Properties

Label 1078.2.a.o.1.2
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $2$
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] = [1078,2,Mod(1,1078)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1078, 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("1078.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1078 = 2 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1078.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: \(8.60787333789\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1078.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} -3.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} +2.82843 q^{5} -1.00000 q^{8} -3.00000 q^{9} -2.82843 q^{10} +1.00000 q^{11} -5.65685 q^{13} +1.00000 q^{16} -2.82843 q^{17} +3.00000 q^{18} -8.48528 q^{19} +2.82843 q^{20} -1.00000 q^{22} -8.00000 q^{23} +3.00000 q^{25} +5.65685 q^{26} -6.00000 q^{29} +8.48528 q^{31} -1.00000 q^{32} +2.82843 q^{34} -3.00000 q^{36} -6.00000 q^{37} +8.48528 q^{38} -2.82843 q^{40} +8.48528 q^{41} -4.00000 q^{43} +1.00000 q^{44} -8.48528 q^{45} +8.00000 q^{46} +2.82843 q^{47} -3.00000 q^{50} -5.65685 q^{52} +6.00000 q^{53} +2.82843 q^{55} +6.00000 q^{58} +5.65685 q^{59} +5.65685 q^{61} -8.48528 q^{62} +1.00000 q^{64} -16.0000 q^{65} -4.00000 q^{67} -2.82843 q^{68} +3.00000 q^{72} +8.48528 q^{73} +6.00000 q^{74} -8.48528 q^{76} +2.82843 q^{80} +9.00000 q^{81} -8.48528 q^{82} +2.82843 q^{83} -8.00000 q^{85} +4.00000 q^{86} -1.00000 q^{88} -11.3137 q^{89} +8.48528 q^{90} -8.00000 q^{92} -2.82843 q^{94} -24.0000 q^{95} -11.3137 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 6 q^{9} + 2 q^{11} + 2 q^{16} + 6 q^{18} - 2 q^{22} - 16 q^{23} + 6 q^{25} - 12 q^{29} - 2 q^{32} - 6 q^{36} - 12 q^{37} - 8 q^{43} + 2 q^{44} + 16 q^{46} - 6 q^{50} + 12 q^{53} + 12 q^{58} + 2 q^{64} - 32 q^{65} - 8 q^{67} + 6 q^{72} + 12 q^{74} + 18 q^{81} - 16 q^{85} + 8 q^{86} - 2 q^{88} - 16 q^{92} - 48 q^{95} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 1.00000 0.500000
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −3.00000 −1.00000
\(10\) −2.82843 −0.894427
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 3.00000 0.707107
\(19\) −8.48528 −1.94666 −0.973329 0.229416i \(-0.926318\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 2.82843 0.632456
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −8.00000 −1.66812 −0.834058 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 5.65685 1.10940
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) −3.00000 −0.500000
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 8.48528 1.37649
\(39\) 0 0
\(40\) −2.82843 −0.447214
\(41\) 8.48528 1.32518 0.662589 0.748983i \(-0.269458\pi\)
0.662589 + 0.748983i \(0.269458\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 1.00000 0.150756
\(45\) −8.48528 −1.26491
\(46\) 8.00000 1.17954
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −3.00000 −0.424264
\(51\) 0 0
\(52\) −5.65685 −0.784465
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) 6.00000 0.787839
\(59\) 5.65685 0.736460 0.368230 0.929735i \(-0.379964\pi\)
0.368230 + 0.929735i \(0.379964\pi\)
\(60\) 0 0
\(61\) 5.65685 0.724286 0.362143 0.932123i \(-0.382045\pi\)
0.362143 + 0.932123i \(0.382045\pi\)
\(62\) −8.48528 −1.07763
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −16.0000 −1.98456
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −2.82843 −0.342997
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 3.00000 0.353553
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) −8.48528 −0.973329
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 2.82843 0.316228
\(81\) 9.00000 1.00000
