Properties

Label 480.2.k.b.241.2
Level $480$
Weight $2$
Character 480.241
Analytic conductor $3.833$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [480,2,Mod(241,480)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(480, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("480.241");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 480.k (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.83281929702\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 241.2
Root \(0.264658 + 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 480.241
Dual form 480.2.k.b.241.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +1.00000i q^{5} -0.941367 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +1.00000i q^{5} -0.941367 q^{7} -1.00000 q^{9} -4.49828i q^{11} -5.55691i q^{13} +1.00000 q^{15} +7.55691 q^{17} -1.05863i q^{19} +0.941367i q^{21} +1.05863 q^{23} -1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} -3.55691 q^{31} -4.49828 q^{33} -0.941367i q^{35} -7.43965i q^{37} -5.55691 q^{39} -3.88273 q^{41} +1.88273i q^{43} -1.00000i q^{45} +10.0552 q^{47} -6.11383 q^{49} -7.55691i q^{51} +2.00000i q^{53} +4.49828 q^{55} -1.05863 q^{57} +8.49828i q^{59} +8.99656i q^{61} +0.941367 q^{63} +5.55691 q^{65} +4.00000i q^{67} -1.05863i q^{69} +12.9966 q^{71} -6.00000 q^{73} +1.00000i q^{75} +4.23453i q^{77} -11.5569 q^{79} +1.00000 q^{81} -5.88273i q^{83} +7.55691i q^{85} -2.00000 q^{87} -4.11727 q^{89} +5.23109i q^{91} +3.55691i q^{93} +1.05863 q^{95} +17.1138 q^{97} +4.49828i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{7} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{7} - 6 q^{9} + 6 q^{15} + 12 q^{17} + 8 q^{23} - 6 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} - 36 q^{73} - 36 q^{79} + 6 q^{81} - 12 q^{87} - 28 q^{89} + 8 q^{95} + 36 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/480\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(421\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) −0.941367 −0.355803 −0.177902 0.984048i \(-0.556931\pi\)
−0.177902 + 0.984048i \(0.556931\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) − 4.49828i − 1.35628i −0.734931 0.678141i \(-0.762786\pi\)
0.734931 0.678141i \(-0.237214\pi\)
\(12\) 0 0
\(13\) − 5.55691i − 1.54121i −0.637313 0.770605i \(-0.719954\pi\)
0.637313 0.770605i \(-0.280046\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 7.55691 1.83282 0.916410 0.400240i \(-0.131073\pi\)
0.916410 + 0.400240i \(0.131073\pi\)
\(18\) 0 0
\(19\) − 1.05863i − 0.242867i −0.992600 0.121434i \(-0.961251\pi\)
0.992600 0.121434i \(-0.0387491\pi\)
\(20\) 0 0
\(21\) 0.941367i 0.205423i
\(22\) 0 0
\(23\) 1.05863 0.220740 0.110370 0.993891i \(-0.464796\pi\)
0.110370 + 0.993891i \(0.464796\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 2.00000i − 0.371391i −0.982607 0.185695i \(-0.940546\pi\)
0.982607 0.185695i \(-0.0594537\pi\)
\(30\) 0 0
\(31\) −3.55691 −0.638841 −0.319420 0.947613i \(-0.603488\pi\)
−0.319420 + 0.947613i \(0.603488\pi\)
\(32\) 0 0
\(33\) −4.49828 −0.783050
\(34\) 0 0
\(35\) − 0.941367i − 0.159120i
\(36\) 0 0
\(37\) − 7.43965i − 1.22307i −0.791217 0.611535i \(-0.790552\pi\)
0.791217 0.611535i \(-0.209448\pi\)
\(38\) 0 0
\(39\) −5.55691 −0.889818
\(40\) 0 0
\(41\) −3.88273 −0.606381 −0.303191 0.952930i \(-0.598052\pi\)
−0.303191 + 0.952930i \(0.598052\pi\)
\(42\) 0 0
\(43\) 1.88273i 0.287114i 0.989642 + 0.143557i \(0.0458541\pi\)
−0.989642 + 0.143557i \(0.954146\pi\)
\(44\) 0 0
\(45\) − 1.00000i − 0.149071i
\(46\) 0 0
\(47\) 10.0552 1.46670 0.733350 0.679851i \(-0.237955\pi\)
0.733350 + 0.679851i \(0.237955\pi\)
\(48\) 0 0
\(49\) −6.11383 −0.873404
\(50\) 0 0
\(51\) − 7.55691i − 1.05818i
\(52\) 0 0
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) 0 0
\(55\) 4.49828 0.606548
\(56\) 0 0
\(57\) −1.05863 −0.140219
\(58\) 0 0
\(59\) 8.49828i 1.10638i 0.833054 + 0.553191i \(0.186590\pi\)
−0.833054 + 0.553191i \(0.813410\pi\)
\(60\) 0 0
\(61\) 8.99656i 1.15189i 0.817488 + 0.575946i \(0.195366\pi\)
−0.817488 + 0.575946i \(0.804634\pi\)
\(62\) 0 0
\(63\) 0.941367 0.118601
\(64\) 0 0
\(65\) 5.55691 0.689250
\(66\) 0 0
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) 0 0
\(69\) − 1.05863i − 0.127444i
\(70\) 0 0
\(71\) 12.9966 1.54241 0.771204 0.636588i \(-0.219655\pi\)
0.771204 + 0.636588i \(0.219655\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 1.00000i 0.115470i
\(76\) 0 0
\(77\) 4.23453i 0.482570i
\(78\) 0 0
\(79\) −11.5569 −1.30025 −0.650127 0.759825i \(-0.725284\pi\)
