Properties

Label 1323.2.a.bc.1.4
Level $1323$
Weight $2$
Character 1323.1
Self dual yes
Analytic conductor $10.564$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,2,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, 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("1323.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.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: \(10.5642081874\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.7168.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.10100\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.10100 q^{2} +2.41421 q^{4} -2.10100 q^{5} +0.870264 q^{8} -4.41421 q^{10} -3.84153 q^{11} -6.82843 q^{13} -3.00000 q^{16} -1.23074 q^{17} -4.41421 q^{19} -5.07227 q^{20} -8.07107 q^{22} +6.81280 q^{23} -0.585786 q^{25} -14.3465 q^{26} +8.40401 q^{29} +0.656854 q^{31} -8.04354 q^{32} -2.58579 q^{34} +3.24264 q^{37} -9.27428 q^{38} -1.82843 q^{40} +5.07227 q^{41} -7.07107 q^{43} -9.27428 q^{44} +14.3137 q^{46} +10.6543 q^{47} -1.23074 q^{50} -16.4853 q^{52} -10.1445 q^{53} +8.07107 q^{55} +17.6569 q^{58} -2.97127 q^{59} +6.48528 q^{61} +1.38005 q^{62} -10.8995 q^{64} +14.3465 q^{65} -5.65685 q^{67} -2.97127 q^{68} -14.7070 q^{71} -11.4142 q^{73} +6.81280 q^{74} -10.6569 q^{76} +5.89949 q^{79} +6.30301 q^{80} +10.6569 q^{82} +0.720950 q^{83} +2.58579 q^{85} -14.8563 q^{86} -3.34315 q^{88} -0.149314 q^{89} +16.4475 q^{92} +22.3848 q^{94} +9.27428 q^{95} +1.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 12 q^{10} - 16 q^{13} - 12 q^{16} - 12 q^{19} - 4 q^{22} - 8 q^{25} - 20 q^{31} - 16 q^{34} - 4 q^{37} + 4 q^{40} + 12 q^{46} - 32 q^{52} + 4 q^{55} + 48 q^{58} - 8 q^{61} - 4 q^{64} - 40 q^{73}+ \cdots - 16 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.10100 1.48563 0.742817 0.669495i \(-0.233489\pi\)
0.742817 + 0.669495i \(0.233489\pi\)
\(3\) 0 0
\(4\) 2.41421 1.20711
\(5\) −2.10100 −0.939597 −0.469799 0.882774i \(-0.655673\pi\)
−0.469799 + 0.882774i \(0.655673\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0.870264 0.307685
\(9\) 0 0
\(10\) −4.41421 −1.39590
\(11\) −3.84153 −1.15827 −0.579133 0.815233i \(-0.696609\pi\)
−0.579133 + 0.815233i \(0.696609\pi\)
\(12\) 0 0
\(13\) −6.82843 −1.89386 −0.946932 0.321433i \(-0.895836\pi\)
−0.946932 + 0.321433i \(0.895836\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −3.00000 −0.750000
\(17\) −1.23074 −0.298498 −0.149249 0.988800i \(-0.547686\pi\)
−0.149249 + 0.988800i \(0.547686\pi\)
\(18\) 0 0
\(19\) −4.41421 −1.01269 −0.506345 0.862331i \(-0.669004\pi\)
−0.506345 + 0.862331i \(0.669004\pi\)
\(20\) −5.07227 −1.13419
\(21\) 0 0
\(22\) −8.07107 −1.72076
\(23\) 6.81280 1.42057 0.710283 0.703916i \(-0.248567\pi\)
0.710283 + 0.703916i \(0.248567\pi\)
\(24\) 0 0
\(25\) −0.585786 −0.117157
\(26\) −14.3465 −2.81359
\(27\) 0 0
\(28\) 0 0
\(29\) 8.40401 1.56059 0.780293 0.625414i \(-0.215070\pi\)
0.780293 + 0.625414i \(0.215070\pi\)
\(30\) 0 0
\(31\) 0.656854 0.117975 0.0589873 0.998259i \(-0.481213\pi\)
0.0589873 + 0.998259i \(0.481213\pi\)
\(32\) −8.04354 −1.42191
\(33\) 0 0
\(34\) −2.58579 −0.443459
\(35\) 0 0
\(36\) 0 0
\(37\) 3.24264 0.533087 0.266543 0.963823i \(-0.414118\pi\)
0.266543 + 0.963823i \(0.414118\pi\)
\(38\) −9.27428 −1.50449
\(39\) 0 0
\(40\) −1.82843 −0.289100
\(41\) 5.07227 0.792155 0.396078 0.918217i \(-0.370371\pi\)
0.396078 + 0.918217i \(0.370371\pi\)
\(42\) 0 0
\(43\) −7.07107 −1.07833 −0.539164 0.842201i \(-0.681260\pi\)
−0.539164 + 0.842201i \(0.681260\pi\)
\(44\) −9.27428 −1.39815
\(45\) 0 0
\(46\) 14.3137 2.11044
\(47\) 10.6543 1.55409 0.777047 0.629443i \(-0.216717\pi\)
0.777047 + 0.629443i \(0.216717\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.23074 −0.174053
\(51\) 0 0
\(52\) −16.4853 −2.28610
\(53\) −10.1445 −1.39346 −0.696730 0.717334i \(-0.745362\pi\)
−0.696730 + 0.717334i \(0.745362\pi\)
\(54\) 0 0
\(55\) 8.07107 1.08830
\(56\) 0 0
\(57\) 0 0
\(58\) 17.6569 2.31846
\(59\) −2.97127 −0.386826 −0.193413 0.981117i \(-0.561956\pi\)
−0.193413 + 0.981117i \(0.561956\pi\)
\(60\) 0 0
\(61\) 6.48528 0.830355 0.415178 0.909740i \(-0.363719\pi\)
0.415178 + 0.909740i \(0.363719\pi\)
\(62\) 1.38005 0.175267
\(63\) 0 0
\(64\) −10.8995 −1.36244
\(65\) 14.3465 1.77947
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) −2.97127 −0.360319
\(69\) 0 0
\(70\) 0 0
