527123 −0.263561 0.964643i \(-0.584897\pi\)
−0.263561 + 0.964643i \(0.584897\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) 51.9615 1.97243
\(695\) −6.92820 −0.262802
\(696\) −10.3923 −0.393919
\(697\) −15.5885 −0.590455
\(698\) −24.0000 −0.908413
\(699\) 12.0000 0.453882
\(700\) 0 0
\(701\) −18.0000 −0.679851 −0.339925 0.940452i \(-0.610402\pi\)
−0.339925 + 0.940452i \(0.610402\pi\)
\(702\) 0 0
\(703\) 30.0000 1.13147
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 57.0000 2.14522
\(707\) 0 0
\(708\) 13.8564 0.520756
\(709\) −5.19615 −0.195146 −0.0975728 0.995228i \(-0.531108\pi\)
−0.0975728 + 0.995228i \(0.531108\pi\)
\(710\) −10.3923 −0.390016
\(711\) 4.00000 0.150012
\(712\) −12.0000 −0.449719
\(713\) 20.7846 0.778390
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −41.5692 −1.55243
\(718\) 12.0000 0.447836
\(719\) −48.0000 −1.79010 −0.895049 0.445968i \(-0.852860\pi\)
−0.895049 + 0.445968i \(0.852860\pi\)
\(720\) 8.66025 0.322749
\(721\) 0 0
\(722\) 12.1244 0.451222
\(723\) −3.46410 −0.128831
\(724\) 11.0000 0.408812
\(725\) −6.00000 −0.222834
\(726\) −38.1051 −1.41421
\(727\) −32.0000 −1.18681 −0.593407 0.804902i \(-0.702218\pi\)
−0.593407 + 0.804902i \(0.702218\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) −5.19615 −0.192318
\(731\) 24.0000 0.887672
\(732\) 2.00000 0.0739221
\(733\) 12.1244 0.447823 0.223912 0.974609i \(-0.428117\pi\)
0.223912 + 0.974609i \(0.428117\pi\)
\(734\) −38.1051 −1.40649
\(735\) 0 0
\(736\) 31.1769 1.14920
\(737\) 0 0
\(738\) −9.00000 −0.331295
\(739\) 20.7846 0.764574 0.382287 0.924044i \(-0.375137\pi\)
0.382287 + 0.924044i \(0.375137\pi\)
\(740\) 15.0000 0.551411
\(741\) 0 0
\(742\) 0 0
\(743\) −34.6410 −1.27086 −0.635428 0.772160i \(-0.719176\pi\)
−0.635428 + 0.772160i \(0.719176\pi\)
\(744\) −12.0000 −0.439941
\(745\) −33.0000 −1.20903
\(746\) −32.9090 −1.20488
\(747\) 13.8564 0.506979
\(748\) 0 0
\(749\) 0 0
\(750\) 42.0000 1.53362
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) −17.3205 −0.631614
\(753\) 36.0000 1.31191
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 42.0000 1.52551
\(759\) 0 0
\(760\) 10.3923 0.376969
\(761\) −34.6410 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(762\) 6.92820 0.250982
\(763\) 0 0
\(764\) 18.0000 0.651217
\(765\) 5.19615 0.187867
\(766\) 36.0000 1.30073
\(767\) 0 0
\(768\) −38.0000 −1.37121
\(769\) −6.92820 −0.249837 −0.124919 0.992167i \(-0.539867\pi\)
−0.124919 + 0.992167i \(0.539867\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 5.19615 0.187014
\(773\) −34.6410 −1.24595 −0.622975 0.782241i \(-0.714076\pi\)
−0.622975 + 0.782241i \(0.714076\pi\)
\(774\) 13.8564 0.498058
\(775\) −6.92820 −0.248868
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) −15.5885 −0.558873
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 31.1769 1.11488
\(783\) 12.0000 0.428845
\(784\) 0 0
\(785\) −22.5167 −0.803654
\(786\) −62.3538 −2.22409
\(787\) −38.1051 −1.35830 −0.679150 0.733999i \(-0.737652\pi\)
−0.679150 + 0.733999i \(0.737652\pi\)
\(788\) 13.8564 0.493614
\(789\) −24.0000 −0.854423
\(790\) 12.0000 0.426941
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) −24.0000 −0.851728
\(795\) −10.3923 −0.368577
