Properties

Label 6160.2.a.bi.1.3
Level $6160$
Weight $2$
Character 6160.1
Self dual yes
Analytic conductor $49.188$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6160,2,Mod(1,6160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6160.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6160.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: \(49.1878476451\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.892.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 8x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.31955\) of defining polynomial
Character \(\chi\) \(=\) 6160.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.93923 q^{3} -1.00000 q^{5} +1.00000 q^{7} +5.63910 q^{9} +O(q^{10})\) \(q+2.93923 q^{3} -1.00000 q^{5} +1.00000 q^{7} +5.63910 q^{9} -1.00000 q^{11} -5.87847 q^{13} -2.93923 q^{15} -4.63910 q^{17} +5.57834 q^{19} +2.93923 q^{21} -5.57834 q^{23} +1.00000 q^{25} +7.75694 q^{27} -9.45681 q^{29} -6.00000 q^{31} -2.93923 q^{33} -1.00000 q^{35} +2.30013 q^{37} -17.2782 q^{39} -3.06077 q^{41} -10.5176 q^{43} -5.63910 q^{45} -8.51757 q^{47} +1.00000 q^{49} -13.6354 q^{51} -9.45681 q^{53} +1.00000 q^{55} +16.3960 q^{57} -7.23937 q^{59} +14.5176 q^{61} +5.63910 q^{63} +5.87847 q^{65} -8.00000 q^{67} -16.3960 q^{69} +7.15667 q^{71} +12.3960 q^{73} +2.93923 q^{75} -1.00000 q^{77} +10.8565 q^{79} +5.88216 q^{81} +13.8785 q^{83} +4.63910 q^{85} -27.7958 q^{87} +3.87847 q^{89} -5.87847 q^{91} -17.6354 q^{93} -5.57834 q^{95} +7.57834 q^{97} -5.63910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{5} + 3 q^{7} + 11 q^{9} - 3 q^{11} - 8 q^{17} + 2 q^{19} - 2 q^{23} + 3 q^{25} - 12 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{35} + 4 q^{37} - 40 q^{39} - 18 q^{41} - 8 q^{43} - 11 q^{45} - 2 q^{47} + 3 q^{49} + 12 q^{51} + 4 q^{53} + 3 q^{55} + 8 q^{57} - 10 q^{59} + 20 q^{61} + 11 q^{63} - 24 q^{67} - 8 q^{69} - 8 q^{71} - 4 q^{73} - 3 q^{77} + 6 q^{79} + 47 q^{81} + 24 q^{83} + 8 q^{85} - 48 q^{87} - 6 q^{89} - 2 q^{95} + 8 q^{97} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.93923 1.69697 0.848484 0.529221i \(-0.177516\pi\)
0.848484 + 0.529221i \(0.177516\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 5.63910 1.87970
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −5.87847 −1.63039 −0.815197 0.579184i \(-0.803371\pi\)
−0.815197 + 0.579184i \(0.803371\pi\)
\(14\) 0 0
\(15\) −2.93923 −0.758907
\(16\) 0 0
\(17\) −4.63910 −1.12515 −0.562574 0.826747i \(-0.690189\pi\)
−0.562574 + 0.826747i \(0.690189\pi\)
\(18\) 0 0
\(19\) 5.57834 1.27976 0.639879 0.768476i \(-0.278984\pi\)
0.639879 + 0.768476i \(0.278984\pi\)
\(20\) 0 0
\(21\) 2.93923 0.641394
\(22\) 0 0
\(23\) −5.57834 −1.16316 −0.581582 0.813488i \(-0.697566\pi\)
−0.581582 + 0.813488i \(0.697566\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 7.75694 1.49282
\(28\) 0 0
\(29\) −9.45681 −1.75608 −0.878042 0.478583i \(-0.841151\pi\)
−0.878042 + 0.478583i \(0.841151\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) −2.93923 −0.511655
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 2.30013 0.378140 0.189070 0.981964i \(-0.439453\pi\)
0.189070 + 0.981964i \(0.439453\pi\)
\(38\) 0 0
\(39\) −17.2782 −2.76673
\(40\) 0 0
\(41\) −3.06077 −0.478011 −0.239006 0.971018i \(-0.576821\pi\)
−0.239006 + 0.971018i \(0.576821\pi\)
\(42\) 0 0
\(43\) −10.5176 −1.60391 −0.801957 0.597381i \(-0.796208\pi\)
−0.801957 + 0.597381i \(0.796208\pi\)
\(44\) 0 0
\(45\) −5.63910 −0.840628
\(46\) 0 0
\(47\) −8.51757 −1.24242 −0.621208 0.783646i \(-0.713358\pi\)
−0.621208 + 0.783646i \(0.713358\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −13.6354 −1.90934
\(52\) 0 0
\(53\) −9.45681 −1.29899 −0.649496 0.760365i \(-0.725020\pi\)
−0.649496 + 0.760365i \(0.725020\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 16.3960 2.17171
\(58\) 0 0
\(59\) −7.23937 −0.942485 −0.471243 0.882004i \(-0.656194\pi\)
−0.471243 + 0.882004i \(0.656194\pi\)
\(60\) 0 0
\(61\) 14.5176 1.85878 0.929392 0.369093i \(-0.120332\pi\)
0.929392 + 0.369093i \(0.120332\pi\)
\(62\) 0 0
\(63\) 5.63910 0.710460
\(64\) 0 0
\(65\) 5.87847 0.729134
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 0 0