\(71\) −14.7070 −1.74540 −0.872701 0.488255i \(-0.837634\pi\)
−0.872701 + 0.488255i \(0.837634\pi\)
\(72\) 0 0
\(73\) −11.4142 −1.33593 −0.667966 0.744192i \(-0.732835\pi\)
−0.667966 + 0.744192i \(0.732835\pi\)
\(74\) 6.81280 0.791972
\(75\) 0 0
\(76\) −10.6569 −1.22243
\(77\) 0 0
\(78\) 0 0
\(79\) 5.89949 0.663745 0.331873 0.943324i \(-0.392320\pi\)
0.331873 + 0.943324i \(0.392320\pi\)
\(80\) 6.30301 0.704698
\(81\) 0 0
\(82\) 10.6569 1.17685
\(83\) 0.720950 0.0791346 0.0395673 0.999217i \(-0.487402\pi\)
0.0395673 + 0.999217i \(0.487402\pi\)
\(84\) 0 0
\(85\) 2.58579 0.280468
\(86\) −14.8563 −1.60200
\(87\) 0 0
\(88\) −3.34315 −0.356381
\(89\) −0.149314 −0.0158272 −0.00791361 0.999969i \(-0.502519\pi\)
−0.00791361 + 0.999969i \(0.502519\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 16.4475 1.71478
\(93\) 0 0
\(94\) 22.3848 2.30881
\(95\) 9.27428 0.951521
\(96\) 0 0
\(97\) 1.65685 0.168228 0.0841140 0.996456i \(-0.473194\pi\)
0.0841140 + 0.996456i \(0.473194\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1.41421 −0.141421
\(101\) −2.25032 −0.223915 −0.111957 0.993713i \(-0.535712\pi\)
−0.111957 + 0.993713i \(0.535712\pi\)
\(102\) 0 0
\(103\) −13.4853 −1.32874 −0.664372 0.747402i \(-0.731301\pi\)
−0.664372 + 0.747402i \(0.731301\pi\)
\(104\) −5.94253 −0.582713
\(105\) 0 0
\(106\) −21.3137 −2.07017
\(107\) −5.43275 −0.525203 −0.262602 0.964904i \(-0.584581\pi\)
−0.262602 + 0.964904i \(0.584581\pi\)
\(108\) 0 0
\(109\) −8.65685 −0.829176 −0.414588 0.910009i \(-0.636074\pi\)
−0.414588 + 0.910009i \(0.636074\pi\)
\(110\) 16.9573 1.61682
\(111\) 0 0
\(112\) 0 0
\(113\) −10.6543 −1.00227 −0.501137 0.865368i \(-0.667085\pi\)
−0.501137 + 0.865368i \(0.667085\pi\)
\(114\) 0 0
\(115\) −14.3137 −1.33476
\(116\) 20.2891 1.88379
\(117\) 0 0
\(118\) −6.24264 −0.574682
\(119\) 0 0
\(120\) 0 0
\(121\) 3.75736 0.341578
\(122\) 13.6256 1.23360
\(123\) 0 0
\(124\) 1.58579 0.142408
\(125\) 11.7358 1.04968
\(126\) 0 0
\(127\) 0.828427 0.0735110 0.0367555 0.999324i \(-0.488298\pi\)
0.0367555 + 0.999324i \(0.488298\pi\)
\(128\) −6.81280 −0.602172
\(129\) 0 0
\(130\) 30.1421 2.64364
\(131\) 7.89422 0.689721 0.344861 0.938654i \(-0.387926\pi\)
0.344861 + 0.938654i \(0.387926\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −11.8851 −1.02671
\(135\) 0 0
\(136\) −1.07107 −0.0918433
\(137\) 16.0871 1.37441 0.687206 0.726463i \(-0.258837\pi\)
0.687206 + 0.726463i \(0.258837\pi\)
\(138\) 0 0
\(139\) 3.17157 0.269009 0.134505 0.990913i \(-0.457056\pi\)
0.134505 + 0.990913i \(0.457056\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −30.8995 −2.59303
\(143\) 26.2316 2.19360
\(144\) 0 0
\(145\) −17.6569 −1.46632
\(146\) −23.9813 −1.98471
\(147\) 0 0
\(148\) 7.82843 0.643493
\(149\) 5.43275 0.445068 0.222534 0.974925i \(-0.428567\pi\)
0.222534 + 0.974925i \(0.428567\pi\)
\(150\) 0 0
\(151\) −4.24264 −0.345261 −0.172631 0.984987i \(-0.555227\pi\)
−0.172631 + 0.984987i \(0.555227\pi\)
\(152\) −3.84153 −0.311589
\(153\) 0 0
\(154\) 0 0
\(155\) −1.38005 −0.110849
\(156\) 0 0
\(157\) −1.51472 −0.120888 −0.0604439 0.998172i \(-0.519252\pi\)
−0.0604439 + 0.998172i \(0.519252\pi\)
\(158\) 12.3949 0.986082
\(159\) 0 0
\(160\) 16.8995 1.33602
\(161\) 0 0
\(162\) 0 0
\(163\) −0.343146 −0.0268772 −0.0134386 0.999910i \(-0.504278\pi\)
−0.0134386 + 0.999910i \(0.504278\pi\)
\(164\) 12.2455 0.956216
\(165\) 0 0
\(166\) 1.51472 0.117565
\(167\) 10.6543 0.824457 0.412228 0.911081i \(-0.364751\pi\)
0.412228 + 0.911081i \(0.364751\pi\)
\(168\) 0 0
\(169\) 33.6274 2.58672
\(170\) 5.43275 0.416673
\(171\) 0 0
\(172\) −17.0711 −1.30166
\(173\) 15.9378 1.21173 0.605863 0.795569i \(-0.292828\pi\)
0.605863 + 0.795569i \(0.292828\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 11.5246 0.868699
\(177\) 0 0
\(178\) −0.313708 −0.0235134
\(179\) −12.0962 −0.904115 −0.452057 0.891989i \(-0.649310\pi\)
−0.452057 + 0.891989i \(0.649310\pi\)
\(180\) 0 0
\(181\) −23.8995 −1.77644 −0.888218 0.459423i \(-0.848056\pi\)
−0.888218 + 0.459423i \(0.848056\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 5.92893 0.437087
\(185\) −6.81280 −0.500887
\(186\) 0 0
\(187\) 4.72792 0.345740
\(188\) 25.7218 1.87596
\(189\) 0 0
\(190\) 19.4853 1.41361
\(191\) −2.61079 −0.188910 −0.0944551 0.995529i \(-0.530111\pi\)
−0.0944551 + 0.995529i \(0.530111\pi\)
\(192\) 0 0
\(193\) 12.3431 0.888479 0.444240 0.895908i \(-0.353474\pi\)