−0.650127 + 0.759825i \(0.725284\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 5.88273i − 0.645714i −0.946448 0.322857i \(-0.895357\pi\)
0.946448 0.322857i \(-0.104643\pi\)
\(84\) 0 0
\(85\) 7.55691i 0.819662i
\(86\) 0 0
\(87\) −2.00000 −0.214423
\(88\) 0 0
\(89\) −4.11727 −0.436429 −0.218215 0.975901i \(-0.570023\pi\)
−0.218215 + 0.975901i \(0.570023\pi\)
\(90\) 0 0
\(91\) 5.23109i 0.548368i
\(92\) 0 0
\(93\) 3.55691i 0.368835i
\(94\) 0 0
\(95\) 1.05863 0.108613
\(96\) 0 0
\(97\) 17.1138 1.73765 0.868823 0.495123i \(-0.164877\pi\)
0.868823 + 0.495123i \(0.164877\pi\)
\(98\) 0 0
\(99\) 4.49828i 0.452094i
\(100\) 0 0
\(101\) 2.00000i 0.199007i 0.995037 + 0.0995037i \(0.0317255\pi\)
−0.995037 + 0.0995037i \(0.968274\pi\)
\(102\) 0 0
\(103\) −10.1725 −1.00232 −0.501161 0.865354i \(-0.667094\pi\)
−0.501161 + 0.865354i \(0.667094\pi\)
\(104\) 0 0
\(105\) −0.941367 −0.0918680
\(106\) 0 0
\(107\) 17.2311i 1.66579i 0.553429 + 0.832896i \(0.313319\pi\)
−0.553429 + 0.832896i \(0.686681\pi\)
\(108\) 0 0
\(109\) 1.88273i 0.180333i 0.995927 + 0.0901666i \(0.0287399\pi\)
−0.995927 + 0.0901666i \(0.971260\pi\)
\(110\) 0 0
\(111\) −7.43965 −0.706140
\(112\) 0 0
\(113\) 15.3224 1.44141 0.720704 0.693243i \(-0.243819\pi\)
0.720704 + 0.693243i \(0.243819\pi\)
\(114\) 0 0
\(115\) 1.05863i 0.0987181i
\(116\) 0 0
\(117\) 5.55691i 0.513737i
\(118\) 0 0
\(119\) −7.11383 −0.652124
\(120\) 0 0
\(121\) −9.23453 −0.839503
\(122\) 0 0
\(123\) 3.88273i 0.350094i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) 18.1725 1.61255 0.806273 0.591544i \(-0.201481\pi\)
0.806273 + 0.591544i \(0.201481\pi\)
\(128\) 0 0
\(129\) 1.88273 0.165765
\(130\) 0 0
\(131\) 6.38101i 0.557512i 0.960362 + 0.278756i \(0.0899220\pi\)
−0.960362 + 0.278756i \(0.910078\pi\)
\(132\) 0 0
\(133\) 0.996562i 0.0864129i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −4.44309 −0.379598 −0.189799 0.981823i \(-0.560784\pi\)
−0.189799 + 0.981823i \(0.560784\pi\)
\(138\) 0 0
\(139\) − 20.1725i − 1.71101i −0.517798 0.855503i \(-0.673248\pi\)
0.517798 0.855503i \(-0.326752\pi\)
\(140\) 0 0
\(141\) − 10.0552i − 0.846800i
\(142\) 0 0
\(143\) −24.9966 −2.09032
\(144\) 0 0
\(145\) 2.00000 0.166091
\(146\) 0 0
\(147\) 6.11383i 0.504260i
\(148\) 0 0
\(149\) − 2.00000i − 0.163846i −0.996639 0.0819232i \(-0.973894\pi\)
0.996639 0.0819232i \(-0.0261062\pi\)
\(150\) 0 0
\(151\) −9.67418 −0.787274 −0.393637 0.919266i \(-0.628783\pi\)
−0.393637 + 0.919266i \(0.628783\pi\)
\(152\) 0 0
\(153\) −7.55691 −0.610940
\(154\) 0 0
\(155\) − 3.55691i − 0.285698i
\(156\) 0 0
\(157\) 4.32582i 0.345238i 0.984989 + 0.172619i \(0.0552229\pi\)
−0.984989 + 0.172619i \(0.944777\pi\)
\(158\) 0 0
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) −0.996562 −0.0785401
\(162\) 0 0
\(163\) 6.11727i 0.479141i 0.970879 + 0.239571i \(0.0770067\pi\)
−0.970879 + 0.239571i \(0.922993\pi\)
\(164\) 0 0
\(165\) − 4.49828i − 0.350191i
\(166\) 0 0
\(167\) 6.05520 0.468565 0.234283 0.972169i \(-0.424726\pi\)
0.234283 + 0.972169i \(0.424726\pi\)
\(168\) 0 0
\(169\) −17.8793 −1.37533
\(170\) 0 0
\(171\) 1.05863i 0.0809557i
\(172\) 0 0
\(173\) 16.8793i 1.28331i 0.766994 + 0.641655i \(0.221752\pi\)
−0.766994 + 0.641655i \(0.778248\pi\)
\(174\) 0 0
\(175\) 0.941367 0.0711606
\(176\) 0 0
\(177\) 8.49828 0.638770
\(178\) 0 0
\(179\) − 10.6155i − 0.793443i −0.917939 0.396722i \(-0.870148\pi\)
0.917939 0.396722i \(-0.129852\pi\)
\(180\) 0 0
\(181\) 14.1173i 1.04933i 0.851309 + 0.524664i \(0.175809\pi\)
−0.851309 + 0.524664i \(0.824191\pi\)
\(182\) 0 0
\(183\) 8.99656 0.665045
\(184\) 0 0
\(185\) 7.43965 0.546974
\(186\) 0 0
\(187\) − 33.9931i − 2.48582i
\(188\) 0 0
\(189\) − 0.941367i − 0.0684744i
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −4.87930 −0.351219 −0.175610 0.984460i \(-0.556190\pi\)
−0.175610 + 0.984460i \(0.556190\pi\)
\(194\) 0 0
\(195\) − 5.55691i − 0.397939i
\(196\) 0 0
\(197\) − 2.88617i − 0.205631i −0.994700 0.102816i \(-0.967215\pi\)
0.994700 0.102816i \(-0.0327852\pi\)
\(198\) 0 0
\(199\) 17.6742 1.25289 0.626445 0.779466i \(-0.284509\pi\)
0.626445 + 0.779466i \(0.284509\pi\)
\(200\) 0 0
\(201\) 4.00000 0.282138
\(202\) 0 0
\(203\) 1.88273i 0.132142i
\(204\) 0 0
\(205\) − 3.88273i − 0.271182i
\(206\) 0 0
\(207\) −1.05863 −0.0735801
\(208\) 0 0
\(209\) −4.76203 −0.329396
\(210\) 0 0
\(211\) 23.9379i 1.64795i 0.566623 + 0.823977i \(0.308250\pi\)
−0.566623 + 0.823977i \(0.691750\pi\)
\(212\) 0 0
\(213\) − 12.9966i − 0.890510i
\(214\) 0 0