\(69\) −16.3960 −1.97385
\(70\) 0 0
\(71\) 7.15667 0.849341 0.424670 0.905348i \(-0.360390\pi\)
0.424670 + 0.905348i \(0.360390\pi\)
\(72\) 0 0
\(73\) 12.3960 1.45085 0.725423 0.688303i \(-0.241644\pi\)
0.725423 + 0.688303i \(0.241644\pi\)
\(74\) 0 0
\(75\) 2.93923 0.339394
\(76\) 0 0
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 10.8565 1.22146 0.610728 0.791840i \(-0.290877\pi\)
0.610728 + 0.791840i \(0.290877\pi\)
\(80\) 0 0
\(81\) 5.88216 0.653574
\(82\) 0 0
\(83\) 13.8785 1.52336 0.761680 0.647953i \(-0.224375\pi\)
0.761680 + 0.647953i \(0.224375\pi\)
\(84\) 0 0
\(85\) 4.63910 0.503181
\(86\) 0 0
\(87\) −27.7958 −2.98002
\(88\) 0 0
\(89\) 3.87847 0.411117 0.205558 0.978645i \(-0.434099\pi\)
0.205558 + 0.978645i \(0.434099\pi\)
\(90\) 0 0
\(91\) −5.87847 −0.616231
\(92\) 0 0
\(93\) −17.6354 −1.82871
\(94\) 0 0
\(95\) −5.57834 −0.572325
\(96\) 0 0
\(97\) 7.57834 0.769463 0.384732 0.923028i \(-0.374294\pi\)
0.384732 + 0.923028i \(0.374294\pi\)
\(98\) 0 0
\(99\) −5.63910 −0.566751
\(100\) 0 0
\(101\) 11.7958 1.17372 0.586862 0.809687i \(-0.300363\pi\)
0.586862 + 0.809687i \(0.300363\pi\)
\(102\) 0 0
\(103\) −17.1178 −1.68667 −0.843335 0.537388i \(-0.819411\pi\)
−0.843335 + 0.537388i \(0.819411\pi\)
\(104\) 0 0
\(105\) −2.93923 −0.286840
\(106\) 0 0
\(107\) −2.51757 −0.243383 −0.121691 0.992568i \(-0.538832\pi\)
−0.121691 + 0.992568i \(0.538832\pi\)
\(108\) 0 0
\(109\) 10.3001 0.986574 0.493287 0.869867i \(-0.335795\pi\)
0.493287 + 0.869867i \(0.335795\pi\)
\(110\) 0 0
\(111\) 6.76063 0.641691
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 5.57834 0.520183
\(116\) 0 0
\(117\) −33.1493 −3.06465
\(118\) 0 0
\(119\) −4.63910 −0.425266
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −8.99631 −0.811170
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −10.4787 −0.929837 −0.464919 0.885353i \(-0.653916\pi\)
−0.464919 + 0.885353i \(0.653916\pi\)
\(128\) 0 0
\(129\) −30.9136 −2.72179
\(130\) 0 0
\(131\) 1.57834 0.137900 0.0689499 0.997620i \(-0.478035\pi\)
0.0689499 + 0.997620i \(0.478035\pi\)
\(132\) 0 0
\(133\) 5.57834 0.483703
\(134\) 0 0
\(135\) −7.75694 −0.667611
\(136\) 0 0
\(137\) −3.27820 −0.280076 −0.140038 0.990146i \(-0.544722\pi\)
−0.140038 + 0.990146i \(0.544722\pi\)
\(138\) 0 0
\(139\) 16.0571 1.36194 0.680972 0.732310i \(-0.261558\pi\)
0.680972 + 0.732310i \(0.261558\pi\)
\(140\) 0 0
\(141\) −25.0351 −2.10834
\(142\) 0 0
\(143\) 5.87847 0.491582
\(144\) 0 0
\(145\) 9.45681 0.785345
\(146\) 0 0
\(147\) 2.93923 0.242424
\(148\) 0 0
\(149\) 10.7350 0.879446 0.439723 0.898133i \(-0.355077\pi\)
0.439723 + 0.898133i \(0.355077\pi\)
\(150\) 0 0
\(151\) 2.17860 0.177292 0.0886461 0.996063i \(-0.471746\pi\)
0.0886461 + 0.996063i \(0.471746\pi\)
\(152\) 0 0
\(153\) −26.1604 −2.11494
\(154\) 0 0
\(155\) 6.00000 0.481932
\(156\) 0 0
\(157\) −17.1567 −1.36925 −0.684626 0.728895i \(-0.740034\pi\)
−0.684626 + 0.728895i \(0.740034\pi\)
\(158\) 0 0
\(159\) −27.7958 −2.20435
\(160\) 0 0
\(161\) −5.57834 −0.439635
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 2.93923 0.228819
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) 21.5564 1.65819
\(170\) 0 0
\(171\) 31.4568 2.40556
\(172\) 0 0
\(173\) −5.48243 −0.416821 −0.208411 0.978041i \(-0.566829\pi\)
−0.208411 + 0.978041i \(0.566829\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) −21.2782 −1.59937
\(178\) 0 0
\(179\) −1.23937 −0.0926347 −0.0463174 0.998927i \(-0.514749\pi\)
−0.0463174 + 0.998927i \(0.514749\pi\)
\(180\) 0 0
\(181\) 8.55641 0.635993 0.317996 0.948092i \(-0.396990\pi\)
0.317996 + 0.948092i \(0.396990\pi\)
\(182\) 0 0
\(183\) 42.6706 3.15430
\(184\) 0 0
\(185\) −2.30013 −0.169109
\(186\) 0 0
\(187\) 4.63910 0.339245
\(188\) 0 0
\(189\) 7.75694 0.564234
\(190\) 0 0
\(191\) 1.27820 0.0924875 0.0462438 0.998930i \(-0.485275\pi\)
0.0462438 + 0.998930i \(0.485275\pi\)
\(192\) 0 0
\(193\) −10.6391 −0.765819 −0.382910 0.923786i \(-0.625078\pi\)
−0.382910 + 0.923786i \(0.625078\pi\)
\(194\) 0 0
\(195\) 17.2782 1.23732
\(196\) 0 0
\(197\) 7.87847 0.561318 0.280659 0.959808i \(-0.409447\pi\)
0.280659 + 0.959808i \(0.409447\pi\)