0.444240 + 0.895908i \(0.353474\pi\)
\(194\) 3.48106 0.249925
\(195\) 0 0
\(196\) 0 0
\(197\) −11.3753 −0.810455 −0.405228 0.914216i \(-0.632808\pi\)
−0.405228 + 0.914216i \(0.632808\pi\)
\(198\) 0 0
\(199\) 18.0711 1.28102 0.640512 0.767948i \(-0.278722\pi\)
0.640512 + 0.767948i \(0.278722\pi\)
\(200\) −0.509789 −0.0360475
\(201\) 0 0
\(202\) −4.72792 −0.332655
\(203\) 0 0
\(204\) 0 0
\(205\) −10.6569 −0.744307
\(206\) −28.3326 −1.97403
\(207\) 0 0
\(208\) 20.4853 1.42040
\(209\) 16.9573 1.17296
\(210\) 0 0
\(211\) 0.242641 0.0167041 0.00835204 0.999965i \(-0.497341\pi\)
0.00835204 + 0.999965i \(0.497341\pi\)
\(212\) −24.4911 −1.68205
\(213\) 0 0
\(214\) −11.4142 −0.780260
\(215\) 14.8563 1.01319
\(216\) 0 0
\(217\) 0 0
\(218\) −18.1881 −1.23185
\(219\) 0 0
\(220\) 19.4853 1.31370
\(221\) 8.40401 0.565315
\(222\) 0 0
\(223\) −11.7279 −0.785360 −0.392680 0.919675i \(-0.628452\pi\)
−0.392680 + 0.919675i \(0.628452\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −22.3848 −1.48901
\(227\) 4.71179 0.312733 0.156366 0.987699i \(-0.450022\pi\)
0.156366 + 0.987699i \(0.450022\pi\)
\(228\) 0 0
\(229\) −18.4853 −1.22154 −0.610771 0.791807i \(-0.709140\pi\)
−0.610771 + 0.791807i \(0.709140\pi\)
\(230\) −30.0731 −1.98296
\(231\) 0 0
\(232\) 7.31371 0.480168
\(233\) 8.19285 0.536731 0.268366 0.963317i \(-0.413516\pi\)
0.268366 + 0.963317i \(0.413516\pi\)
\(234\) 0 0
\(235\) −22.3848 −1.46022
\(236\) −7.17327 −0.466940
\(237\) 0 0
\(238\) 0 0
\(239\) 2.97127 0.192195 0.0960976 0.995372i \(-0.469364\pi\)
0.0960976 + 0.995372i \(0.469364\pi\)
\(240\) 0 0
\(241\) −0.727922 −0.0468896 −0.0234448 0.999725i \(-0.507463\pi\)
−0.0234448 + 0.999725i \(0.507463\pi\)
\(242\) 7.89422 0.507460
\(243\) 0 0
\(244\) 15.6569 1.00233
\(245\) 0 0
\(246\) 0 0
\(247\) 30.1421 1.91790
\(248\) 0.571637 0.0362990
\(249\) 0 0
\(250\) 24.6569 1.55944
\(251\) −27.9721 −1.76559 −0.882793 0.469762i \(-0.844340\pi\)
−0.882793 + 0.469762i \(0.844340\pi\)
\(252\) 0 0
\(253\) −26.1716 −1.64539
\(254\) 1.74053 0.109210
\(255\) 0 0
\(256\) 7.48528 0.467830
\(257\) −29.0536 −1.81231 −0.906156 0.422944i \(-0.860997\pi\)
−0.906156 + 0.422944i \(0.860997\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 34.6356 2.14801
\(261\) 0 0
\(262\) 16.5858 1.02467
\(263\) −31.3039 −1.93028 −0.965140 0.261734i \(-0.915706\pi\)
−0.965140 + 0.261734i \(0.915706\pi\)
\(264\) 0 0
\(265\) 21.3137 1.30929
\(266\) 0 0
\(267\) 0 0
\(268\) −13.6569 −0.834225
\(269\) −5.07227 −0.309262 −0.154631 0.987972i \(-0.549419\pi\)
−0.154631 + 0.987972i \(0.549419\pi\)
\(270\) 0 0
\(271\) 11.1716 0.678625 0.339312 0.940674i \(-0.389806\pi\)
0.339312 + 0.940674i \(0.389806\pi\)
\(272\) 3.69222 0.223874
\(273\) 0 0
\(274\) 33.7990 2.04187
\(275\) 2.25032 0.135699
\(276\) 0 0
\(277\) −22.5563 −1.35528 −0.677640 0.735394i \(-0.736997\pi\)
−0.677640 + 0.735394i \(0.736997\pi\)
\(278\) 6.66348 0.399649
\(279\) 0 0
\(280\) 0 0
\(281\) 3.99084 0.238074 0.119037 0.992890i \(-0.462019\pi\)
0.119037 + 0.992890i \(0.462019\pi\)
\(282\) 0 0
\(283\) 8.48528 0.504398 0.252199 0.967675i \(-0.418846\pi\)
0.252199 + 0.967675i \(0.418846\pi\)
\(284\) −35.5059 −2.10689
\(285\) 0 0
\(286\) 55.1127 3.25888
\(287\) 0 0
\(288\) 0 0
\(289\) −15.4853 −0.910899
\(290\) −37.0971 −2.17842
\(291\) 0 0
\(292\) −27.5563 −1.61261
\(293\) 25.0009 1.46057 0.730283 0.683144i \(-0.239388\pi\)
0.730283 + 0.683144i \(0.239388\pi\)
\(294\) 0 0
\(295\) 6.24264 0.363461
\(296\) 2.82195 0.164023
\(297\) 0 0
\(298\) 11.4142 0.661208
\(299\) −46.5207 −2.69036
\(300\) 0 0
\(301\) 0 0
\(302\) −8.91380 −0.512932
\(303\) 0 0
\(304\) 13.2426 0.759518
\(305\) −13.6256 −0.780199
\(306\) 0 0
\(307\) 24.0711 1.37381 0.686904 0.726748i \(-0.258969\pi\)
0.686904 + 0.726748i \(0.258969\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −2.89949 −0.164680
\(311\) −13.3270 −0.755703 −0.377852 0.925866i \(-0.623337\pi\)
−0.377852 + 0.925866i \(0.623337\pi\)
\(312\) 0 0
\(313\) −19.1716 −1.08364 −0.541821 0.840494i \(-0.682265\pi\)
−0.541821 + 0.840494i \(0.682265\pi\)
\(314\) −3.18243 −0.179595
\(315\) 0 0
\(316\) 14.2426 0.801211
\(317\) −26.4428 −1.48517 −0.742587 0.669750i \(-0.766401\pi\)
−0.742587 + 0.669750i \(0.766401\pi\)
\(318\) 0 0
\(319\) −32.2843 −1.80757
\(320\) 22.8999 1.28014
\(321\) 0 0
\(322\) 0 0
\(323\) 5.43275 0.302286