\(215\) −1.88273 −0.128401
\(216\) 0 0
\(217\) 3.34836 0.227302
\(218\) 0 0
\(219\) 6.00000i 0.405442i
\(220\) 0 0
\(221\) − 41.9931i − 2.82476i
\(222\) 0 0
\(223\) 24.0552 1.61086 0.805428 0.592694i \(-0.201936\pi\)
0.805428 + 0.592694i \(0.201936\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) − 11.1138i − 0.737651i −0.929499 0.368825i \(-0.879760\pi\)
0.929499 0.368825i \(-0.120240\pi\)
\(228\) 0 0
\(229\) − 17.2311i − 1.13866i −0.822108 0.569331i \(-0.807202\pi\)
0.822108 0.569331i \(-0.192798\pi\)
\(230\) 0 0
\(231\) 4.23453 0.278612
\(232\) 0 0
\(233\) −8.44309 −0.553125 −0.276562 0.960996i \(-0.589195\pi\)
−0.276562 + 0.960996i \(0.589195\pi\)
\(234\) 0 0
\(235\) 10.0552i 0.655929i
\(236\) 0 0
\(237\) 11.5569i 0.750702i
\(238\) 0 0
\(239\) −10.1173 −0.654432 −0.327216 0.944950i \(-0.606110\pi\)
−0.327216 + 0.944950i \(0.606110\pi\)
\(240\) 0 0
\(241\) 16.8793 1.08729 0.543646 0.839315i \(-0.317044\pi\)
0.543646 + 0.839315i \(0.317044\pi\)
\(242\) 0 0
\(243\) − 1.00000i − 0.0641500i
\(244\) 0 0
\(245\) − 6.11383i − 0.390598i
\(246\) 0 0
\(247\) −5.88273 −0.374309
\(248\) 0 0
\(249\) −5.88273 −0.372803
\(250\) 0 0
\(251\) − 11.8466i − 0.747753i −0.927478 0.373877i \(-0.878028\pi\)
0.927478 0.373877i \(-0.121972\pi\)
\(252\) 0 0
\(253\) − 4.76203i − 0.299386i
\(254\) 0 0
\(255\) 7.55691 0.473232
\(256\) 0 0
\(257\) −10.6707 −0.665623 −0.332811 0.942993i \(-0.607997\pi\)
−0.332811 + 0.942993i \(0.607997\pi\)
\(258\) 0 0
\(259\) 7.00344i 0.435172i
\(260\) 0 0
\(261\) 2.00000i 0.123797i
\(262\) 0 0
\(263\) 1.94480 0.119922 0.0599609 0.998201i \(-0.480902\pi\)
0.0599609 + 0.998201i \(0.480902\pi\)
\(264\) 0 0
\(265\) −2.00000 −0.122859
\(266\) 0 0
\(267\) 4.11727i 0.251973i
\(268\) 0 0
\(269\) 9.76547i 0.595411i 0.954658 + 0.297706i \(0.0962214\pi\)
−0.954658 + 0.297706i \(0.903779\pi\)
\(270\) 0 0
\(271\) −3.44652 −0.209361 −0.104681 0.994506i \(-0.533382\pi\)
−0.104681 + 0.994506i \(0.533382\pi\)
\(272\) 0 0
\(273\) 5.23109 0.316600
\(274\) 0 0
\(275\) 4.49828i 0.271257i
\(276\) 0 0
\(277\) − 18.7880i − 1.12886i −0.825480 0.564431i \(-0.809096\pi\)
0.825480 0.564431i \(-0.190904\pi\)
\(278\) 0 0
\(279\) 3.55691 0.212947
\(280\) 0 0
\(281\) −16.8793 −1.00693 −0.503467 0.864014i \(-0.667943\pi\)
−0.503467 + 0.864014i \(0.667943\pi\)
\(282\) 0 0
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 0 0
\(285\) − 1.05863i − 0.0627080i
\(286\) 0 0
\(287\) 3.65508 0.215752
\(288\) 0 0
\(289\) 40.1070 2.35923
\(290\) 0 0
\(291\) − 17.1138i − 1.00323i
\(292\) 0 0
\(293\) − 20.2277i − 1.18171i −0.806777 0.590856i \(-0.798790\pi\)
0.806777 0.590856i \(-0.201210\pi\)
\(294\) 0 0
\(295\) −8.49828 −0.494789
\(296\) 0 0
\(297\) 4.49828 0.261017
\(298\) 0 0
\(299\) − 5.88273i − 0.340207i
\(300\) 0 0
\(301\) − 1.77234i − 0.102156i
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 0 0
\(305\) −8.99656 −0.515142
\(306\) 0 0
\(307\) − 8.11039i − 0.462884i −0.972849 0.231442i \(-0.925656\pi\)
0.972849 0.231442i \(-0.0743444\pi\)
\(308\) 0 0
\(309\) 10.1725i 0.578691i
\(310\) 0 0
\(311\) −31.8759 −1.80751 −0.903757 0.428046i \(-0.859202\pi\)
−0.903757 + 0.428046i \(0.859202\pi\)
\(312\) 0 0
\(313\) −5.11383 −0.289051 −0.144525 0.989501i \(-0.546166\pi\)
−0.144525 + 0.989501i \(0.546166\pi\)
\(314\) 0 0
\(315\) 0.941367i 0.0530400i
\(316\) 0 0
\(317\) 24.6448i 1.38419i 0.721807 + 0.692094i \(0.243312\pi\)
−0.721807 + 0.692094i \(0.756688\pi\)
\(318\) 0 0
\(319\) −8.99656 −0.503711
\(320\) 0 0
\(321\) 17.2311 0.961746
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) 5.55691i 0.308242i
\(326\) 0 0
\(327\) 1.88273 0.104115
\(328\) 0 0
\(329\) −9.46563 −0.521857
\(330\) 0 0
\(331\) 11.0518i 0.607460i 0.952758 + 0.303730i \(0.0982320\pi\)
−0.952758 + 0.303730i \(0.901768\pi\)
\(332\) 0 0
\(333\) 7.43965i 0.407690i
\(334\) 0 0
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) −19.9931 −1.08909 −0.544547 0.838730i \(-0.683299\pi\)
−0.544547 + 0.838730i \(0.683299\pi\)
\(338\) 0 0
\(339\) − 15.3224i − 0.832198i
\(340\) 0 0
\(341\) 16.0000i 0.866449i
\(342\) 0 0
\(343\) 12.3449 0.666563
\(344\) 0 0
\(345\) 1.05863 0.0569949
\(346\) 0 0
\(347\) 6.87930i 0.369300i 0.982804 + 0.184650i \(0.0591151\pi\)
−0.982804 + 0.184650i \(0.940885\pi\)
\(348\) 0 0
\(349\) − 4.76203i − 0.254906i −0.991845 0.127453i \(-0.959320\pi\)
0.991845 0.127453i \(-0.0406801\pi\)
\(350\) 0 0
\(351\) 5.55691 0.296606
\(352\) 0 0
\(353\) −3.79145 −0.201798 −0.100899 0.994897i \(-0.532172\pi\)