\(198\) 0 0
\(199\) −0.517571 −0.0366897 −0.0183448 0.999832i \(-0.505840\pi\)
−0.0183448 + 0.999832i \(0.505840\pi\)
\(200\) 0 0
\(201\) −23.5139 −1.65854
\(202\) 0 0
\(203\) −9.45681 −0.663738
\(204\) 0 0
\(205\) 3.06077 0.213773
\(206\) 0 0
\(207\) −31.4568 −2.18640
\(208\) 0 0
\(209\) −5.57834 −0.385862
\(210\) 0 0
\(211\) −2.72180 −0.187376 −0.0936881 0.995602i \(-0.529866\pi\)
−0.0936881 + 0.995602i \(0.529866\pi\)
\(212\) 0 0
\(213\) 21.0351 1.44130
\(214\) 0 0
\(215\) 10.5176 0.717292
\(216\) 0 0
\(217\) −6.00000 −0.407307
\(218\) 0 0
\(219\) 36.4349 2.46204
\(220\) 0 0
\(221\) 27.2708 1.83443
\(222\) 0 0
\(223\) 22.3960 1.49975 0.749875 0.661580i \(-0.230114\pi\)
0.749875 + 0.661580i \(0.230114\pi\)
\(224\) 0 0
\(225\) 5.63910 0.375940
\(226\) 0 0
\(227\) −13.0351 −0.865173 −0.432586 0.901592i \(-0.642399\pi\)
−0.432586 + 0.901592i \(0.642399\pi\)
\(228\) 0 0
\(229\) −6.63910 −0.438724 −0.219362 0.975644i \(-0.570398\pi\)
−0.219362 + 0.975644i \(0.570398\pi\)
\(230\) 0 0
\(231\) −2.93923 −0.193387
\(232\) 0 0
\(233\) 19.0351 1.24703 0.623517 0.781810i \(-0.285703\pi\)
0.623517 + 0.781810i \(0.285703\pi\)
\(234\) 0 0
\(235\) 8.51757 0.555625
\(236\) 0 0
\(237\) 31.9099 2.07277
\(238\) 0 0
\(239\) −22.6135 −1.46274 −0.731372 0.681979i \(-0.761120\pi\)
−0.731372 + 0.681979i \(0.761120\pi\)
\(240\) 0 0
\(241\) −17.9744 −1.15783 −0.578916 0.815387i \(-0.696524\pi\)
−0.578916 + 0.815387i \(0.696524\pi\)
\(242\) 0 0
\(243\) −5.98176 −0.383730
\(244\) 0 0
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −32.7921 −2.08651
\(248\) 0 0
\(249\) 40.7921 2.58509
\(250\) 0 0
\(251\) −8.76063 −0.552966 −0.276483 0.961019i \(-0.589169\pi\)
−0.276483 + 0.961019i \(0.589169\pi\)
\(252\) 0 0
\(253\) 5.57834 0.350707
\(254\) 0 0
\(255\) 13.6354 0.853882
\(256\) 0 0
\(257\) −5.09960 −0.318104 −0.159052 0.987270i \(-0.550844\pi\)
−0.159052 + 0.987270i \(0.550844\pi\)
\(258\) 0 0
\(259\) 2.30013 0.142923
\(260\) 0 0
\(261\) −53.3279 −3.30091
\(262\) 0 0
\(263\) 11.7569 0.724964 0.362482 0.931991i \(-0.381929\pi\)
0.362482 + 0.931991i \(0.381929\pi\)
\(264\) 0 0
\(265\) 9.45681 0.580927
\(266\) 0 0
\(267\) 11.3997 0.697652
\(268\) 0 0
\(269\) −11.6742 −0.711791 −0.355896 0.934526i \(-0.615824\pi\)
−0.355896 + 0.934526i \(0.615824\pi\)
\(270\) 0 0
\(271\) −17.6354 −1.07127 −0.535637 0.844448i \(-0.679929\pi\)
−0.535637 + 0.844448i \(0.679929\pi\)
\(272\) 0 0
\(273\) −17.2782 −1.04572
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 22.0000 1.32185 0.660926 0.750451i \(-0.270164\pi\)
0.660926 + 0.750451i \(0.270164\pi\)
\(278\) 0 0
\(279\) −33.8346 −2.02563
\(280\) 0 0
\(281\) 1.15667 0.0690013 0.0345007 0.999405i \(-0.489016\pi\)
0.0345007 + 0.999405i \(0.489016\pi\)
\(282\) 0 0
\(283\) −17.2782 −1.02708 −0.513541 0.858065i \(-0.671667\pi\)
−0.513541 + 0.858065i \(0.671667\pi\)
\(284\) 0 0
\(285\) −16.3960 −0.971218
\(286\) 0 0
\(287\) −3.06077 −0.180671
\(288\) 0 0
\(289\) 4.52126 0.265957
\(290\) 0 0
\(291\) 22.2745 1.30575
\(292\) 0 0
\(293\) 1.87847 0.109741 0.0548707 0.998493i \(-0.482525\pi\)
0.0548707 + 0.998493i \(0.482525\pi\)
\(294\) 0 0
\(295\) 7.23937 0.421492
\(296\) 0 0
\(297\) −7.75694 −0.450103
\(298\) 0 0
\(299\) 32.7921 1.89642
\(300\) 0 0
\(301\) −10.5176 −0.606223
\(302\) 0 0
\(303\) 34.6706 1.99177
\(304\) 0 0
\(305\) −14.5176 −0.831274
\(306\) 0 0
\(307\) −8.60027 −0.490843 −0.245422 0.969416i \(-0.578926\pi\)
−0.245422 + 0.969416i \(0.578926\pi\)
\(308\) 0 0
\(309\) −50.3133 −2.86223
\(310\) 0 0
\(311\) −13.7569 −0.780084 −0.390042 0.920797i \(-0.627540\pi\)
−0.390042 + 0.920797i \(0.627540\pi\)
\(312\) 0 0
\(313\) 2.90040 0.163940 0.0819701 0.996635i \(-0.473879\pi\)
0.0819701 + 0.996635i \(0.473879\pi\)
\(314\) 0 0
\(315\) −5.63910 −0.317727
\(316\) 0 0
\(317\) 19.3353 1.08598 0.542989 0.839740i \(-0.317293\pi\)
0.542989 + 0.839740i \(0.317293\pi\)
\(318\) 0 0
\(319\) 9.45681 0.529479
\(320\) 0 0
\(321\) −7.39973 −0.413013
\(322\) 0 0
\(323\) −25.8785 −1.43992
\(324\) 0 0
\(325\) −5.87847 −0.326079
\(326\) 0 0
\(327\) 30.2745 1.67418
\(328\) 0 0
\(329\) −8.51757 −0.469589
\(330\) 0 0
\(331\) −10.5176 −0.578098 −0.289049 0.957314i \(-0.593339\pi\)