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) −0.720950 −0.0399297
\(327\) 0 0
\(328\) 4.41421 0.243734
\(329\) 0 0
\(330\) 0 0
\(331\) 30.9706 1.70230 0.851148 0.524926i \(-0.175907\pi\)
0.851148 + 0.524926i \(0.175907\pi\)
\(332\) 1.74053 0.0955239
\(333\) 0 0
\(334\) 22.3848 1.22484
\(335\) 11.8851 0.649351
\(336\) 0 0
\(337\) 10.7990 0.588258 0.294129 0.955766i \(-0.404970\pi\)
0.294129 + 0.955766i \(0.404970\pi\)
\(338\) 70.6513 3.84292
\(339\) 0 0
\(340\) 6.24264 0.338555
\(341\) −2.52333 −0.136646
\(342\) 0 0
\(343\) 0 0
\(344\) −6.15370 −0.331785
\(345\) 0 0
\(346\) 33.4853 1.80018
\(347\) 1.59121 0.0854209 0.0427104 0.999087i \(-0.486401\pi\)
0.0427104 + 0.999087i \(0.486401\pi\)
\(348\) 0 0
\(349\) −13.0711 −0.699678 −0.349839 0.936810i \(-0.613764\pi\)
−0.349839 + 0.936810i \(0.613764\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 30.8995 1.64695
\(353\) 17.9769 0.956815 0.478407 0.878138i \(-0.341214\pi\)
0.478407 + 0.878138i \(0.341214\pi\)
\(354\) 0 0
\(355\) 30.8995 1.63997
\(356\) −0.360475 −0.0191051
\(357\) 0 0
\(358\) −25.4142 −1.34318
\(359\) 14.5577 0.768326 0.384163 0.923265i \(-0.374490\pi\)
0.384163 + 0.923265i \(0.374490\pi\)
\(360\) 0 0
\(361\) 0.485281 0.0255411
\(362\) −50.2129 −2.63913
\(363\) 0 0
\(364\) 0 0
\(365\) 23.9813 1.25524
\(366\) 0 0
\(367\) −29.3848 −1.53387 −0.766936 0.641723i \(-0.778220\pi\)
−0.766936 + 0.641723i \(0.778220\pi\)
\(368\) −20.4384 −1.06542
\(369\) 0 0
\(370\) −14.3137 −0.744134
\(371\) 0 0
\(372\) 0 0
\(373\) −15.1421 −0.784030 −0.392015 0.919959i \(-0.628222\pi\)
−0.392015 + 0.919959i \(0.628222\pi\)
\(374\) 9.93338 0.513643
\(375\) 0 0
\(376\) 9.27208 0.478171
\(377\) −57.3862 −2.95554
\(378\) 0 0
\(379\) 17.8995 0.919435 0.459718 0.888065i \(-0.347951\pi\)
0.459718 + 0.888065i \(0.347951\pi\)
\(380\) 22.3901 1.14859
\(381\) 0 0
\(382\) −5.48528 −0.280651
\(383\) 5.43275 0.277600 0.138800 0.990320i \(-0.455675\pi\)
0.138800 + 0.990320i \(0.455675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 25.9330 1.31995
\(387\) 0 0
\(388\) 4.00000 0.203069
\(389\) 6.66348 0.337852 0.168926 0.985629i \(-0.445970\pi\)
0.168926 + 0.985629i \(0.445970\pi\)
\(390\) 0 0
\(391\) −8.38478 −0.424036
\(392\) 0 0
\(393\) 0 0
\(394\) −23.8995 −1.20404
\(395\) −12.3949 −0.623653
\(396\) 0 0
\(397\) −23.0711 −1.15790 −0.578952 0.815362i \(-0.696538\pi\)
−0.578952 + 0.815362i \(0.696538\pi\)
\(398\) 37.9674 1.90313
\(399\) 0 0
\(400\) 1.75736 0.0878680
\(401\) −11.1641 −0.557509 −0.278755 0.960362i \(-0.589922\pi\)
−0.278755 + 0.960362i \(0.589922\pi\)
\(402\) 0 0
\(403\) −4.48528 −0.223428
\(404\) −5.43275 −0.270289
\(405\) 0 0
\(406\) 0 0
\(407\) −12.4567 −0.617456
\(408\) 0 0
\(409\) −5.07107 −0.250748 −0.125374 0.992110i \(-0.540013\pi\)
−0.125374 + 0.992110i \(0.540013\pi\)
\(410\) −22.3901 −1.10577
\(411\) 0 0
\(412\) −32.5563 −1.60394
\(413\) 0 0
\(414\) 0 0
\(415\) −1.51472 −0.0743546
\(416\) 54.9247 2.69291
\(417\) 0 0
\(418\) 35.6274 1.74259
\(419\) −12.3949 −0.605528 −0.302764 0.953066i \(-0.597909\pi\)
−0.302764 + 0.953066i \(0.597909\pi\)
\(420\) 0 0
\(421\) 7.92893 0.386433 0.193216 0.981156i \(-0.438108\pi\)
0.193216 + 0.981156i \(0.438108\pi\)
\(422\) 0.509789 0.0248161
\(423\) 0 0
\(424\) −8.82843 −0.428746
\(425\) 0.720950 0.0349712
\(426\) 0 0
\(427\) 0 0
\(428\) −13.1158 −0.633976
\(429\) 0 0
\(430\) 31.2132 1.50523
\(431\) 0.149314 0.00719219 0.00359609 0.999994i \(-0.498855\pi\)
0.00359609 + 0.999994i \(0.498855\pi\)
\(432\) 0 0
\(433\) 8.97056 0.431098 0.215549 0.976493i \(-0.430846\pi\)
0.215549 + 0.976493i \(0.430846\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −20.8995 −1.00090
\(437\) −30.0731 −1.43859
\(438\) 0 0
\(439\) 19.7990 0.944954 0.472477 0.881343i \(-0.343360\pi\)
0.472477 + 0.881343i \(0.343360\pi\)
\(440\) 7.02396 0.334854
\(441\) 0 0
\(442\) 17.6569 0.839851
\(443\) 11.5246 0.547550 0.273775 0.961794i \(-0.411728\pi\)
0.273775 + 0.961794i \(0.411728\pi\)
\(444\) 0 0
\(445\) 0.313708 0.0148712
\(446\) −24.6404 −1.16676
\(447\) 0 0
\(448\) 0 0
\(449\) 25.0009 1.17986 0.589932 0.807453i \(-0.299154\pi\)
0.589932 + 0.807453i \(0.299154\pi\)
\(450\) 0 0
\(451\) −19.4853 −0.917526
\(452\) −25.7218 −1.20985
\(453\) 0 0
\(454\) 9.89949 0.464606
\(455\) 0 0
\(456\) 0 0
\(457\) −13.9706 −0.653515 −0.326758 0.945108i \(-0.605956\pi\)
−0.326758 + 0.945108i \(0.605956\pi\)