\(796\) −2.00000 −0.0708881
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 0 0
\(799\) −10.3923 −0.367653
\(800\) −10.3923 −0.367423
\(801\) −6.92820 −0.244796
\(802\) −3.00000 −0.105934
\(803\) 0 0
\(804\) −6.92820 −0.244339
\(805\) 0 0
\(806\) 0 0
\(807\) −12.0000 −0.422420
\(808\) −5.19615 −0.182800
\(809\) 33.0000 1.16022 0.580109 0.814539i \(-0.303010\pi\)
0.580109 + 0.814539i \(0.303010\pi\)
\(810\) −33.0000 −1.15950
\(811\) −38.1051 −1.33805 −0.669026 0.743239i \(-0.733288\pi\)
−0.669026 + 0.743239i \(0.733288\pi\)
\(812\) 0 0
\(813\) −41.5692 −1.45790
\(814\) 0 0
\(815\) −36.0000 −1.26102
\(816\) −30.0000 −1.05021
\(817\) 27.7128 0.969549
\(818\) −27.0000 −0.944033
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) −41.5692 −1.45078 −0.725388 0.688340i \(-0.758340\pi\)
−0.725388 + 0.688340i \(0.758340\pi\)
\(822\) −54.0000 −1.88347
\(823\) 4.00000 0.139431 0.0697156 0.997567i \(-0.477791\pi\)
0.0697156 + 0.997567i \(0.477791\pi\)
\(824\) −17.3205 −0.603388
\(825\) 0 0
\(826\) 0 0
\(827\) 20.7846 0.722752 0.361376 0.932420i \(-0.382307\pi\)
0.361376 + 0.932420i \(0.382307\pi\)
\(828\) 6.00000 0.208514
\(829\) 25.0000 0.868286 0.434143 0.900844i \(-0.357051\pi\)
0.434143 + 0.900844i \(0.357051\pi\)
\(830\) 41.5692 1.44289
\(831\) −14.0000 −0.485655
\(832\) 0 0
\(833\) 0 0
\(834\) 13.8564 0.479808
\(835\) 24.0000 0.830554
\(836\) 0 0
\(837\) 13.8564 0.478947
\(838\) 31.1769 1.07699
\(839\) −45.0333 −1.55472 −0.777361 0.629054i \(-0.783442\pi\)
−0.777361 + 0.629054i \(0.783442\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 27.0000 0.930481
\(843\) 45.0333 1.55103
\(844\) 10.0000 0.344214
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 0 0
\(848\) 15.0000 0.515102
\(849\) −8.00000 −0.274559
\(850\) −10.3923 −0.356453
\(851\) −51.9615 −1.78122
\(852\) 6.92820 0.237356
\(853\) −25.9808 −0.889564 −0.444782 0.895639i \(-0.646719\pi\)
−0.444782 + 0.895639i \(0.646719\pi\)
\(854\) 0 0
\(855\) 6.00000 0.205196
\(856\) 10.3923 0.355202
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 13.8564 0.472500
\(861\) 0 0
\(862\) 12.0000 0.408722
\(863\) 27.7128 0.943355 0.471678 0.881771i \(-0.343649\pi\)
0.471678 + 0.881771i \(0.343649\pi\)
\(864\) 20.7846 0.707107
\(865\) −10.3923 −0.353349
\(866\) 29.4449 1.00058
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) 0 0
\(870\) −18.0000 −0.610257
\(871\) 0 0
\(872\) 24.0000 0.812743
\(873\) 6.92820 0.234484
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) 3.46410 0.117041
\(877\) −12.1244 −0.409410 −0.204705 0.978824i \(-0.565624\pi\)
−0.204705 + 0.978824i \(0.565624\pi\)
\(878\) 48.4974 1.63671
\(879\) 10.3923 0.350524
\(880\) 0 0
\(881\) 27.0000 0.909653 0.454827 0.890580i \(-0.349701\pi\)
0.454827 + 0.890580i \(0.349701\pi\)
\(882\) 0 0
\(883\) −10.0000 −0.336527 −0.168263 0.985742i \(-0.553816\pi\)
−0.168263 + 0.985742i \(0.553816\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 20.7846 0.698273
\(887\) −36.0000 −1.20876 −0.604381 0.796696i \(-0.706579\pi\)
−0.604381 + 0.796696i \(0.706579\pi\)
\(888\) 30.0000 1.00673
\(889\) 0 0
\(890\) −20.7846 −0.696702
\(891\) 0 0
\(892\) 10.3923 0.347960
\(893\) −12.0000 −0.401565
\(894\) 66.0000 2.20737