−0.100899 + 0.994897i \(0.532172\pi\)
\(354\) 0 0
\(355\) 12.9966i 0.689786i
\(356\) 0 0
\(357\) 7.11383i 0.376504i
\(358\) 0 0
\(359\) 12.9966 0.685932 0.342966 0.939348i \(-0.388568\pi\)
0.342966 + 0.939348i \(0.388568\pi\)
\(360\) 0 0
\(361\) 17.8793 0.941016
\(362\) 0 0
\(363\) 9.23453i 0.484687i
\(364\) 0 0
\(365\) − 6.00000i − 0.314054i
\(366\) 0 0
\(367\) −22.9345 −1.19717 −0.598585 0.801059i \(-0.704270\pi\)
−0.598585 + 0.801059i \(0.704270\pi\)
\(368\) 0 0
\(369\) 3.88273 0.202127
\(370\) 0 0
\(371\) − 1.88273i − 0.0977467i
\(372\) 0 0
\(373\) 15.4396i 0.799435i 0.916638 + 0.399717i \(0.130892\pi\)
−0.916638 + 0.399717i \(0.869108\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −11.1138 −0.572391
\(378\) 0 0
\(379\) 6.28973i 0.323082i 0.986866 + 0.161541i \(0.0516463\pi\)
−0.986866 + 0.161541i \(0.948354\pi\)
\(380\) 0 0
\(381\) − 18.1725i − 0.931003i
\(382\) 0 0
\(383\) −2.94137 −0.150297 −0.0751484 0.997172i \(-0.523943\pi\)
−0.0751484 + 0.997172i \(0.523943\pi\)
\(384\) 0 0
\(385\) −4.23453 −0.215812
\(386\) 0 0
\(387\) − 1.88273i − 0.0957047i
\(388\) 0 0
\(389\) 12.2277i 0.619967i 0.950742 + 0.309983i \(0.100324\pi\)
−0.950742 + 0.309983i \(0.899676\pi\)
\(390\) 0 0
\(391\) 8.00000 0.404577
\(392\) 0 0
\(393\) 6.38101 0.321880
\(394\) 0 0
\(395\) − 11.5569i − 0.581491i
\(396\) 0 0
\(397\) − 5.32238i − 0.267123i −0.991041 0.133561i \(-0.957359\pi\)
0.991041 0.133561i \(-0.0426413\pi\)
\(398\) 0 0
\(399\) 0.996562 0.0498905
\(400\) 0 0
\(401\) 6.99656 0.349392 0.174696 0.984622i \(-0.444106\pi\)
0.174696 + 0.984622i \(0.444106\pi\)
\(402\) 0 0
\(403\) 19.7655i 0.984588i
\(404\) 0 0
\(405\) 1.00000i 0.0496904i
\(406\) 0 0
\(407\) −33.4656 −1.65883
\(408\) 0 0
\(409\) 16.2277 0.802406 0.401203 0.915989i \(-0.368592\pi\)
0.401203 + 0.915989i \(0.368592\pi\)
\(410\) 0 0
\(411\) 4.44309i 0.219161i
\(412\) 0 0
\(413\) − 8.00000i − 0.393654i
\(414\) 0 0
\(415\) 5.88273 0.288772
\(416\) 0 0
\(417\) −20.1725 −0.987850
\(418\) 0 0
\(419\) 15.6121i 0.762701i 0.924430 + 0.381351i \(0.124541\pi\)
−0.924430 + 0.381351i \(0.875459\pi\)
\(420\) 0 0
\(421\) 33.2311i 1.61958i 0.586717 + 0.809792i \(0.300420\pi\)
−0.586717 + 0.809792i \(0.699580\pi\)
\(422\) 0 0
\(423\) −10.0552 −0.488900
\(424\) 0 0
\(425\) −7.55691 −0.366564
\(426\) 0 0
\(427\) − 8.46907i − 0.409847i
\(428\) 0 0
\(429\) 24.9966i 1.20685i
\(430\) 0 0
\(431\) 12.9966 0.626022 0.313011 0.949749i \(-0.398662\pi\)
0.313011 + 0.949749i \(0.398662\pi\)
\(432\) 0 0
\(433\) 20.2277 0.972079 0.486040 0.873937i \(-0.338441\pi\)
0.486040 + 0.873937i \(0.338441\pi\)
\(434\) 0 0
\(435\) − 2.00000i − 0.0958927i
\(436\) 0 0
\(437\) − 1.12070i − 0.0536106i
\(438\) 0 0
\(439\) 5.43965 0.259620 0.129810 0.991539i \(-0.458563\pi\)
0.129810 + 0.991539i \(0.458563\pi\)
\(440\) 0 0
\(441\) 6.11383 0.291135
\(442\) 0 0
\(443\) 15.3484i 0.729223i 0.931160 + 0.364611i \(0.118798\pi\)
−0.931160 + 0.364611i \(0.881202\pi\)
\(444\) 0 0
\(445\) − 4.11727i − 0.195177i
\(446\) 0 0
\(447\) −2.00000 −0.0945968
\(448\) 0 0
\(449\) 4.22766 0.199515 0.0997577 0.995012i \(-0.468193\pi\)
0.0997577 + 0.995012i \(0.468193\pi\)
\(450\) 0 0
\(451\) 17.4656i 0.822424i
\(452\) 0 0
\(453\) 9.67418i 0.454533i
\(454\) 0 0
\(455\) −5.23109 −0.245238
\(456\) 0 0
\(457\) 2.65164 0.124038 0.0620192 0.998075i \(-0.480246\pi\)
0.0620192 + 0.998075i \(0.480246\pi\)
\(458\) 0 0
\(459\) 7.55691i 0.352727i
\(460\) 0 0
\(461\) − 10.2345i − 0.476670i −0.971183 0.238335i \(-0.923398\pi\)
0.971183 0.238335i \(-0.0766016\pi\)
\(462\) 0 0
\(463\) −19.0586 −0.885730 −0.442865 0.896588i \(-0.646038\pi\)
−0.442865 + 0.896588i \(0.646038\pi\)
\(464\) 0 0
\(465\) −3.55691 −0.164948
\(466\) 0 0
\(467\) 4.11039i 0.190206i 0.995467 + 0.0951031i \(0.0303181\pi\)
−0.995467 + 0.0951031i \(0.969682\pi\)
\(468\) 0 0
\(469\) − 3.76547i − 0.173873i
\(470\) 0 0
\(471\) 4.32582 0.199323
\(472\) 0 0
\(473\) 8.46907 0.389408
\(474\) 0 0
\(475\) 1.05863i 0.0485734i
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) 25.2311 1.15284 0.576419 0.817154i \(-0.304450\pi\)
0.576419 + 0.817154i \(0.304450\pi\)
\(480\) 0 0
\(481\) −41.3415 −1.88501
\(482\) 0 0
\(483\) 0.996562i 0.0453451i
\(484\) 0 0
\(485\) 17.1138i 0.777099i
\(486\) 0 0
\(487\) 21.9379 0.994102 0.497051 0.867721i \(-0.334416\pi\)
0.497051 + 0.867721i \(0.334416\pi\)
\(488\) 0 0
\(489\) 6.11727 0.276632
\(490\) 0 0
\(491\) 7.50172i 0.338548i 0.985569 + 0.169274i \(0.0541423\pi\)
−0.985569 + 0.169274i \(0.945858\pi\)
\(492\) 0 0