\(458\) −38.8376 −1.81476
\(459\) 0 0
\(460\) −34.5563 −1.61120
\(461\) −20.8607 −0.971580 −0.485790 0.874075i \(-0.661468\pi\)
−0.485790 + 0.874075i \(0.661468\pi\)
\(462\) 0 0
\(463\) −7.89949 −0.367121 −0.183560 0.983008i \(-0.558762\pi\)
−0.183560 + 0.983008i \(0.558762\pi\)
\(464\) −25.2120 −1.17044
\(465\) 0 0
\(466\) 17.2132 0.797386
\(467\) 13.1158 0.606927 0.303464 0.952843i \(-0.401857\pi\)
0.303464 + 0.952843i \(0.401857\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −47.0305 −2.16935
\(471\) 0 0
\(472\) −2.58579 −0.119020
\(473\) 27.1637 1.24899
\(474\) 0 0
\(475\) 2.58579 0.118644
\(476\) 0 0
\(477\) 0 0
\(478\) 6.24264 0.285532
\(479\) −27.2512 −1.24514 −0.622569 0.782565i \(-0.713911\pi\)
−0.622569 + 0.782565i \(0.713911\pi\)
\(480\) 0 0
\(481\) −22.1421 −1.00959
\(482\) −1.52937 −0.0696607
\(483\) 0 0
\(484\) 9.07107 0.412321
\(485\) −3.48106 −0.158067
\(486\) 0 0
\(487\) 35.4142 1.60477 0.802386 0.596806i \(-0.203564\pi\)
0.802386 + 0.596806i \(0.203564\pi\)
\(488\) 5.64391 0.255488
\(489\) 0 0
\(490\) 0 0
\(491\) 22.8999 1.03346 0.516728 0.856149i \(-0.327150\pi\)
0.516728 + 0.856149i \(0.327150\pi\)
\(492\) 0 0
\(493\) −10.3431 −0.465832
\(494\) 63.3287 2.84929
\(495\) 0 0
\(496\) −1.97056 −0.0884809
\(497\) 0 0
\(498\) 0 0
\(499\) −10.4437 −0.467522 −0.233761 0.972294i \(-0.575103\pi\)
−0.233761 + 0.972294i \(0.575103\pi\)
\(500\) 28.3326 1.26707
\(501\) 0 0
\(502\) −58.7696 −2.62301
\(503\) −5.22158 −0.232819 −0.116409 0.993201i \(-0.537138\pi\)
−0.116409 + 0.993201i \(0.537138\pi\)
\(504\) 0 0
\(505\) 4.72792 0.210390
\(506\) −54.9866 −2.44445
\(507\) 0 0
\(508\) 2.00000 0.0887357
\(509\) 14.8563 0.658495 0.329248 0.944244i \(-0.393205\pi\)
0.329248 + 0.944244i \(0.393205\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 29.3522 1.29720
\(513\) 0 0
\(514\) −61.0416 −2.69243
\(515\) 28.3326 1.24848
\(516\) 0 0
\(517\) −40.9289 −1.80005
\(518\) 0 0
\(519\) 0 0
\(520\) 12.4853 0.547516
\(521\) 30.0731 1.31753 0.658764 0.752349i \(-0.271079\pi\)
0.658764 + 0.752349i \(0.271079\pi\)
\(522\) 0 0
\(523\) −43.7696 −1.91391 −0.956954 0.290238i \(-0.906265\pi\)
−0.956954 + 0.290238i \(0.906265\pi\)
\(524\) 19.0583 0.832567
\(525\) 0 0
\(526\) −65.7696 −2.86769
\(527\) −0.808416 −0.0352152
\(528\) 0 0
\(529\) 23.4142 1.01801
\(530\) 44.7802 1.94513
\(531\) 0 0
\(532\) 0 0
\(533\) −34.6356 −1.50024
\(534\) 0 0
\(535\) 11.4142 0.493479
\(536\) −4.92296 −0.212639
\(537\) 0 0
\(538\) −10.6569 −0.459450
\(539\) 0 0
\(540\) 0 0
\(541\) −21.3848 −0.919403 −0.459702 0.888073i \(-0.652044\pi\)
−0.459702 + 0.888073i \(0.652044\pi\)
\(542\) 23.4715 1.00819
\(543\) 0 0
\(544\) 9.89949 0.424437
\(545\) 18.1881 0.779092
\(546\) 0 0
\(547\) 7.51472 0.321306 0.160653 0.987011i \(-0.448640\pi\)
0.160653 + 0.987011i \(0.448640\pi\)
\(548\) 38.8376 1.65906
\(549\) 0 0
\(550\) 4.72792 0.201599
\(551\) −37.0971 −1.58039
\(552\) 0 0
\(553\) 0 0
\(554\) −47.3910 −2.01345
\(555\) 0 0
\(556\) 7.65685 0.324723
\(557\) −0.932112 −0.0394948 −0.0197474 0.999805i \(-0.506286\pi\)
−0.0197474 + 0.999805i \(0.506286\pi\)
\(558\) 0 0
\(559\) 48.2843 2.04221
\(560\) 0 0
\(561\) 0 0
\(562\) 8.38478 0.353690
\(563\) 8.10538 0.341601 0.170801 0.985306i \(-0.445365\pi\)
0.170801 + 0.985306i \(0.445365\pi\)
\(564\) 0 0
\(565\) 22.3848 0.941735
\(566\) 17.8276 0.749350
\(567\) 0 0
\(568\) −12.7990 −0.537034
\(569\) −24.2799 −1.01787 −0.508934 0.860806i \(-0.669960\pi\)
−0.508934 + 0.860806i \(0.669960\pi\)
\(570\) 0 0
\(571\) 16.0416 0.671321 0.335661 0.941983i \(-0.391040\pi\)
0.335661 + 0.941983i \(0.391040\pi\)
\(572\) 63.3287 2.64791
\(573\) 0 0
\(574\) 0 0
\(575\) −3.99084 −0.166430
\(576\) 0 0
\(577\) 29.3553 1.22208 0.611039 0.791600i \(-0.290752\pi\)
0.611039 + 0.791600i \(0.290752\pi\)
\(578\) −32.5346 −1.35326
\(579\) 0 0
\(580\) −42.6274 −1.77001
\(581\) 0 0
\(582\) 0 0
\(583\) 38.9706 1.61400
\(584\) −9.93338 −0.411046
\(585\) 0 0
\(586\) 52.5269 2.16987
\(587\) −17.0192 −0.702457 −0.351228 0.936290i \(-0.614236\pi\)
−0.351228 + 0.936290i \(0.614236\pi\)
\(588\) 0 0
\(589\) −2.89949 −0.119472
\(590\) 13.1158 0.539969
\(591\) 0 0
\(592\) −9.72792 −0.399815
\(593\) 32.5346 1.33604 0.668018 0.744145i \(-0.267143\pi\)
0.668018 + 0.744145i \(0.267143\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 13.1158 0.537244
\(597\) 0 0
\(598\) −97.7401 −3.99689