\(82\) −8.48528 −0.937043
\(83\) 2.82843 0.310460 0.155230 0.987878i \(-0.450388\pi\)
0.155230 + 0.987878i \(0.450388\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 4.00000 0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) −11.3137 −1.19925 −0.599625 0.800281i \(-0.704684\pi\)
−0.599625 + 0.800281i \(0.704684\pi\)
\(90\) 8.48528 0.894427
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) −2.82843 −0.291730
\(95\) −24.0000 −2.46235
\(96\) 0 0
\(97\) −11.3137 −1.14873 −0.574367 0.818598i \(-0.694752\pi\)
−0.574367 + 0.818598i \(0.694752\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 3.00000 0.300000
\(101\) −11.3137 −1.12576 −0.562878 0.826540i \(-0.690306\pi\)
−0.562878 + 0.826540i \(0.690306\pi\)
\(102\) 0 0
\(103\) 14.1421 1.39347 0.696733 0.717331i \(-0.254636\pi\)
0.696733 + 0.717331i \(0.254636\pi\)
\(104\) 5.65685 0.554700
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) −2.82843 −0.269680
\(111\) 0 0
\(112\) 0 0
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) 0 0
\(115\) −22.6274 −2.11002
\(116\) −6.00000 −0.557086
\(117\) 16.9706 1.56893
\(118\) −5.65685 −0.520756
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.65685 −0.512148
\(123\) 0 0
\(124\) 8.48528 0.762001
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 16.0000 1.40329
\(131\) −2.82843 −0.247121 −0.123560 0.992337i \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 2.82843 0.242536
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) 8.48528 0.719712 0.359856 0.933008i \(-0.382826\pi\)
0.359856 + 0.933008i \(0.382826\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) −3.00000 −0.250000
\(145\) −16.9706 −1.40933
\(146\) −8.48528 −0.702247
\(147\) 0 0
\(148\) −6.00000 −0.493197
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 8.48528 0.688247
\(153\) 8.48528 0.685994
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 19.7990 1.58013 0.790066 0.613022i \(-0.210046\pi\)
0.790066 + 0.613022i \(0.210046\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −2.82843 −0.223607
\(161\) 0 0
\(162\) −9.00000 −0.707107
\(163\) 12.0000 0.939913 0.469956 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 8.48528 0.662589
\(165\) 0 0
\(166\) −2.82843 −0.219529
\(167\) 5.65685 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 8.00000 0.613572
\(171\) 25.4558 1.94666
\(172\) −4.00000 −0.304997
\(173\) −5.65685 −0.430083 −0.215041 0.976605i \(-0.568989\pi\)
−0.215041 + 0.976605i \(0.568989\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) 11.3137 0.847998
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) −8.48528 −0.632456
\(181\) −8.48528 −0.630706 −0.315353 0.948974i \(-0.602123\pi\)
−0.315353 + 0.948974i \(0.602123\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 8.00000 0.589768
\(185\) −16.9706 −1.24770
\(186\) 0 0
\(187\) −2.82843 −0.206835
\(188\) 2.82843 0.206284
\(189\) 0 0
\(190\) 24.0000 1.74114
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 11.3137 0.812277
\(195\) 0 0
\(196\) 0 0
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 3.00000 0.213201
\(199\) 14.1421 1.00251 0.501255 0.865300i \(-0.332872\pi\)
0.501255 + 0.865300i \(0.332872\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 11.3137 0.796030
\(203\) 0 0
\(204\) 0 0
\(205\) 24.0000 1.67623
\(206\) −14.1421 −0.985329
\(207\) 24.0000 1.66812
\(208\) −5.65685 −0.392232
\(209\) −8.48528 −0.586939
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) −4.00000 −0.273434
\(215\) −11.3137 −0.771589
\(216\) 0 0
\(217\) 0 0
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) 2.82843 0.190693
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) −2.82843 −0.189405 −0.0947027 0.995506i \(-0.530190\pi\)