\(493\) − 15.1138i − 0.680693i
\(494\) 0 0
\(495\) −4.49828 −0.202183
\(496\) 0 0
\(497\) −12.2345 −0.548794
\(498\) 0 0
\(499\) 29.1690i 1.30578i 0.757451 + 0.652892i \(0.226445\pi\)
−0.757451 + 0.652892i \(0.773555\pi\)
\(500\) 0 0
\(501\) − 6.05520i − 0.270526i
\(502\) 0 0
\(503\) 23.9379 1.06734 0.533670 0.845693i \(-0.320813\pi\)
0.533670 + 0.845693i \(0.320813\pi\)
\(504\) 0 0
\(505\) −2.00000 −0.0889988
\(506\) 0 0
\(507\) 17.8793i 0.794047i
\(508\) 0 0
\(509\) 28.6967i 1.27196i 0.771706 + 0.635980i \(0.219404\pi\)
−0.771706 + 0.635980i \(0.780596\pi\)
\(510\) 0 0
\(511\) 5.64820 0.249862
\(512\) 0 0
\(513\) 1.05863 0.0467398
\(514\) 0 0
\(515\) − 10.1725i − 0.448252i
\(516\) 0 0
\(517\) − 45.2311i − 1.98926i
\(518\) 0 0
\(519\) 16.8793 0.740919
\(520\) 0 0
\(521\) 10.0000 0.438108 0.219054 0.975713i \(-0.429703\pi\)
0.219054 + 0.975713i \(0.429703\pi\)
\(522\) 0 0
\(523\) − 25.7586i − 1.12634i −0.826340 0.563172i \(-0.809581\pi\)
0.826340 0.563172i \(-0.190419\pi\)
\(524\) 0 0
\(525\) − 0.941367i − 0.0410846i
\(526\) 0 0
\(527\) −26.8793 −1.17088
\(528\) 0 0
\(529\) −21.8793 −0.951274
\(530\) 0 0
\(531\) − 8.49828i − 0.368794i
\(532\) 0 0
\(533\) 21.5760i 0.934561i
\(534\) 0 0
\(535\) −17.2311 −0.744965
\(536\) 0 0
\(537\) −10.6155 −0.458095
\(538\) 0 0
\(539\) 27.5017i 1.18458i
\(540\) 0 0
\(541\) 12.3449i 0.530750i 0.964145 + 0.265375i \(0.0854957\pi\)
−0.964145 + 0.265375i \(0.914504\pi\)
\(542\) 0 0
\(543\) 14.1173 0.605830
\(544\) 0 0
\(545\) −1.88273 −0.0806475
\(546\) 0 0
\(547\) 19.8759i 0.849830i 0.905233 + 0.424915i \(0.139696\pi\)
−0.905233 + 0.424915i \(0.860304\pi\)
\(548\) 0 0
\(549\) − 8.99656i − 0.383964i
\(550\) 0 0
\(551\) −2.11727 −0.0901986
\(552\) 0 0
\(553\) 10.8793 0.462635
\(554\) 0 0
\(555\) − 7.43965i − 0.315795i
\(556\) 0 0
\(557\) 3.12070i 0.132228i 0.997812 + 0.0661142i \(0.0210602\pi\)
−0.997812 + 0.0661142i \(0.978940\pi\)
\(558\) 0 0
\(559\) 10.4622 0.442503
\(560\) 0 0
\(561\) −33.9931 −1.43519
\(562\) 0 0
\(563\) 0.651639i 0.0274633i 0.999906 + 0.0137317i \(0.00437106\pi\)
−0.999906 + 0.0137317i \(0.995629\pi\)
\(564\) 0 0
\(565\) 15.3224i 0.644617i
\(566\) 0 0
\(567\) −0.941367 −0.0395337
\(568\) 0 0
\(569\) 26.9966 1.13175 0.565877 0.824489i \(-0.308538\pi\)
0.565877 + 0.824489i \(0.308538\pi\)
\(570\) 0 0
\(571\) − 14.9414i − 0.625277i −0.949872 0.312638i \(-0.898787\pi\)
0.949872 0.312638i \(-0.101213\pi\)
\(572\) 0 0
\(573\) − 8.00000i − 0.334205i
\(574\) 0 0
\(575\) −1.05863 −0.0441481
\(576\) 0 0
\(577\) −8.87930 −0.369650 −0.184825 0.982771i \(-0.559172\pi\)
−0.184825 + 0.982771i \(0.559172\pi\)
\(578\) 0 0
\(579\) 4.87930i 0.202777i
\(580\) 0 0
\(581\) 5.53781i 0.229747i
\(582\) 0 0
\(583\) 8.99656 0.372600
\(584\) 0 0
\(585\) −5.55691 −0.229750
\(586\) 0 0
\(587\) − 1.23109i − 0.0508127i −0.999677 0.0254064i \(-0.991912\pi\)
0.999677 0.0254064i \(-0.00808797\pi\)
\(588\) 0 0
\(589\) 3.76547i 0.155153i
\(590\) 0 0
\(591\) −2.88617 −0.118721
\(592\) 0 0
\(593\) 3.55691 0.146065 0.0730325 0.997330i \(-0.476732\pi\)
0.0730325 + 0.997330i \(0.476732\pi\)
\(594\) 0 0
\(595\) − 7.11383i − 0.291639i
\(596\) 0 0
\(597\) − 17.6742i − 0.723356i
\(598\) 0 0
\(599\) 19.2242 0.785480 0.392740 0.919649i \(-0.371527\pi\)
0.392740 + 0.919649i \(0.371527\pi\)
\(600\) 0 0
\(601\) −27.7586 −1.13230 −0.566148 0.824303i \(-0.691567\pi\)
−0.566148 + 0.824303i \(0.691567\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) − 9.23453i − 0.375437i
\(606\) 0 0
\(607\) −7.16902 −0.290982 −0.145491 0.989360i \(-0.546476\pi\)
−0.145491 + 0.989360i \(0.546476\pi\)
\(608\) 0 0
\(609\) 1.88273 0.0762922
\(610\) 0 0
\(611\) − 55.8759i − 2.26050i
\(612\) 0 0
\(613\) 9.55691i 0.386000i 0.981199 + 0.193000i \(0.0618218\pi\)
−0.981199 + 0.193000i \(0.938178\pi\)
\(614\) 0 0
\(615\) −3.88273 −0.156567
\(616\) 0 0
\(617\) 1.32926 0.0535139 0.0267569 0.999642i \(-0.491482\pi\)
0.0267569 + 0.999642i \(0.491482\pi\)
\(618\) 0 0
\(619\) − 28.1725i − 1.13235i −0.824286 0.566173i \(-0.808423\pi\)
0.824286 0.566173i \(-0.191577\pi\)
\(620\) 0 0
\(621\) 1.05863i 0.0424815i
\(622\) 0 0
\(623\) 3.87586 0.155283
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 4.76203i 0.190177i
\(628\) 0 0
\(629\) − 56.2208i − 2.24167i
\(630\) 0 0
\(631\) 23.3224 0.928449 0.464225 0.885717i \(-0.346333\pi\)
0.464225 + 0.885717i \(0.346333\pi\)
\(632\) 0 0
\(633\) 23.9379 0.951447
\(634\) 0 0
\(635\) 18.1725i 0.721152i
\(636\) 0 0
\(637\) 33.9740i 1.34610i
\(638\) 0 0
\(639\) −12.9966 −0.514136