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 12.0000 0.400445
\(899\) 10.3923 0.346603
\(900\) −2.00000 −0.0666667
\(901\) 9.00000 0.299833
\(902\) 0 0
\(903\) 0 0
\(904\) −25.9808 −0.864107
\(905\) −19.0526 −0.633328
\(906\) 60.0000 1.99337
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 24.2487 0.804722
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) −34.6410 −1.14708
\(913\) 0 0
\(914\) −3.00000 −0.0992312
\(915\) −3.46410 −0.114520
\(916\) 0 0
\(917\) 0 0
\(918\) 20.7846 0.685994
\(919\) 22.0000 0.725713 0.362857 0.931845i \(-0.381802\pi\)
0.362857 + 0.931845i \(0.381802\pi\)
\(920\) −18.0000 −0.593442
\(921\) −34.6410 −1.14146
\(922\) −39.0000 −1.28440
\(923\) 0 0
\(924\) 0 0
\(925\) 17.3205 0.569495
\(926\) −24.0000 −0.788689
\(927\) −10.0000 −0.328443
\(928\) 15.5885 0.511716
\(929\) −46.7654 −1.53432 −0.767161 0.641455i \(-0.778331\pi\)
−0.767161 + 0.641455i \(0.778331\pi\)
\(930\) −20.7846 −0.681554
\(931\) 0 0
\(932\) −6.00000 −0.196537
\(933\) 60.0000 1.96431
\(934\) −20.7846 −0.680093
\(935\) 0 0
\(936\) 0 0
\(937\) −7.00000 −0.228680 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(938\) 0 0
\(939\) 20.0000 0.652675
\(940\) −6.00000 −0.195698
\(941\) 20.7846 0.677559 0.338779 0.940866i \(-0.389986\pi\)
0.338779 + 0.940866i \(0.389986\pi\)
\(942\) 45.0333 1.46726
\(943\) 31.1769 1.01526
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) 0 0
\(947\) 17.3205 0.562841 0.281420 0.959585i \(-0.409194\pi\)
0.281420 + 0.959585i \(0.409194\pi\)
\(948\) −8.00000 −0.259828
\(949\) 0 0
\(950\) −12.0000 −0.389331
\(951\) −10.3923 −0.336994
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 5.19615 0.168232
\(955\) −31.1769 −1.00886
\(956\) 20.7846 0.672222
\(957\) 0 0
\(958\) −42.0000 −1.35696
\(959\) 0 0
\(960\) 3.46410 0.111803
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) 1.73205 0.0557856
\(965\) −9.00000 −0.289720
\(966\) 0 0
\(967\) −58.8897 −1.89377 −0.946883 0.321578i \(-0.895787\pi\)
−0.946883 + 0.321578i \(0.895787\pi\)
\(968\) −19.0526 −0.612372
\(969\) −20.7846 −0.667698
\(970\) 20.7846 0.667354
\(971\) −6.00000 −0.192549 −0.0962746 0.995355i \(-0.530693\pi\)
−0.0962746 + 0.995355i \(0.530693\pi\)
\(972\) 10.0000 0.320750
\(973\) 0 0
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 5.00000 0.160046
\(977\) 43.3013 1.38533 0.692665 0.721259i \(-0.256436\pi\)
0.692665 + 0.721259i \(0.256436\pi\)
\(978\) 72.0000 2.30231
\(979\) 0 0
\(980\) 0 0
\(981\) 13.8564 0.442401
\(982\) 20.7846 0.663264
\(983\) −51.9615 −1.65732 −0.828658 0.559756i \(-0.810895\pi\)
−0.828658 + 0.559756i \(0.810895\pi\)
\(984\) −18.0000 −0.573819
\(985\) −24.0000 −0.764704
\(986\) 15.5885 0.496438
\(987\) 0 0
\(988\) 0 0
\(989\) −48.0000 −1.52631
\(990\) 0 0
\(991\) 2.00000 0.0635321 0.0317660 0.999495i \(-0.489887\pi\)
0.0317660 + 0.999495i \(0.489887\pi\)
\(992\) 18.0000 0.571501
\(993\) −55.4256 −1.75888
\(994\) 0 0
\(995\) 3.46410 0.109819
\(996\) −27.7128 −0.878114
\(997\) −17.0000 −0.538395 −0.269198 0.963085i \(-0.586759\pi\)
−0.269198 + 0.963085i \(0.586759\pi\)
\(998\) 54.0000 1.70934
\(999\) −34.6410 −1.09599
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 8281.2.a.q.1.1 2