−0.289049 + 0.957314i \(0.593339\pi\)
\(332\) 0 0
\(333\) 12.9707 0.710789
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) 7.27820 0.396469 0.198234 0.980155i \(-0.436479\pi\)
0.198234 + 0.980155i \(0.436479\pi\)
\(338\) 0 0
\(339\) −41.1493 −2.23492
\(340\) 0 0
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 16.3960 0.882733
\(346\) 0 0
\(347\) 14.2745 0.766296 0.383148 0.923687i \(-0.374840\pi\)
0.383148 + 0.923687i \(0.374840\pi\)
\(348\) 0 0
\(349\) 0.396041 0.0211996 0.0105998 0.999944i \(-0.496626\pi\)
0.0105998 + 0.999944i \(0.496626\pi\)
\(350\) 0 0
\(351\) −45.5989 −2.43389
\(352\) 0 0
\(353\) −5.09960 −0.271424 −0.135712 0.990748i \(-0.543332\pi\)
−0.135712 + 0.990748i \(0.543332\pi\)
\(354\) 0 0
\(355\) −7.15667 −0.379837
\(356\) 0 0
\(357\) −13.6354 −0.721662
\(358\) 0 0
\(359\) 18.3704 0.969554 0.484777 0.874638i \(-0.338901\pi\)
0.484777 + 0.874638i \(0.338901\pi\)
\(360\) 0 0
\(361\) 12.1178 0.637781
\(362\) 0 0
\(363\) 2.93923 0.154270
\(364\) 0 0
\(365\) −12.3960 −0.648838
\(366\) 0 0
\(367\) −17.1178 −0.893544 −0.446772 0.894648i \(-0.647426\pi\)
−0.446772 + 0.894648i \(0.647426\pi\)
\(368\) 0 0
\(369\) −17.2600 −0.898518
\(370\) 0 0
\(371\) −9.45681 −0.490973
\(372\) 0 0
\(373\) −6.35721 −0.329164 −0.164582 0.986363i \(-0.552627\pi\)
−0.164582 + 0.986363i \(0.552627\pi\)
\(374\) 0 0
\(375\) −2.93923 −0.151781
\(376\) 0 0
\(377\) 55.5915 2.86311
\(378\) 0 0
\(379\) 5.03514 0.258638 0.129319 0.991603i \(-0.458721\pi\)
0.129319 + 0.991603i \(0.458721\pi\)
\(380\) 0 0
\(381\) −30.7995 −1.57790
\(382\) 0 0
\(383\) 7.43118 0.379716 0.189858 0.981812i \(-0.439197\pi\)
0.189858 + 0.981812i \(0.439197\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) −59.3097 −3.01488
\(388\) 0 0
\(389\) −5.15667 −0.261454 −0.130727 0.991418i \(-0.541731\pi\)
−0.130727 + 0.991418i \(0.541731\pi\)
\(390\) 0 0
\(391\) 25.8785 1.30873
\(392\) 0 0
\(393\) 4.63910 0.234012
\(394\) 0 0
\(395\) −10.8565 −0.546252
\(396\) 0 0
\(397\) −19.6354 −0.985473 −0.492736 0.870179i \(-0.664003\pi\)
−0.492736 + 0.870179i \(0.664003\pi\)
\(398\) 0 0
\(399\) 16.3960 0.820829
\(400\) 0 0
\(401\) −5.11784 −0.255573 −0.127786 0.991802i \(-0.540787\pi\)
−0.127786 + 0.991802i \(0.540787\pi\)
\(402\) 0 0
\(403\) 35.2708 1.75696
\(404\) 0 0
\(405\) −5.88216 −0.292287
\(406\) 0 0
\(407\) −2.30013 −0.114013
\(408\) 0 0
\(409\) 2.21744 0.109645 0.0548226 0.998496i \(-0.482541\pi\)
0.0548226 + 0.998496i \(0.482541\pi\)
\(410\) 0 0
\(411\) −9.63541 −0.475280
\(412\) 0 0
\(413\) −7.23937 −0.356226
\(414\) 0 0
\(415\) −13.8785 −0.681267
\(416\) 0 0
\(417\) 47.1955 2.31117
\(418\) 0 0
\(419\) 4.51757 0.220698 0.110349 0.993893i \(-0.464803\pi\)
0.110349 + 0.993893i \(0.464803\pi\)
\(420\) 0 0
\(421\) −12.3133 −0.600116 −0.300058 0.953921i \(-0.597006\pi\)
−0.300058 + 0.953921i \(0.597006\pi\)
\(422\) 0 0
\(423\) −48.0315 −2.33537
\(424\) 0 0
\(425\) −4.63910 −0.225029
\(426\) 0 0
\(427\) 14.5176 0.702555
\(428\) 0 0
\(429\) 17.2782 0.834200
\(430\) 0 0
\(431\) 2.17860 0.104940 0.0524698 0.998623i \(-0.483291\pi\)
0.0524698 + 0.998623i \(0.483291\pi\)
\(432\) 0 0
\(433\) 5.02193 0.241339 0.120669 0.992693i \(-0.461496\pi\)
0.120669 + 0.992693i \(0.461496\pi\)
\(434\) 0 0
\(435\) 27.7958 1.33271
\(436\) 0 0
\(437\) −31.1178 −1.48857
\(438\) 0 0
\(439\) −7.15667 −0.341569 −0.170785 0.985308i \(-0.554630\pi\)
−0.170785 + 0.985308i \(0.554630\pi\)
\(440\) 0 0
\(441\) 5.63910 0.268529
\(442\) 0 0
\(443\) 19.5139 0.927132 0.463566 0.886062i \(-0.346570\pi\)
0.463566 + 0.886062i \(0.346570\pi\)
\(444\) 0 0
\(445\) −3.87847 −0.183857
\(446\) 0 0
\(447\) 31.5527 1.49239
\(448\) 0 0
\(449\) 1.36090 0.0642248 0.0321124 0.999484i \(-0.489777\pi\)
0.0321124 + 0.999484i \(0.489777\pi\)
\(450\) 0 0
\(451\) 3.06077 0.144126
\(452\) 0 0
\(453\) 6.40343 0.300859
\(454\) 0 0
\(455\) 5.87847 0.275587
\(456\) 0 0
\(457\) 4.47874 0.209506 0.104753 0.994498i \(-0.466595\pi\)
0.104753 + 0.994498i \(0.466595\pi\)
\(458\) 0 0
\(459\) −35.9852 −1.67965
\(460\) 0 0
\(461\) −36.1530 −1.68381 −0.841906 0.539624i \(-0.818566\pi\)
−0.841906 + 0.539624i \(0.818566\pi\)
\(462\) 0 0