\(640\) 0 0
\(641\) 27.1070 1.07066 0.535330 0.844643i \(-0.320187\pi\)
0.535330 + 0.844643i \(0.320187\pi\)
\(642\) 0 0
\(643\) − 20.3449i − 0.802325i −0.916007 0.401163i \(-0.868606\pi\)
0.916007 0.401163i \(-0.131394\pi\)
\(644\) 0 0
\(645\) 1.88273i 0.0741326i
\(646\) 0 0
\(647\) −37.6965 −1.48200 −0.741002 0.671503i \(-0.765649\pi\)
−0.741002 + 0.671503i \(0.765649\pi\)
\(648\) 0 0
\(649\) 38.2277 1.50057
\(650\) 0 0
\(651\) − 3.34836i − 0.131233i
\(652\) 0 0
\(653\) − 8.64476i − 0.338296i −0.985591 0.169148i \(-0.945898\pi\)
0.985591 0.169148i \(-0.0541015\pi\)
\(654\) 0 0
\(655\) −6.38101 −0.249327
\(656\) 0 0
\(657\) 6.00000 0.234082
\(658\) 0 0
\(659\) − 29.2603i − 1.13982i −0.821707 0.569910i \(-0.806978\pi\)
0.821707 0.569910i \(-0.193022\pi\)
\(660\) 0 0
\(661\) 28.7620i 1.11871i 0.828927 + 0.559357i \(0.188952\pi\)
−0.828927 + 0.559357i \(0.811048\pi\)
\(662\) 0 0
\(663\) −41.9931 −1.63088
\(664\) 0 0
\(665\) −0.996562 −0.0386450
\(666\) 0 0
\(667\) − 2.11727i − 0.0819809i
\(668\) 0 0
\(669\) − 24.0552i − 0.930028i
\(670\) 0 0
\(671\) 40.4691 1.56229
\(672\) 0 0
\(673\) −18.0000 −0.693849 −0.346925 0.937893i \(-0.612774\pi\)
−0.346925 + 0.937893i \(0.612774\pi\)
\(674\) 0 0
\(675\) − 1.00000i − 0.0384900i
\(676\) 0 0
\(677\) − 42.8724i − 1.64772i −0.566793 0.823860i \(-0.691816\pi\)
0.566793 0.823860i \(-0.308184\pi\)
\(678\) 0 0
\(679\) −16.1104 −0.618260
\(680\) 0 0
\(681\) −11.1138 −0.425883
\(682\) 0 0
\(683\) 26.1173i 0.999349i 0.866213 + 0.499675i \(0.166547\pi\)
−0.866213 + 0.499675i \(0.833453\pi\)
\(684\) 0 0
\(685\) − 4.44309i − 0.169762i
\(686\) 0 0
\(687\) −17.2311 −0.657407
\(688\) 0 0
\(689\) 11.1138 0.423403
\(690\) 0 0
\(691\) 5.29317i 0.201362i 0.994919 + 0.100681i \(0.0321021\pi\)
−0.994919 + 0.100681i \(0.967898\pi\)
\(692\) 0 0
\(693\) − 4.23453i − 0.160857i
\(694\) 0 0
\(695\) 20.1725 0.765185
\(696\) 0 0
\(697\) −29.3415 −1.11139
\(698\) 0 0
\(699\) 8.44309i 0.319347i
\(700\) 0 0
\(701\) 7.99312i 0.301896i 0.988542 + 0.150948i \(0.0482326\pi\)
−0.988542 + 0.150948i \(0.951767\pi\)
\(702\) 0 0
\(703\) −7.87586 −0.297044
\(704\) 0 0
\(705\) 10.0552 0.378701
\(706\) 0 0
\(707\) − 1.88273i − 0.0708075i
\(708\) 0 0
\(709\) − 28.9966i − 1.08899i −0.838764 0.544494i \(-0.816722\pi\)
0.838764 0.544494i \(-0.183278\pi\)
\(710\) 0 0
\(711\) 11.5569 0.433418
\(712\) 0 0
\(713\) −3.76547 −0.141018
\(714\) 0 0
\(715\) − 24.9966i − 0.934818i
\(716\) 0 0
\(717\) 10.1173i 0.377836i
\(718\) 0 0
\(719\) −26.8793 −1.00243 −0.501214 0.865323i \(-0.667113\pi\)
−0.501214 + 0.865323i \(0.667113\pi\)
\(720\) 0 0
\(721\) 9.57602 0.356630
\(722\) 0 0
\(723\) − 16.8793i − 0.627748i
\(724\) 0 0
\(725\) 2.00000i 0.0742781i
\(726\) 0 0
\(727\) −41.8138 −1.55079 −0.775394 0.631478i \(-0.782449\pi\)
−0.775394 + 0.631478i \(0.782449\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 14.2277i 0.526229i
\(732\) 0 0
\(733\) 30.0844i 1.11119i 0.831452 + 0.555597i \(0.187510\pi\)
−0.831452 + 0.555597i \(0.812490\pi\)
\(734\) 0 0
\(735\) −6.11383 −0.225512
\(736\) 0 0
\(737\) 17.9931 0.662785
\(738\) 0 0
\(739\) − 29.0449i − 1.06843i −0.845348 0.534217i \(-0.820607\pi\)
0.845348 0.534217i \(-0.179393\pi\)
\(740\) 0 0
\(741\) 5.88273i 0.216108i
\(742\) 0 0
\(743\) 43.2863 1.58802 0.794010 0.607905i \(-0.207990\pi\)
0.794010 + 0.607905i \(0.207990\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 0 0
\(747\) 5.88273i 0.215238i
\(748\) 0 0
\(749\) − 16.2208i − 0.592694i
\(750\) 0 0
\(751\) −41.7846 −1.52474 −0.762370 0.647141i \(-0.775964\pi\)
−0.762370 + 0.647141i \(0.775964\pi\)
\(752\) 0 0
\(753\) −11.8466 −0.431716
\(754\) 0 0
\(755\) − 9.67418i − 0.352079i
\(756\) 0 0
\(757\) − 16.3258i − 0.593372i −0.954975 0.296686i \(-0.904119\pi\)
0.954975 0.296686i \(-0.0958815\pi\)
\(758\) 0 0
\(759\) −4.76203 −0.172851
\(760\) 0 0
\(761\) 50.2208 1.82050 0.910251 0.414057i \(-0.135889\pi\)
0.910251 + 0.414057i \(0.135889\pi\)
\(762\) 0 0
\(763\) − 1.77234i − 0.0641631i
\(764\) 0 0
\(765\) − 7.55691i − 0.273221i
\(766\) 0 0
\(767\) 47.2242 1.70517
\(768\) 0 0
\(769\) −31.3415 −1.13020 −0.565101 0.825021i \(-0.691163\pi\)
−0.565101 + 0.825021i \(0.691163\pi\)
\(770\) 0 0
\(771\) 10.6707i 0.384297i
\(772\) 0 0
\(773\) − 9.11383i − 0.327802i −0.986477 0.163901i \(-0.947592\pi\)
0.986477 0.163901i \(-0.0524077\pi\)
\(774\) 0 0
\(775\) 3.55691 0.127768
\(776\) 0 0
\(777\) 7.00344 0.251247
\(778\) 0 0
\(779\) 4.11039i 0.147270i
\(780\) 0 0
\(781\) − 58.4622i − 2.09194i
\(782\) 0 0