\(599\) 2.90942 0.118876 0.0594378 0.998232i \(-0.481069\pi\)
0.0594378 + 0.998232i \(0.481069\pi\)
\(600\) 0 0
\(601\) −26.6274 −1.08615 −0.543077 0.839683i \(-0.682741\pi\)
−0.543077 + 0.839683i \(0.682741\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10.2426 −0.416767
\(605\) −7.89422 −0.320946
\(606\) 0 0
\(607\) −6.68629 −0.271388 −0.135694 0.990751i \(-0.543326\pi\)
−0.135694 + 0.990751i \(0.543326\pi\)
\(608\) 35.5059 1.43995
\(609\) 0 0
\(610\) −28.6274 −1.15909
\(611\) −72.7523 −2.94324
\(612\) 0 0
\(613\) 5.62742 0.227289 0.113645 0.993521i \(-0.463747\pi\)
0.113645 + 0.993521i \(0.463747\pi\)
\(614\) 50.5734 2.04098
\(615\) 0 0
\(616\) 0 0
\(617\) 13.8368 0.557047 0.278523 0.960429i \(-0.410155\pi\)
0.278523 + 0.960429i \(0.410155\pi\)
\(618\) 0 0
\(619\) 34.4558 1.38490 0.692449 0.721467i \(-0.256532\pi\)
0.692449 + 0.721467i \(0.256532\pi\)
\(620\) −3.33174 −0.133806
\(621\) 0 0
\(622\) −28.0000 −1.12270
\(623\) 0 0
\(624\) 0 0
\(625\) −21.7279 −0.869117
\(626\) −40.2795 −1.60989
\(627\) 0 0
\(628\) −3.65685 −0.145924
\(629\) −3.99084 −0.159125
\(630\) 0 0
\(631\) −41.3553 −1.64633 −0.823165 0.567802i \(-0.807794\pi\)
−0.823165 + 0.567802i \(0.807794\pi\)
\(632\) 5.13412 0.204224
\(633\) 0 0
\(634\) −55.5563 −2.20642
\(635\) −1.74053 −0.0690707
\(636\) 0 0
\(637\) 0 0
\(638\) −67.8294 −2.68539
\(639\) 0 0
\(640\) 14.3137 0.565799
\(641\) 25.2995 0.999270 0.499635 0.866236i \(-0.333467\pi\)
0.499635 + 0.866236i \(0.333467\pi\)
\(642\) 0 0
\(643\) 11.2426 0.443366 0.221683 0.975119i \(-0.428845\pi\)
0.221683 + 0.975119i \(0.428845\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 11.4142 0.449086
\(647\) −5.43275 −0.213583 −0.106792 0.994281i \(-0.534058\pi\)
−0.106792 + 0.994281i \(0.534058\pi\)
\(648\) 0 0
\(649\) 11.4142 0.448047
\(650\) 8.40401 0.329632
\(651\) 0 0
\(652\) −0.828427 −0.0324437
\(653\) −21.2212 −0.830449 −0.415225 0.909719i \(-0.636297\pi\)
−0.415225 + 0.909719i \(0.636297\pi\)
\(654\) 0 0
\(655\) −16.5858 −0.648060
\(656\) −15.2168 −0.594117
\(657\) 0 0
\(658\) 0 0
\(659\) 23.1110 0.900278 0.450139 0.892958i \(-0.351374\pi\)
0.450139 + 0.892958i \(0.351374\pi\)
\(660\) 0 0
\(661\) −30.1421 −1.17239 −0.586197 0.810169i \(-0.699375\pi\)
−0.586197 + 0.810169i \(0.699375\pi\)
\(662\) 65.0692 2.52899
\(663\) 0 0
\(664\) 0.627417 0.0243485
\(665\) 0 0
\(666\) 0 0
\(667\) 57.2548 2.21692
\(668\) 25.7218 0.995207
\(669\) 0 0
\(670\) 24.9706 0.964697
\(671\) −24.9134 −0.961771
\(672\) 0 0
\(673\) 20.8284 0.802877 0.401438 0.915886i \(-0.368510\pi\)
0.401438 + 0.915886i \(0.368510\pi\)
\(674\) 22.6887 0.873936
\(675\) 0 0
\(676\) 81.1838 3.12245
\(677\) 15.4280 0.592945 0.296473 0.955041i \(-0.404190\pi\)
0.296473 + 0.955041i \(0.404190\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 2.25032 0.0862957
\(681\) 0 0
\(682\) −5.30152 −0.203006
\(683\) 23.1110 0.884319 0.442160 0.896936i \(-0.354212\pi\)
0.442160 + 0.896936i \(0.354212\pi\)
\(684\) 0 0
\(685\) −33.7990 −1.29139
\(686\) 0 0
\(687\) 0 0
\(688\) 21.2132 0.808746
\(689\) 69.2713 2.63902
\(690\) 0 0
\(691\) 39.1716 1.49016 0.745078 0.666977i \(-0.232412\pi\)
0.745078 + 0.666977i \(0.232412\pi\)
\(692\) 38.4772 1.46268
\(693\) 0 0
\(694\) 3.34315 0.126904
\(695\) −6.66348 −0.252760
\(696\) 0 0
\(697\) −6.24264 −0.236457
\(698\) −27.4624 −1.03947
\(699\) 0 0
\(700\) 0 0
\(701\) 4.20201 0.158708 0.0793538 0.996847i \(-0.474714\pi\)
0.0793538 + 0.996847i \(0.474714\pi\)
\(702\) 0 0
\(703\) −14.3137 −0.539852
\(704\) 41.8707 1.57806
\(705\) 0 0
\(706\) 37.7696 1.42148
\(707\) 0 0
\(708\) 0 0
\(709\) 42.6569 1.60201 0.801006 0.598656i \(-0.204299\pi\)
0.801006 + 0.598656i \(0.204299\pi\)
\(710\) 64.9199 2.43640
\(711\) 0 0
\(712\) −0.129942 −0.00486979
\(713\) 4.47502 0.167591
\(714\) 0 0
\(715\) −55.1127 −2.06110
\(716\) −29.2029 −1.09136
\(717\) 0 0
\(718\) 30.5858 1.14145
\(719\) 18.7597 0.699619 0.349810 0.936821i \(-0.386246\pi\)
0.349810 + 0.936821i \(0.386246\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.01958 0.0379447
\(723\) 0 0
\(724\) −57.6985 −2.14435
\(725\) −4.92296 −0.182834
\(726\) 0 0
\(727\) 26.0000 0.964287 0.482143 0.876092i \(-0.339858\pi\)
0.482143 + 0.876092i \(0.339858\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 50.3848 1.86482
\(731\) 8.70264 0.321879
\(732\) 0 0
\(733\) −15.7574 −0.582011 −0.291006 0.956721i \(-0.593990\pi\)