\(463\) −21.0922 −0.980238 −0.490119 0.871655i \(-0.663047\pi\)
−0.490119 + 0.871655i \(0.663047\pi\)
\(464\) 0 0
\(465\) 17.6354 0.817823
\(466\) 0 0
\(467\) −19.1311 −0.885279 −0.442640 0.896700i \(-0.645958\pi\)
−0.442640 + 0.896700i \(0.645958\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) −50.4275 −2.32358
\(472\) 0 0
\(473\) 10.5176 0.483598
\(474\) 0 0
\(475\) 5.57834 0.255952
\(476\) 0 0
\(477\) −53.3279 −2.44172
\(478\) 0 0
\(479\) −5.27820 −0.241167 −0.120584 0.992703i \(-0.538477\pi\)
−0.120584 + 0.992703i \(0.538477\pi\)
\(480\) 0 0
\(481\) −13.5213 −0.616517
\(482\) 0 0
\(483\) −16.3960 −0.746046
\(484\) 0 0
\(485\) −7.57834 −0.344115
\(486\) 0 0
\(487\) 19.8140 0.897859 0.448929 0.893567i \(-0.351806\pi\)
0.448929 + 0.893567i \(0.351806\pi\)
\(488\) 0 0
\(489\) 23.5139 1.06333
\(490\) 0 0
\(491\) 6.47874 0.292381 0.146191 0.989256i \(-0.453299\pi\)
0.146191 + 0.989256i \(0.453299\pi\)
\(492\) 0 0
\(493\) 43.8711 1.97585
\(494\) 0 0
\(495\) 5.63910 0.253459
\(496\) 0 0
\(497\) 7.15667 0.321021
\(498\) 0 0
\(499\) 21.2394 0.950805 0.475402 0.879768i \(-0.342302\pi\)
0.475402 + 0.879768i \(0.342302\pi\)
\(500\) 0 0
\(501\) −23.5139 −1.05052
\(502\) 0 0
\(503\) −42.2357 −1.88320 −0.941598 0.336739i \(-0.890676\pi\)
−0.941598 + 0.336739i \(0.890676\pi\)
\(504\) 0 0
\(505\) −11.7958 −0.524905
\(506\) 0 0
\(507\) 63.3593 2.81389
\(508\) 0 0
\(509\) 38.8698 1.72287 0.861436 0.507867i \(-0.169566\pi\)
0.861436 + 0.507867i \(0.169566\pi\)
\(510\) 0 0
\(511\) 12.3960 0.548369
\(512\) 0 0
\(513\) 43.2708 1.91045
\(514\) 0 0
\(515\) 17.1178 0.754302
\(516\) 0 0
\(517\) 8.51757 0.374602
\(518\) 0 0
\(519\) −16.1141 −0.707332
\(520\) 0 0
\(521\) 25.9488 1.13684 0.568418 0.822740i \(-0.307556\pi\)
0.568418 + 0.822740i \(0.307556\pi\)
\(522\) 0 0
\(523\) −42.7482 −1.86925 −0.934625 0.355636i \(-0.884264\pi\)
−0.934625 + 0.355636i \(0.884264\pi\)
\(524\) 0 0
\(525\) 2.93923 0.128279
\(526\) 0 0
\(527\) 27.8346 1.21249
\(528\) 0 0
\(529\) 8.11784 0.352949
\(530\) 0 0
\(531\) −40.8235 −1.77159
\(532\) 0 0
\(533\) 17.9926 0.779347
\(534\) 0 0
\(535\) 2.51757 0.108844
\(536\) 0 0
\(537\) −3.64279 −0.157198
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 0 0
\(541\) −6.73501 −0.289561 −0.144780 0.989464i \(-0.546248\pi\)
−0.144780 + 0.989464i \(0.546248\pi\)
\(542\) 0 0
\(543\) 25.1493 1.07926
\(544\) 0 0
\(545\) −10.3001 −0.441209
\(546\) 0 0
\(547\) −23.7569 −1.01577 −0.507887 0.861424i \(-0.669573\pi\)
−0.507887 + 0.861424i \(0.669573\pi\)
\(548\) 0 0
\(549\) 81.8661 3.49396
\(550\) 0 0
\(551\) −52.7532 −2.24736
\(552\) 0 0
\(553\) 10.8565 0.461667
\(554\) 0 0
\(555\) −6.76063 −0.286973
\(556\) 0 0
\(557\) −34.1918 −1.44875 −0.724377 0.689404i \(-0.757872\pi\)
−0.724377 + 0.689404i \(0.757872\pi\)
\(558\) 0 0
\(559\) 61.8272 2.61501
\(560\) 0 0
\(561\) 13.6354 0.575687
\(562\) 0 0
\(563\) 31.3485 1.32118 0.660591 0.750746i \(-0.270306\pi\)
0.660591 + 0.750746i \(0.270306\pi\)
\(564\) 0 0
\(565\) 14.0000 0.588984
\(566\) 0 0
\(567\) 5.88216 0.247028
\(568\) 0 0
\(569\) −38.5490 −1.61606 −0.808030 0.589142i \(-0.799466\pi\)
−0.808030 + 0.589142i \(0.799466\pi\)
\(570\) 0 0
\(571\) −26.9136 −1.12630 −0.563150 0.826355i \(-0.690411\pi\)
−0.563150 + 0.826355i \(0.690411\pi\)
\(572\) 0 0
\(573\) 3.75694 0.156948
\(574\) 0 0
\(575\) −5.57834 −0.232633
\(576\) 0 0
\(577\) −15.3353 −0.638416 −0.319208 0.947685i \(-0.603417\pi\)
−0.319208 + 0.947685i \(0.603417\pi\)
\(578\) 0 0
\(579\) −31.2708 −1.29957
\(580\) 0 0
\(581\) 13.8785 0.575776
\(582\) 0 0
\(583\) 9.45681 0.391661
\(584\) 0 0
\(585\) 33.1493 1.37055
\(586\) 0 0
\(587\) 21.2526 0.877188 0.438594 0.898685i \(-0.355477\pi\)
0.438594 + 0.898685i \(0.355477\pi\)
\(588\) 0 0
\(589\) −33.4700 −1.37911
\(590\) 0 0
\(591\) 23.1567 0.952538
\(592\) 0 0
\(593\) 19.1955 0.788265 0.394133 0.919054i \(-0.371045\pi\)
0.394133 + 0.919054i \(0.371045\pi\)
\(594\) 0 0
\(595\) 4.63910 0.190185
\(596\) 0 0
\(597\) −1.52126 −0.0622612
\(598\) 0 0
\(599\) −3.39973 −0.138909 −0.0694547 0.997585i \(-0.522126\pi\)
−0.0694547 + 0.997585i \(0.522126\pi\)
\(600\) 0 0
\(601\) −7.90409 −0.322415 −0.161207 0.986921i \(-0.551539\pi\)