\(783\) 2.00000 0.0714742
\(784\) 0 0
\(785\) −4.32582 −0.154395
\(786\) 0 0
\(787\) − 36.2208i − 1.29113i −0.763705 0.645566i \(-0.776622\pi\)
0.763705 0.645566i \(-0.223378\pi\)
\(788\) 0 0
\(789\) − 1.94480i − 0.0692369i
\(790\) 0 0
\(791\) −14.4240 −0.512858
\(792\) 0 0
\(793\) 49.9931 1.77531
\(794\) 0 0
\(795\) 2.00000i 0.0709327i
\(796\) 0 0
\(797\) 10.0000i 0.354218i 0.984191 + 0.177109i \(0.0566745\pi\)
−0.984191 + 0.177109i \(0.943325\pi\)
\(798\) 0 0
\(799\) 75.9862 2.68820
\(800\) 0 0
\(801\) 4.11727 0.145476
\(802\) 0 0
\(803\) 26.9897i 0.952445i
\(804\) 0 0
\(805\) − 0.996562i − 0.0351242i
\(806\) 0 0
\(807\) 9.76547 0.343761
\(808\) 0 0
\(809\) 47.5760 1.67268 0.836342 0.548208i \(-0.184690\pi\)
0.836342 + 0.548208i \(0.184690\pi\)
\(810\) 0 0
\(811\) − 20.5174i − 0.720463i −0.932863 0.360231i \(-0.882698\pi\)
0.932863 0.360231i \(-0.117302\pi\)
\(812\) 0 0
\(813\) 3.44652i 0.120875i
\(814\) 0 0
\(815\) −6.11727 −0.214278
\(816\) 0 0
\(817\) 1.99312 0.0697306
\(818\) 0 0
\(819\) − 5.23109i − 0.182789i
\(820\) 0 0
\(821\) 44.4622i 1.55174i 0.630892 + 0.775871i \(0.282689\pi\)
−0.630892 + 0.775871i \(0.717311\pi\)
\(822\) 0 0
\(823\) −32.1656 −1.12122 −0.560611 0.828079i \(-0.689434\pi\)
−0.560611 + 0.828079i \(0.689434\pi\)
\(824\) 0 0
\(825\) 4.49828 0.156610
\(826\) 0 0
\(827\) 20.0000i 0.695468i 0.937593 + 0.347734i \(0.113049\pi\)
−0.937593 + 0.347734i \(0.886951\pi\)
\(828\) 0 0
\(829\) 33.8827i 1.17680i 0.808571 + 0.588398i \(0.200241\pi\)
−0.808571 + 0.588398i \(0.799759\pi\)
\(830\) 0 0
\(831\) −18.7880 −0.651749
\(832\) 0 0
\(833\) −46.2017 −1.60079
\(834\) 0 0
\(835\) 6.05520i 0.209549i
\(836\) 0 0
\(837\) − 3.55691i − 0.122945i
\(838\) 0 0
\(839\) −4.52750 −0.156307 −0.0781533 0.996941i \(-0.524902\pi\)
−0.0781533 + 0.996941i \(0.524902\pi\)
\(840\) 0 0
\(841\) 25.0000 0.862069
\(842\) 0 0
\(843\) 16.8793i 0.581354i
\(844\) 0 0
\(845\) − 17.8793i − 0.615066i
\(846\) 0 0
\(847\) 8.69308 0.298698
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) − 7.87586i − 0.269981i
\(852\) 0 0
\(853\) − 50.4293i − 1.72667i −0.504633 0.863334i \(-0.668372\pi\)
0.504633 0.863334i \(-0.331628\pi\)
\(854\) 0 0
\(855\) −1.05863 −0.0362045
\(856\) 0 0
\(857\) 26.4362 0.903044 0.451522 0.892260i \(-0.350881\pi\)
0.451522 + 0.892260i \(0.350881\pi\)
\(858\) 0 0
\(859\) − 0.406994i − 0.0138865i −0.999976 0.00694323i \(-0.997790\pi\)
0.999976 0.00694323i \(-0.00221012\pi\)
\(860\) 0 0
\(861\) − 3.65508i − 0.124565i
\(862\) 0 0
\(863\) 29.9311 1.01886 0.509432 0.860511i \(-0.329855\pi\)
0.509432 + 0.860511i \(0.329855\pi\)
\(864\) 0 0
\(865\) −16.8793 −0.573913
\(866\) 0 0
\(867\) − 40.1070i − 1.36210i
\(868\) 0 0
\(869\) 51.9862i 1.76351i
\(870\) 0 0
\(871\) 22.2277 0.753155
\(872\) 0 0
\(873\) −17.1138 −0.579215
\(874\) 0 0
\(875\) 0.941367i 0.0318240i
\(876\) 0 0
\(877\) − 11.2051i − 0.378370i −0.981941 0.189185i \(-0.939415\pi\)
0.981941 0.189185i \(-0.0605846\pi\)
\(878\) 0 0
\(879\) −20.2277 −0.682262
\(880\) 0 0
\(881\) −48.3380 −1.62855 −0.814275 0.580479i \(-0.802865\pi\)
−0.814275 + 0.580479i \(0.802865\pi\)
\(882\) 0 0
\(883\) 50.5726i 1.70190i 0.525244 + 0.850951i \(0.323974\pi\)
−0.525244 + 0.850951i \(0.676026\pi\)
\(884\) 0 0
\(885\) 8.49828i 0.285667i
\(886\) 0 0
\(887\) 48.0483 1.61330 0.806652 0.591026i \(-0.201277\pi\)
0.806652 + 0.591026i \(0.201277\pi\)
\(888\) 0 0
\(889\) −17.1070 −0.573749
\(890\) 0 0
\(891\) − 4.49828i − 0.150698i
\(892\) 0 0
\(893\) − 10.6448i − 0.356213i
\(894\) 0 0
\(895\) 10.6155 0.354839
\(896\) 0 0
\(897\) −5.88273 −0.196419
\(898\) 0 0
\(899\) 7.11383i 0.237259i
\(900\) 0 0
\(901\) 15.1138i 0.503515i
\(902\) 0 0
\(903\) −1.77234 −0.0589799
\(904\) 0 0
\(905\) −14.1173 −0.469274
\(906\) 0 0
\(907\) 6.46219i 0.214573i 0.994228 + 0.107287i \(0.0342163\pi\)
−0.994228 + 0.107287i \(0.965784\pi\)
\(908\) 0 0
\(909\) − 2.00000i − 0.0663358i
\(910\) 0 0
\(911\) −50.3380 −1.66777 −0.833887 0.551935i \(-0.813890\pi\)
−0.833887 + 0.551935i \(0.813890\pi\)
\(912\) 0 0
\(913\) −26.4622 −0.875771
\(914\) 0 0
\(915\) 8.99656i 0.297417i
\(916\) 0 0
\(917\) − 6.00688i − 0.198365i
\(918\) 0 0
\(919\) −46.4362 −1.53179 −0.765895 0.642966i \(-0.777704\pi\)
−0.765895 + 0.642966i \(0.777704\pi\)
\(920\) 0 0
\(921\) −8.11039 −0.267246
\(922\) 0 0
\(923\) − 72.2208i − 2.37718i
\(924\) 0 0
\(925\) 7.43965i 0.244614i
\(926\) 0 0
\(927\) 10.1725 0.334107
\(928\) 0 0
\(929\) −35.9931 −1.18090 −0.590448 0.807076i \(-0.701049\pi\)
−0.590448 + 0.807076i \(0.701049\pi\)