−0.291006 + 0.956721i \(0.593990\pi\)
\(734\) −61.7375 −2.27877
\(735\) 0 0
\(736\) −54.7990 −2.01992
\(737\) 21.7310 0.800471
\(738\) 0 0
\(739\) −29.5563 −1.08725 −0.543624 0.839329i \(-0.682948\pi\)
−0.543624 + 0.839329i \(0.682948\pi\)
\(740\) −16.4475 −0.604624
\(741\) 0 0
\(742\) 0 0
\(743\) 29.0536 1.06587 0.532936 0.846156i \(-0.321089\pi\)
0.532936 + 0.846156i \(0.321089\pi\)
\(744\) 0 0
\(745\) −11.4142 −0.418184
\(746\) −31.8137 −1.16478
\(747\) 0 0
\(748\) 11.4142 0.417345
\(749\) 0 0
\(750\) 0 0
\(751\) −12.1005 −0.441554 −0.220777 0.975324i \(-0.570859\pi\)
−0.220777 + 0.975324i \(0.570859\pi\)
\(752\) −31.9630 −1.16557
\(753\) 0 0
\(754\) −120.569 −4.39085
\(755\) 8.91380 0.324406
\(756\) 0 0
\(757\) −9.65685 −0.350984 −0.175492 0.984481i \(-0.556152\pi\)
−0.175492 + 0.984481i \(0.556152\pi\)
\(758\) 37.6069 1.36594
\(759\) 0 0
\(760\) 8.07107 0.292768
\(761\) −17.3178 −0.627770 −0.313885 0.949461i \(-0.601631\pi\)
−0.313885 + 0.949461i \(0.601631\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −6.30301 −0.228035
\(765\) 0 0
\(766\) 11.4142 0.412412
\(767\) 20.2891 0.732596
\(768\) 0 0
\(769\) 10.9706 0.395609 0.197804 0.980242i \(-0.436619\pi\)
0.197804 + 0.980242i \(0.436619\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 29.7990 1.07249
\(773\) −34.9961 −1.25872 −0.629361 0.777113i \(-0.716683\pi\)
−0.629361 + 0.777113i \(0.716683\pi\)
\(774\) 0 0
\(775\) −0.384776 −0.0138216
\(776\) 1.44190 0.0517612
\(777\) 0 0
\(778\) 14.0000 0.501924
\(779\) −22.3901 −0.802208
\(780\) 0 0
\(781\) 56.4975 2.02164
\(782\) −17.6164 −0.629963
\(783\) 0 0
\(784\) 0 0
\(785\) 3.18243 0.113586
\(786\) 0 0
\(787\) 6.34315 0.226109 0.113054 0.993589i \(-0.463937\pi\)
0.113054 + 0.993589i \(0.463937\pi\)
\(788\) −27.4624 −0.978306
\(789\) 0 0
\(790\) −26.0416 −0.926520
\(791\) 0 0
\(792\) 0 0
\(793\) −44.2843 −1.57258
\(794\) −48.4724 −1.72022
\(795\) 0 0
\(796\) 43.6274 1.54633
\(797\) −40.9386 −1.45012 −0.725060 0.688685i \(-0.758188\pi\)
−0.725060 + 0.688685i \(0.758188\pi\)
\(798\) 0 0
\(799\) −13.1127 −0.463894
\(800\) 4.71179 0.166587
\(801\) 0 0
\(802\) −23.4558 −0.828255
\(803\) 43.8481 1.54736
\(804\) 0 0
\(805\) 0 0
\(806\) −9.42359 −0.331932
\(807\) 0 0
\(808\) −1.95837 −0.0688952
\(809\) −33.1937 −1.16703 −0.583515 0.812103i \(-0.698323\pi\)
−0.583515 + 0.812103i \(0.698323\pi\)
\(810\) 0 0
\(811\) 7.62742 0.267835 0.133917 0.990992i \(-0.457244\pi\)
0.133917 + 0.990992i \(0.457244\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −26.1716 −0.917313
\(815\) 0.720950 0.0252538
\(816\) 0 0
\(817\) 31.2132 1.09201
\(818\) −10.6543 −0.372520
\(819\) 0 0
\(820\) −25.7279 −0.898458
\(821\) 5.13412 0.179182 0.0895910 0.995979i \(-0.471444\pi\)
0.0895910 + 0.995979i \(0.471444\pi\)
\(822\) 0 0
\(823\) −25.2721 −0.880929 −0.440465 0.897770i \(-0.645186\pi\)
−0.440465 + 0.897770i \(0.645186\pi\)
\(824\) −11.7358 −0.408834
\(825\) 0 0
\(826\) 0 0
\(827\) −24.3418 −0.846446 −0.423223 0.906025i \(-0.639101\pi\)
−0.423223 + 0.906025i \(0.639101\pi\)
\(828\) 0 0
\(829\) 3.21320 0.111599 0.0557996 0.998442i \(-0.482229\pi\)
0.0557996 + 0.998442i \(0.482229\pi\)
\(830\) −3.18243 −0.110464
\(831\) 0 0
\(832\) 74.4264 2.58027
\(833\) 0 0
\(834\) 0 0
\(835\) −22.3848 −0.774657
\(836\) 40.9386 1.41589
\(837\) 0 0
\(838\) −26.0416 −0.899593
\(839\) 3.48106 0.120179 0.0600897 0.998193i \(-0.480861\pi\)
0.0600897 + 0.998193i \(0.480861\pi\)
\(840\) 0 0
\(841\) 41.6274 1.43543
\(842\) 16.6587 0.574097
\(843\) 0 0
\(844\) 0.585786 0.0201636
\(845\) −70.6513 −2.43048
\(846\) 0 0
\(847\) 0 0
\(848\) 30.4336 1.04509
\(849\) 0 0
\(850\) 1.51472 0.0519544
\(851\) 22.0915 0.757285
\(852\) 0 0
\(853\) −33.1716 −1.13577 −0.567887 0.823107i \(-0.692239\pi\)
−0.567887 + 0.823107i \(0.692239\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.72792 −0.161597
\(857\) −34.0640 −1.16360 −0.581802 0.813331i \(-0.697652\pi\)
−0.581802 + 0.813331i \(0.697652\pi\)
\(858\) 0 0
\(859\) 31.7279 1.08254 0.541271 0.840848i \(-0.317943\pi\)
0.541271 + 0.840848i \(0.317943\pi\)
\(860\) 35.8664 1.22303
\(861\) 0 0
\(862\) 0.313708 0.0106850
\(863\) 33.7035 1.14728 0.573640 0.819107i \(-0.305531\pi\)
0.573640 + 0.819107i \(0.305531\pi\)
\(864\) 0 0
\(865\) −33.4853 −1.13853
\(866\) 18.8472 0.640453
\(867\) 0 0
\(868\) 0 0
\(869\) −22.6631 −0.768793
\(870\) 0 0