7.6 odd 2 169.2.a.a.1.1 2
13.2 odd 12 637.2.q.a.589.1 2
13.7 odd 12 637.2.q.a.491.1 2
13.12 even 2 inner 8281.2.a.q.1.2 2
21.20 even 2 1521.2.a.k.1.2 2
28.27 even 2 2704.2.a.o.1.2 2
35.34 odd 2 4225.2.a.v.1.2 2
91.2 odd 12 637.2.k.c.459.1 2
91.6 even 12 169.2.e.a.23.1 2
91.20 even 12 13.2.e.a.10.1 yes 2
91.33 even 12 637.2.u.c.361.1 2
91.34 even 4 169.2.b.a.168.1 2
91.41 even 12 13.2.e.a.4.1 2
91.46 odd 12 637.2.k.c.569.1 2
91.48 odd 6 169.2.c.a.146.2 4
91.54 even 12 637.2.k.a.459.1 2
91.55 odd 6 169.2.c.a.22.2 4
91.59 even 12 637.2.k.a.569.1 2
91.62 odd 6 169.2.c.a.22.1 4
91.67 odd 12 637.2.u.b.30.1 2
91.69 odd 6 169.2.c.a.146.1 4
91.72 odd 12 637.2.u.b.361.1 2
91.76 even 12 169.2.e.a.147.1 2
91.80 even 12 637.2.u.c.30.1 2
91.83 even 4 169.2.b.a.168.2 2
91.90 odd 2 169.2.a.a.1.2 2
273.20 odd 12 117.2.q.c.10.1 2
273.41 odd 12 117.2.q.c.82.1 2
273.83 odd 4 1521.2.b.a.1351.1 2
273.125 odd 4 1521.2.b.a.1351.2 2
273.272 even 2 1521.2.a.k.1.1 2
364.83 odd 4 2704.2.f.b.337.1 2
364.111 odd 12 208.2.w.b.49.1 2
364.223 odd 12 208.2.w.b.17.1 2
364.307 odd 4 2704.2.f.b.337.2 2
364.363 even 2 2704.2.a.o.1.1 2
455.132 odd 12 325.2.m.a.199.2 4
455.202 odd 12 325.2.m.a.49.1 4
455.223 odd 12 325.2.m.a.199.1 4
455.293 odd 12 325.2.m.a.49.2 4
455.314 even 12 325.2.n.a.251.1 2
455.384 even 12 325.2.n.a.101.1 2
455.454 odd 2 4225.2.a.v.1.1 2
728.293 even 12 832.2.w.d.257.1 2
728.405 even 12 832.2.w.d.641.1 2
728.475 odd 12 832.2.w.a.257.1 2
728.587 odd 12 832.2.w.a.641.1 2
1092.587 even 12 1872.2.by.d.433.1 2
1092.839 even 12 1872.2.by.d.1297.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.2.e.a.4.1 2 91.41 even 12
13.2.e.a.10.1 yes 2 91.20 even 12
117.2.q.c.10.1 2 273.20 odd 12
117.2.q.c.82.1 2 273.41 odd 12
169.2.a.a.1.1 2 7.6 odd 2
169.2.a.a.1.2 2 91.90 odd 2
169.2.b.a.168.1 2 91.34 even 4
169.2.b.a.168.2 2 91.83 even 4
169.2.c.a.22.1 4 91.62 odd 6
169.2.c.a.22.2 4 91.55 odd 6
169.2.c.a.146.1 4 91.69 odd 6
169.2.c.a.146.2 4 91.48 odd 6
169.2.e.a.23.1 2 91.6 even 12
169.2.e.a.147.1 2 91.76 even 12
208.2.w.b.17.1 2 364.223 odd 12
208.2.w.b.49.1 2 364.111 odd 12
325.2.m.a.49.1 4 455.202 odd 12
325.2.m.a.49.2 4 455.293 odd 12
325.2.m.a.199.1 4 455.223 odd 12
325.2.m.a.199.2 4 455.132 odd 12
325.2.n.a.101.1 2 455.384 even 12
325.2.n.a.251.1 2 455.314 even 12
637.2.k.a.459.1 2 91.54 even 12
637.2.k.a.569.1 2 91.59 even 12
637.2.k.c.459.1 2 91.2 odd 12
637.2.k.c.569.1 2 91.46 odd 12
637.2.q.a.491.1 2 13.7 odd 12
637.2.q.a.589.1 2 13.2 odd 12
637.2.u.b.30.1 2 91.67 odd 12
637.2.u.b.361.1 2 91.72 odd 12
637.2.u.c.30.1 2 91.80 even 12
637.2.u.c.361.1 2 91.33 even 12
832.2.w.a.257.1 2 728.475 odd 12
832.2.w.a.641.1 2 728.587 odd 12
832.2.w.d.257.1 2 728.293 even 12
832.2.w.d.641.1 2 728.405 even 12
1521.2.a.k.1.1 2 273.272 even 2
1521.2.a.k.1.2 2 21.20 even 2
1521.2.b.a.1351.1 2 273.83 odd 4
1521.2.b.a.1351.2 2 273.125 odd 4
1872.2.by.d.433.1 2 1092.587 even 12
1872.2.by.d.1297.1 2 1092.839 even 12
2704.2.a.o.1.1 2 364.363 even 2
2704.2.a.o.1.2 2 28.27 even 2
2704.2.f.b.337.1 2 364.83 odd 4
2704.2.f.b.337.2 2 364.307 odd 4
4225.2.a.v.1.1 2 455.454 odd 2
4225.2.a.v.1.2 2 35.34 odd 2
8281.2.a.q.1.1 2 1.1 even 1 trivial
8281.2.a.q.1.2 2 13.12 even 2 inner