\(930\) 0 0
\(931\) 6.47230i 0.212121i
\(932\) 0 0
\(933\) 31.8759i 1.04357i
\(934\) 0 0
\(935\) 33.9931 1.11169
\(936\) 0 0
\(937\) 2.70360 0.0883227 0.0441613 0.999024i \(-0.485938\pi\)
0.0441613 + 0.999024i \(0.485938\pi\)
\(938\) 0 0
\(939\) 5.11383i 0.166883i
\(940\) 0 0
\(941\) − 17.7655i − 0.579138i −0.957157 0.289569i \(-0.906488\pi\)
0.957157 0.289569i \(-0.0935119\pi\)
\(942\) 0 0
\(943\) −4.11039 −0.133853
\(944\) 0 0
\(945\) 0.941367 0.0306227
\(946\) 0 0
\(947\) 26.2277i 0.852284i 0.904656 + 0.426142i \(0.140128\pi\)
−0.904656 + 0.426142i \(0.859872\pi\)
\(948\) 0 0
\(949\) 33.3415i 1.08231i
\(950\) 0 0
\(951\) 24.6448 0.799161
\(952\) 0 0
\(953\) 9.09472 0.294607 0.147304 0.989091i \(-0.452941\pi\)
0.147304 + 0.989091i \(0.452941\pi\)
\(954\) 0 0
\(955\) 8.00000i 0.258874i
\(956\) 0 0
\(957\) 8.99656i 0.290818i
\(958\) 0 0
\(959\) 4.18257 0.135062
\(960\) 0 0
\(961\) −18.3484 −0.591883
\(962\) 0 0
\(963\) − 17.2311i − 0.555264i
\(964\) 0 0
\(965\) − 4.87930i − 0.157070i
\(966\) 0 0
\(967\) 7.47574 0.240404 0.120202 0.992749i \(-0.461646\pi\)
0.120202 + 0.992749i \(0.461646\pi\)
\(968\) 0 0
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) 41.0777i 1.31825i 0.752035 + 0.659124i \(0.229073\pi\)
−0.752035 + 0.659124i \(0.770927\pi\)
\(972\) 0 0
\(973\) 18.9897i 0.608781i
\(974\) 0 0
\(975\) 5.55691 0.177964
\(976\) 0 0
\(977\) −4.20855 −0.134644 −0.0673218 0.997731i \(-0.521445\pi\)
−0.0673218 + 0.997731i \(0.521445\pi\)
\(978\) 0 0
\(979\) 18.5206i 0.591922i
\(980\) 0 0
\(981\) − 1.88273i − 0.0601111i
\(982\) 0 0
\(983\) 8.35504 0.266484 0.133242 0.991084i \(-0.457461\pi\)
0.133242 + 0.991084i \(0.457461\pi\)
\(984\) 0 0
\(985\) 2.88617 0.0919611
\(986\) 0 0
\(987\) 9.46563i 0.301294i
\(988\) 0 0
\(989\) 1.99312i 0.0633777i
\(990\) 0 0
\(991\) 13.9087 0.441825 0.220912 0.975294i \(-0.429097\pi\)
0.220912 + 0.975294i \(0.429097\pi\)
\(992\) 0 0
\(993\) 11.0518 0.350717
\(994\) 0 0
\(995\) 17.6742i 0.560309i
\(996\) 0 0
\(997\) 34.8984i 1.10524i 0.833432 + 0.552622i \(0.186373\pi\)
−0.833432 + 0.552622i \(0.813627\pi\)
\(998\) 0 0
\(999\) 7.43965 0.235380
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 480.2.k.b.241.2 6
3.2 odd 2 1440.2.k.f.721.2 6
4.3 odd 2 120.2.k.b.61.4 yes 6
5.2 odd 4 2400.2.d.e.49.3 6
5.3 odd 4 2400.2.d.f.49.4 6
5.4 even 2 2400.2.k.c.1201.5 6
8.3 odd 2 120.2.k.b.61.3 6
8.5 even 2 inner 480.2.k.b.241.5 6
12.11 even 2 360.2.k.f.181.3 6
15.2 even 4 7200.2.d.r.2449.3 6
15.8 even 4 7200.2.d.q.2449.4 6
15.14 odd 2 7200.2.k.p.3601.4 6
16.3 odd 4 3840.2.a.bp.1.2 3
16.5 even 4 3840.2.a.bo.1.2 3
16.11 odd 4 3840.2.a.bq.1.2 3
16.13 even 4 3840.2.a.br.1.2 3
20.3 even 4 600.2.d.e.349.5 6
20.7 even 4 600.2.d.f.349.2 6
20.19 odd 2 600.2.k.c.301.3 6
24.5 odd 2 1440.2.k.f.721.5 6
24.11 even 2 360.2.k.f.181.4 6
40.3 even 4 600.2.d.f.349.1 6
40.13 odd 4 2400.2.d.e.49.4 6
40.19 odd 2 600.2.k.c.301.4 6
40.27 even 4 600.2.d.e.349.6 6
40.29 even 2 2400.2.k.c.1201.2 6
40.37 odd 4 2400.2.d.f.49.3 6
60.23 odd 4 1800.2.d.q.1549.2 6
60.47 odd 4 1800.2.d.r.1549.5 6
60.59 even 2 1800.2.k.p.901.4 6
120.29 odd 2 7200.2.k.p.3601.3 6
120.53 even 4 7200.2.d.r.2449.4 6
120.59 even 2 1800.2.k.p.901.3 6
120.77 even 4 7200.2.d.q.2449.3 6
120.83 odd 4 1800.2.d.r.1549.6 6
120.107 odd 4 1800.2.d.q.1549.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.2.k.b.61.3 6 8.3 odd 2
120.2.k.b.61.4 yes 6 4.3 odd 2
360.2.k.f.181.3 6 12.11 even 2
360.2.k.f.181.4 6 24.11 even 2
480.2.k.b.241.2 6 1.1 even 1 trivial
480.2.k.b.241.5 6 8.5 even 2 inner
600.2.d.e.349.5 6 20.3 even 4
600.2.d.e.349.6 6 40.27 even 4
600.2.d.f.349.1 6 40.3 even 4
600.2.d.f.349.2 6 20.7 even 4
600.2.k.c.301.3 6 20.19 odd 2
600.2.k.c.301.4 6 40.19 odd 2
1440.2.k.f.721.2 6 3.2 odd 2
1440.2.k.f.721.5 6 24.5 odd 2
1800.2.d.q.1549.1 6 120.107 odd 4
1800.2.d.q.1549.2 6 60.23 odd 4
1800.2.d.r.1549.5 6 60.47 odd 4
1800.2.d.r.1549.6 6 120.83 odd 4
1800.2.k.p.901.3 6 120.59 even 2
1800.2.k.p.901.4 6 60.59 even 2
2400.2.d.e.49.3 6 5.2 odd 4
2400.2.d.e.49.4 6 40.13 odd 4
2400.2.d.f.49.3 6 40.37 odd 4
2400.2.d.f.49.4 6 5.3 odd 4
2400.2.k.c.1201.2 6 40.29 even 2
2400.2.k.c.1201.5 6 5.4 even 2
3840.2.a.bo.1.2 3 16.5 even 4
3840.2.a.bp.1.2 3 16.3 odd 4
3840.2.a.bq.1.2 3 16.11 odd 4
3840.2.a.br.1.2 3 16.13 even 4
7200.2.d.q.2449.3 6 120.77 even 4
7200.2.d.q.2449.4 6 15.8 even 4
7200.2.d.r.2449.3 6 15.2 even 4
7200.2.d.r.2449.4 6 120.53 even 4
7200.2.k.p.3601.3 6 120.29 odd 2
7200.2.k.p.3601.4 6 15.14 odd 2