\(871\) 38.6274 1.30884
\(872\) −7.53375 −0.255125
\(873\) 0 0
\(874\) −63.1838 −2.13722
\(875\) 0 0
\(876\) 0 0
\(877\) −26.8284 −0.905932 −0.452966 0.891528i \(-0.649634\pi\)
−0.452966 + 0.891528i \(0.649634\pi\)
\(878\) 41.5977 1.40386
\(879\) 0 0
\(880\) −24.2132 −0.816227
\(881\) −9.69660 −0.326687 −0.163343 0.986569i \(-0.552228\pi\)
−0.163343 + 0.986569i \(0.552228\pi\)
\(882\) 0 0
\(883\) −44.6274 −1.50183 −0.750916 0.660398i \(-0.770388\pi\)
−0.750916 + 0.660398i \(0.770388\pi\)
\(884\) 20.2891 0.682396
\(885\) 0 0
\(886\) 24.2132 0.813458
\(887\) −48.6835 −1.63463 −0.817317 0.576189i \(-0.804539\pi\)
−0.817317 + 0.576189i \(0.804539\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0.659102 0.0220932
\(891\) 0 0
\(892\) −28.3137 −0.948013
\(893\) −47.0305 −1.57382
\(894\) 0 0
\(895\) 25.4142 0.849503
\(896\) 0 0
\(897\) 0 0
\(898\) 52.5269 1.75285
\(899\) 5.52021 0.184109
\(900\) 0 0
\(901\) 12.4853 0.415945
\(902\) −40.9386 −1.36311
\(903\) 0 0
\(904\) −9.27208 −0.308385
\(905\) 50.2129 1.66913
\(906\) 0 0
\(907\) 25.2721 0.839146 0.419573 0.907722i \(-0.362180\pi\)
0.419573 + 0.907722i \(0.362180\pi\)
\(908\) 11.3753 0.377502
\(909\) 0 0
\(910\) 0 0
\(911\) 51.9534 1.72129 0.860647 0.509202i \(-0.170059\pi\)
0.860647 + 0.509202i \(0.170059\pi\)
\(912\) 0 0
\(913\) −2.76955 −0.0916588
\(914\) −29.3522 −0.970884
\(915\) 0 0
\(916\) −44.6274 −1.47453
\(917\) 0 0
\(918\) 0 0
\(919\) −18.8284 −0.621093 −0.310546 0.950558i \(-0.600512\pi\)
−0.310546 + 0.950558i \(0.600512\pi\)
\(920\) −12.4567 −0.410685
\(921\) 0 0
\(922\) −43.8284 −1.44341
\(923\) 100.426 3.30556
\(924\) 0 0
\(925\) −1.89949 −0.0624550
\(926\) −16.5969 −0.545407
\(927\) 0 0
\(928\) −67.5980 −2.21901
\(929\) −11.0767 −0.363413 −0.181707 0.983353i \(-0.558162\pi\)
−0.181707 + 0.983353i \(0.558162\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 19.7793 0.647892
\(933\) 0 0
\(934\) 27.5563 0.901671
\(935\) −9.93338 −0.324856
\(936\) 0 0
\(937\) −59.2548 −1.93577 −0.967886 0.251391i \(-0.919112\pi\)
−0.967886 + 0.251391i \(0.919112\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −54.0416 −1.76264
\(941\) −29.0536 −0.947119 −0.473560 0.880762i \(-0.657031\pi\)
−0.473560 + 0.880762i \(0.657031\pi\)
\(942\) 0 0
\(943\) 34.5563 1.12531
\(944\) 8.91380 0.290120
\(945\) 0 0
\(946\) 57.0711 1.85554
\(947\) 31.3039 1.01724 0.508620 0.860991i \(-0.330156\pi\)
0.508620 + 0.860991i \(0.330156\pi\)
\(948\) 0 0
\(949\) 77.9411 2.53008
\(950\) 5.43275 0.176262
\(951\) 0 0
\(952\) 0 0
\(953\) −40.2795 −1.30478 −0.652391 0.757883i \(-0.726234\pi\)
−0.652391 + 0.757883i \(0.726234\pi\)
\(954\) 0 0
\(955\) 5.48528 0.177500
\(956\) 7.17327 0.232000
\(957\) 0 0
\(958\) −57.2548 −1.84982
\(959\) 0 0
\(960\) 0 0
\(961\) −30.5685 −0.986082
\(962\) −46.5207 −1.49989
\(963\) 0 0
\(964\) −1.75736 −0.0566007
\(965\) −25.9330 −0.834812
\(966\) 0 0
\(967\) 1.89949 0.0610836 0.0305418 0.999533i \(-0.490277\pi\)
0.0305418 + 0.999533i \(0.490277\pi\)
\(968\) 3.26989 0.105098
\(969\) 0 0
\(970\) −7.31371 −0.234829
\(971\) −24.4911 −0.785956 −0.392978 0.919548i \(-0.628555\pi\)
−0.392978 + 0.919548i \(0.628555\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 74.4054 2.38410
\(975\) 0 0
\(976\) −19.4558 −0.622766
\(977\) −53.9051 −1.72458 −0.862289 0.506417i \(-0.830970\pi\)
−0.862289 + 0.506417i \(0.830970\pi\)
\(978\) 0 0
\(979\) 0.573593 0.0183321
\(980\) 0 0
\(981\) 0 0
\(982\) 48.1127 1.53534
\(983\) 15.5773 0.496838 0.248419 0.968653i \(-0.420089\pi\)
0.248419 + 0.968653i \(0.420089\pi\)
\(984\) 0 0
\(985\) 23.8995 0.761501
\(986\) −21.7310 −0.692055
\(987\) 0 0
\(988\) 72.7696 2.31511
\(989\) −48.1738 −1.53184
\(990\) 0 0
\(991\) 26.9706 0.856748 0.428374 0.903601i \(-0.359087\pi\)
0.428374 + 0.903601i \(0.359087\pi\)
\(992\) −5.28343 −0.167749
\(993\) 0 0
\(994\) 0 0
\(995\) −37.9674 −1.20365
\(996\) 0 0
\(997\) −52.8701 −1.67441 −0.837206 0.546888i \(-0.815812\pi\)
−0.837206 + 0.546888i \(0.815812\pi\)
\(998\) −21.9421 −0.694566
\(999\) 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 1323.2.a.bc.1.4 yes 4
3.2 odd 2 inner 1323.2.a.bc.1.1 4
7.6 odd 2 1323.2.a.bd.1.4 yes 4
21.20 even 2 1323.2.a.bd.1.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.2.a.bc.1.1 4 3.2 odd 2 inner
1323.2.a.bc.1.4 yes 4 1.1 even 1 trivial
1323.2.a.bd.1.1 yes 4 21.20 even 2
1323.2.a.bd.1.4 yes 4 7.6 odd 2