−0.161207 + 0.986921i \(0.551539\pi\)
\(602\) 0 0
\(603\) −45.1128 −1.83714
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 23.7181 0.962688 0.481344 0.876532i \(-0.340149\pi\)
0.481344 + 0.876532i \(0.340149\pi\)
\(608\) 0 0
\(609\) −27.7958 −1.12634
\(610\) 0 0
\(611\) 50.0703 2.02563
\(612\) 0 0
\(613\) 40.9136 1.65249 0.826243 0.563314i \(-0.190474\pi\)
0.826243 + 0.563314i \(0.190474\pi\)
\(614\) 0 0
\(615\) 8.99631 0.362766
\(616\) 0 0
\(617\) −30.3572 −1.22214 −0.611068 0.791578i \(-0.709260\pi\)
−0.611068 + 0.791578i \(0.709260\pi\)
\(618\) 0 0
\(619\) 34.9963 1.40662 0.703310 0.710883i \(-0.251705\pi\)
0.703310 + 0.710883i \(0.251705\pi\)
\(620\) 0 0
\(621\) −43.2708 −1.73640
\(622\) 0 0
\(623\) 3.87847 0.155388
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −16.3960 −0.654795
\(628\) 0 0
\(629\) −10.6706 −0.425463
\(630\) 0 0
\(631\) −38.2357 −1.52214 −0.761069 0.648671i \(-0.775325\pi\)
−0.761069 + 0.648671i \(0.775325\pi\)
\(632\) 0 0
\(633\) −8.00000 −0.317971
\(634\) 0 0
\(635\) 10.4787 0.415836
\(636\) 0 0
\(637\) −5.87847 −0.232913
\(638\) 0 0
\(639\) 40.3572 1.59651
\(640\) 0 0
\(641\) −25.5139 −1.00774 −0.503869 0.863780i \(-0.668091\pi\)
−0.503869 + 0.863780i \(0.668091\pi\)
\(642\) 0 0
\(643\) 37.8528 1.49277 0.746385 0.665514i \(-0.231788\pi\)
0.746385 + 0.665514i \(0.231788\pi\)
\(644\) 0 0
\(645\) 30.9136 1.21722
\(646\) 0 0
\(647\) 24.7094 0.971426 0.485713 0.874118i \(-0.338560\pi\)
0.485713 + 0.874118i \(0.338560\pi\)
\(648\) 0 0
\(649\) 7.23937 0.284170
\(650\) 0 0
\(651\) −17.6354 −0.691186
\(652\) 0 0
\(653\) 16.7789 0.656608 0.328304 0.944572i \(-0.393523\pi\)
0.328304 + 0.944572i \(0.393523\pi\)
\(654\) 0 0
\(655\) −1.57834 −0.0616707
\(656\) 0 0
\(657\) 69.9025 2.72716
\(658\) 0 0
\(659\) −10.1215 −0.394279 −0.197139 0.980375i \(-0.563165\pi\)
−0.197139 + 0.980375i \(0.563165\pi\)
\(660\) 0 0
\(661\) 23.1128 0.898984 0.449492 0.893284i \(-0.351605\pi\)
0.449492 + 0.893284i \(0.351605\pi\)
\(662\) 0 0
\(663\) 80.1553 3.11298
\(664\) 0 0
\(665\) −5.57834 −0.216319
\(666\) 0 0
\(667\) 52.7532 2.04261
\(668\) 0 0
\(669\) 65.8272 2.54503
\(670\) 0 0
\(671\) −14.5176 −0.560445
\(672\) 0 0
\(673\) 38.0000 1.46479 0.732396 0.680879i \(-0.238402\pi\)
0.732396 + 0.680879i \(0.238402\pi\)
\(674\) 0 0
\(675\) 7.75694 0.298565
\(676\) 0 0
\(677\) 8.83092 0.339400 0.169700 0.985496i \(-0.445720\pi\)
0.169700 + 0.985496i \(0.445720\pi\)
\(678\) 0 0
\(679\) 7.57834 0.290830
\(680\) 0 0
\(681\) −38.3133 −1.46817
\(682\) 0 0
\(683\) −11.3997 −0.436199 −0.218099 0.975927i \(-0.569986\pi\)
−0.218099 + 0.975927i \(0.569986\pi\)
\(684\) 0 0
\(685\) 3.27820 0.125254
\(686\) 0 0
\(687\) −19.5139 −0.744501
\(688\) 0 0
\(689\) 55.5915 2.11787
\(690\) 0 0
\(691\) 0.0826952 0.00314587 0.00157294 0.999999i \(-0.499499\pi\)
0.00157294 + 0.999999i \(0.499499\pi\)
\(692\) 0 0
\(693\) −5.63910 −0.214212
\(694\) 0 0
\(695\) −16.0571 −0.609079
\(696\) 0 0
\(697\) 14.1992 0.537833
\(698\) 0 0
\(699\) 55.9488 2.11618
\(700\) 0 0
\(701\) 38.0571 1.43740 0.718698 0.695322i \(-0.244738\pi\)
0.718698 + 0.695322i \(0.244738\pi\)
\(702\) 0 0
\(703\) 12.8309 0.483927
\(704\) 0 0
\(705\) 25.0351 0.942878
\(706\) 0 0
\(707\) 11.7958 0.443626
\(708\) 0 0
\(709\) 36.4275 1.36806 0.684032 0.729452i \(-0.260225\pi\)
0.684032 + 0.729452i \(0.260225\pi\)
\(710\) 0 0
\(711\) 61.2211 2.29597
\(712\) 0 0
\(713\) 33.4700 1.25346
\(714\) 0 0
\(715\) −5.87847 −0.219842
\(716\) 0 0
\(717\) −66.4663 −2.48223
\(718\) 0 0
\(719\) −37.3097 −1.39142 −0.695708 0.718325i \(-0.744909\pi\)
−0.695708 + 0.718325i \(0.744909\pi\)
\(720\) 0 0
\(721\) −17.1178 −0.637502
\(722\) 0 0
\(723\) −52.8309 −1.96480
\(724\) 0 0
\(725\) −9.45681 −0.351217
\(726\) 0 0
\(727\) −3.23937 −0.120142 −0.0600708 0.998194i \(-0.519133\pi\)
−0.0600708 + 0.998194i \(0.519133\pi\)
\(728\) 0 0
\(729\) −35.2283 −1.30475
\(730\) 0 0
\(731\) 48.7921 1.80464
\(732\) 0 0
\(733\) −26.3522 −0.973340 −0.486670 0.873586i \(-0.661789\pi\)
−0.486670 + 0.873586i \(0.661789\pi\)
\(734\) 0 0
\(735\) −2.93923 −0.108415
\(736\) 0 0
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 11.9223 0.438570 0.219285 0.975661i \(-0.429628\pi\)
0.219285 + 0.975661i \(0.429628\pi\)
\(740\) 0 0
\(741\) −96.3836 −3.54074
\(742\) 0 0
\(743\) 26.0703 0.956426 0.478213 0.878244i \(-0.341285\pi\)
0.478213 + 0.878244i \(0.341285\pi\)
\(744\) 0 0
\(745\) −10.7350 −0.393300
\(746\) 0 0
\(747\) 78.2621 2.86346
\(748\) 0 0
\(749\) −2.51757 −0.0919901
\(750\) 0 0
\(751\) −36.6003 −1.33556 −0.667781 0.744357i \(-0.732756\pi\)
−0.667781 + 0.744357i \(0.732756\pi\)
\(752\) 0 0
\(753\) −25.7496 −0.938366
\(754\) 0 0
\(755\) −2.17860 −0.0792875
\(756\) 0 0
\(757\) −43.0922 −1.56621 −0.783107 0.621887i \(-0.786366\pi\)
−0.783107 + 0.621887i \(0.786366\pi\)
\(758\) 0 0
\(759\) 16.3960 0.595139
\(760\) 0 0
\(761\) −44.0959 −1.59848 −0.799238 0.601015i \(-0.794763\pi\)
−0.799238 + 0.601015i \(0.794763\pi\)
\(762\) 0 0
\(763\) 10.3001 0.372890
\(764\) 0 0
\(765\) 26.1604 0.945830
\(766\) 0 0
\(767\) 42.5564 1.53662
\(768\) 0 0
\(769\) −8.33897 −0.300711 −0.150355 0.988632i \(-0.548042\pi\)
−0.150355 + 0.988632i \(0.548042\pi\)
\(770\) 0 0
\(771\) −14.9889 −0.539813
\(772\) 0 0
\(773\) −19.2782 −0.693389 −0.346694 0.937978i \(-0.612696\pi\)
−0.346694 + 0.937978i \(0.612696\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 6.76063 0.242536
\(778\) 0 0
\(779\) −17.0740 −0.611739
\(780\) 0 0
\(781\) −7.15667 −0.256086
\(782\) 0 0
\(783\) −73.3559 −2.62153
\(784\) 0 0
\(785\) 17.1567 0.612348
\(786\) 0 0
\(787\) −43.5915 −1.55387 −0.776935 0.629580i \(-0.783227\pi\)
−0.776935 + 0.629580i \(0.783227\pi\)
\(788\) 0 0
\(789\) 34.5564 1.23024
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −85.3411 −3.03055
\(794\) 0 0
\(795\) 27.7958 0.985815
\(796\) 0 0
\(797\) −51.1493 −1.81180 −0.905900 0.423491i \(-0.860805\pi\)
−0.905900 + 0.423491i \(0.860805\pi\)
\(798\) 0 0
\(799\) 39.5139 1.39790
\(800\) 0 0
\(801\) 21.8711 0.772777
\(802\) 0 0
\(803\) −12.3960 −0.437447
\(804\) 0 0
\(805\) 5.57834 0.196611
\(806\) 0 0
\(807\) −34.3133 −1.20789
\(808\) 0 0
\(809\) 45.1567 1.58762 0.793812 0.608163i \(-0.208093\pi\)
0.793812 + 0.608163i \(0.208093\pi\)
\(810\) 0 0
\(811\) −39.2914 −1.37971 −0.689854 0.723948i \(-0.742325\pi\)
−0.689854 + 0.723948i \(0.742325\pi\)
\(812\) 0 0
\(813\) −51.8346 −1.81792
\(814\) 0 0
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) −58.6706 −2.05262
\(818\) 0 0
\(819\) −33.1493 −1.15833
\(820\) 0 0
\(821\) 25.6486 0.895143 0.447572 0.894248i \(-0.352289\pi\)
0.447572 + 0.894248i \(0.352289\pi\)
\(822\) 0 0
\(823\) 51.6486 1.80036 0.900179 0.435520i \(-0.143436\pi\)
0.900179 + 0.435520i \(0.143436\pi\)
\(824\) 0 0
\(825\) −2.93923 −0.102331
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) 5.36090 0.186192 0.0930958 0.995657i \(-0.470324\pi\)
0.0930958 + 0.995657i \(0.470324\pi\)
\(830\) 0 0
\(831\) 64.6632 2.24314
\(832\) 0 0
\(833\) −4.63910 −0.160735
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) −46.5416 −1.60871
\(838\) 0 0
\(839\) −16.9649 −0.585692 −0.292846 0.956160i \(-0.594602\pi\)
−0.292846 + 0.956160i \(0.594602\pi\)
\(840\) 0 0
\(841\) 60.4312 2.08383
\(842\) 0 0
\(843\) 3.39973 0.117093
\(844\) 0 0
\(845\) −21.5564 −0.741563
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 0 0
\(849\) −50.7847 −1.74293
\(850\) 0 0
\(851\) −12.8309 −0.439838
\(852\) 0 0
\(853\) 31.0666 1.06370 0.531850 0.846839i \(-0.321497\pi\)
0.531850 + 0.846839i \(0.321497\pi\)
\(854\) 0 0
\(855\) −31.4568 −1.07580
\(856\) 0 0
\(857\) 37.5966 1.28427 0.642137 0.766590i \(-0.278048\pi\)
0.642137 + 0.766590i \(0.278048\pi\)
\(858\) 0 0
\(859\) 14.0388 0.478999 0.239499 0.970897i \(-0.423017\pi\)
0.239499 + 0.970897i \(0.423017\pi\)
\(860\) 0 0
\(861\) −8.99631 −0.306593
\(862\) 0 0
\(863\) −38.8053 −1.32095 −0.660474 0.750849i \(-0.729645\pi\)
−0.660474 + 0.750849i \(0.729645\pi\)
\(864\) 0 0
\(865\) 5.48243 0.186408
\(866\) 0 0
\(867\) 13.2891 0.451320
\(868\) 0 0
\(869\) −10.8565 −0.368283
\(870\) 0 0
\(871\) 47.0278 1.59347
\(872\) 0 0
\(873\) 42.7350 1.44636
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 31.7131 1.07087 0.535437 0.844575i \(-0.320147\pi\)
0.535437 + 0.844575i \(0.320147\pi\)
\(878\) 0 0
\(879\) 5.52126 0.186228
\(880\) 0 0
\(881\) 2.24306 0.0755706 0.0377853 0.999286i \(-0.487970\pi\)
0.0377853 + 0.999286i \(0.487970\pi\)
\(882\) 0 0
\(883\) −24.4349 −0.822299 −0.411150 0.911568i \(-0.634873\pi\)
−0.411150 + 0.911568i \(0.634873\pi\)
\(884\) 0 0
\(885\) 21.2782 0.715259
\(886\) 0 0
\(887\) −35.5915 −1.19505 −0.597524 0.801851i \(-0.703849\pi\)
−0.597524 + 0.801851i \(0.703849\pi\)
\(888\) 0 0
\(889\) −10.4787 −0.351446
\(890\) 0 0
\(891\) −5.88216 −0.197060
\(892\) 0 0
\(893\) −47.5139 −1.58999
\(894\) 0 0
\(895\) 1.23937 0.0414275
\(896\) 0 0
\(897\) 96.3836 3.21816
\(898\) 0 0
\(899\) 56.7408 1.89241
\(900\) 0 0
\(901\) 43.8711 1.46156
\(902\) 0 0
\(903\) −30.9136 −1.02874
\(904\) 0 0
\(905\) −8.55641 −0.284425
\(906\) 0 0
\(907\) −17.3923 −0.577503 −0.288752 0.957404i \(-0.593240\pi\)
−0.288752 + 0.957404i \(0.593240\pi\)
\(908\) 0 0
\(909\) 66.5176 2.20625
\(910\) 0 0
\(911\) −52.1141 −1.72662 −0.863309 0.504675i \(-0.831612\pi\)
−0.863309 + 0.504675i \(0.831612\pi\)
\(912\) 0 0
\(913\) −13.8785 −0.459310
\(914\) 0 0
\(915\) −42.6706 −1.41064
\(916\) 0 0
\(917\) 1.57834 0.0521213
\(918\) 0 0
\(919\) −12.1347 −0.400288 −0.200144 0.979766i \(-0.564141\pi\)
−0.200144 + 0.979766i \(0.564141\pi\)
\(920\) 0 0
\(921\) −25.2782 −0.832945
\(922\) 0 0
\(923\) −42.0703 −1.38476
\(924\) 0 0
\(925\) 2.30013 0.0756279
\(926\) 0 0
\(927\) −96.5292 −3.17044
\(928\) 0 0
\(929\) 23.8008 0.780879 0.390439 0.920629i \(-0.372323\pi\)
0.390439 + 0.920629i \(0.372323\pi\)
\(930\) 0 0
\(931\) 5.57834 0.182823
\(932\) 0 0
\(933\) −40.4349 −1.32378
\(934\) 0 0
\(935\) −4.63910 −0.151715
\(936\) 0 0
\(937\) 9.16170 0.299300 0.149650 0.988739i \(-0.452185\pi\)
0.149650 + 0.988739i \(0.452185\pi\)
\(938\) 0 0
\(939\) 8.52496 0.278201
\(940\) 0 0
\(941\) 0.996308 0.0324787 0.0162393 0.999868i \(-0.494831\pi\)
0.0162393 + 0.999868i \(0.494831\pi\)
\(942\) 0 0
\(943\) 17.0740 0.556005
\(944\) 0 0
\(945\) −7.75694 −0.252333
\(946\) 0 0
\(947\) 18.8359 0.612086 0.306043 0.952018i \(-0.400995\pi\)
0.306043 + 0.952018i \(0.400995\pi\)
\(948\) 0 0
\(949\) −72.8698 −2.36545
\(950\) 0 0
\(951\) 56.8309 1.84287
\(952\) 0 0
\(953\) 21.4386 0.694463 0.347232 0.937779i \(-0.387122\pi\)
0.347232 + 0.937779i \(0.387122\pi\)
\(954\) 0 0
\(955\) −1.27820 −0.0413617
\(956\) 0 0
\(957\) 27.7958 0.898510
\(958\) 0 0
\(959\) −3.27820 −0.105859
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −14.1968 −0.457487
\(964\) 0 0
\(965\) 10.6391 0.342485
\(966\) 0 0
\(967\) −31.5139 −1.01342 −0.506709 0.862117i \(-0.669138\pi\)
−0.506709 + 0.862117i \(0.669138\pi\)
\(968\) 0 0
\(969\) −76.0629 −2.44349
\(970\) 0 0
\(971\) −16.1968 −0.519781 −0.259891 0.965638i \(-0.583687\pi\)
−0.259891 + 0.965638i \(0.583687\pi\)
\(972\) 0 0
\(973\) 16.0571 0.514766
\(974\) 0 0
\(975\) −17.2782 −0.553345
\(976\) 0 0
\(977\) 17.5139 0.560319 0.280159 0.959954i \(-0.409613\pi\)
0.280159 + 0.959954i \(0.409613\pi\)
\(978\) 0 0
\(979\) −3.87847 −0.123956
\(980\) 0 0
\(981\) 58.0835 1.85446
\(982\) 0 0
\(983\) −41.1178 −1.31146 −0.655728 0.754997i \(-0.727638\pi\)
−0.655728 + 0.754997i \(0.727638\pi\)
\(984\) 0 0
\(985\) −7.87847 −0.251029
\(986\) 0 0
\(987\) −25.0351 −0.796877
\(988\) 0 0
\(989\) 58.6706 1.86562
\(990\) 0 0
\(991\) 2.79947 0.0889280 0.0444640 0.999011i \(-0.485842\pi\)
0.0444640 + 0.999011i \(0.485842\pi\)
\(992\) 0 0
\(993\) −30.9136 −0.981014
\(994\) 0 0
\(995\) 0.517571 0.0164081
\(996\) 0 0
\(997\) 11.7181 0.371116 0.185558 0.982633i \(-0.440591\pi\)
0.185558 + 0.982633i \(0.440591\pi\)
\(998\) 0 0
\(999\) 17.8420 0.564496
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 6160.2.a.bi.1.3 3
4.3 odd 2 770.2.a.l.1.1 3
12.11 even 2 6930.2.a.cl.1.1 3
20.3 even 4 3850.2.c.z.1849.4 6
20.7 even 4 3850.2.c.z.1849.3 6
20.19 odd 2 3850.2.a.bu.1.3 3
28.27 even 2 5390.2.a.bz.1.3 3
44.43 even 2 8470.2.a.cl.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.l.1.1 3 4.3 odd 2
3850.2.a.bu.1.3 3 20.19 odd 2
3850.2.c.z.1849.3 6 20.7 even 4
3850.2.c.z.1849.4 6 20.3 even 4
5390.2.a.bz.1.3 3 28.27 even 2
6160.2.a.bi.1.3 3 1.1 even 1 trivial
6930.2.a.cl.1.1 3 12.11 even 2
8470.2.a.cl.1.1 3 44.43 even 2