−0.0947027 + 0.995506i \(0.530190\pi\)
\(224\) 0 0
\(225\) −9.00000 −0.600000
\(226\) 6.00000 0.399114
\(227\) −2.82843 −0.187729 −0.0938647 0.995585i \(-0.529922\pi\)
−0.0938647 + 0.995585i \(0.529922\pi\)
\(228\) 0 0
\(229\) 14.1421 0.934539 0.467269 0.884115i \(-0.345238\pi\)
0.467269 + 0.884115i \(0.345238\pi\)
\(230\) 22.6274 1.49201
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) −26.0000 −1.70332 −0.851658 0.524097i \(-0.824403\pi\)
−0.851658 + 0.524097i \(0.824403\pi\)
\(234\) −16.9706 −1.10940
\(235\) 8.00000 0.521862
\(236\) 5.65685 0.368230
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −2.82843 −0.182195 −0.0910975 0.995842i \(-0.529037\pi\)
−0.0910975 + 0.995842i \(0.529037\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 5.65685 0.362143
\(245\) 0 0
\(246\) 0 0
\(247\) 48.0000 3.05417
\(248\) −8.48528 −0.538816
\(249\) 0 0
\(250\) 5.65685 0.357771
\(251\) 5.65685 0.357057 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(252\) 0 0
\(253\) −8.00000 −0.502956
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −11.3137 −0.705730 −0.352865 0.935674i \(-0.614792\pi\)
−0.352865 + 0.935674i \(0.614792\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −16.0000 −0.992278
\(261\) 18.0000 1.11417
\(262\) 2.82843 0.174741
\(263\) −16.0000 −0.986602 −0.493301 0.869859i \(-0.664210\pi\)
−0.493301 + 0.869859i \(0.664210\pi\)
\(264\) 0 0
\(265\) 16.9706 1.04249
\(266\) 0 0
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 25.4558 1.55207 0.776035 0.630690i \(-0.217228\pi\)
0.776035 + 0.630690i \(0.217228\pi\)
\(270\) 0 0
\(271\) −28.2843 −1.71815 −0.859074 0.511852i \(-0.828960\pi\)
−0.859074 + 0.511852i \(0.828960\pi\)
\(272\) −2.82843 −0.171499
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 3.00000 0.180907
\(276\) 0 0
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) −8.48528 −0.508913
\(279\) −25.4558 −1.52400
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −14.1421 −0.840663 −0.420331 0.907371i \(-0.638086\pi\)
−0.420331 + 0.907371i \(0.638086\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 5.65685 0.334497
\(287\) 0 0
\(288\) 3.00000 0.176777
\(289\) −9.00000 −0.529412
\(290\) 16.9706 0.996546
\(291\) 0 0
\(292\) 8.48528 0.496564
\(293\) −5.65685 −0.330477 −0.165238 0.986254i \(-0.552839\pi\)
−0.165238 + 0.986254i \(0.552839\pi\)
\(294\) 0 0
\(295\) 16.0000 0.931556
\(296\) 6.00000 0.348743
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 45.2548 2.61715
\(300\) 0 0
\(301\) 0 0
\(302\) −8.00000 −0.460348
\(303\) 0 0
\(304\) −8.48528 −0.486664
\(305\) 16.0000 0.916157
\(306\) −8.48528 −0.485071
\(307\) −14.1421 −0.807134 −0.403567 0.914950i \(-0.632230\pi\)
−0.403567 + 0.914950i \(0.632230\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −24.0000 −1.36311
\(311\) −14.1421 −0.801927 −0.400963 0.916094i \(-0.631325\pi\)
−0.400963 + 0.916094i \(0.631325\pi\)
\(312\) 0 0
\(313\) 16.9706 0.959233 0.479616 0.877478i \(-0.340776\pi\)
0.479616 + 0.877478i \(0.340776\pi\)
\(314\) −19.7990 −1.11732
\(315\) 0 0
\(316\) 0 0
\(317\) −2.00000 −0.112331 −0.0561656 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 2.82843 0.158114
\(321\) 0 0
\(322\) 0 0
\(323\) 24.0000 1.33540
\(324\) 9.00000 0.500000
\(325\) −16.9706 −0.941357
\(326\) −12.0000 −0.664619
\(327\) 0 0
\(328\) −8.48528 −0.468521
\(329\) 0 0
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 2.82843 0.155230
\(333\) 18.0000 0.986394
\(334\) −5.65685 −0.309529
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −19.0000 −1.03346
\(339\) 0 0
\(340\) −8.00000 −0.433861
\(341\) 8.48528 0.459504
\(342\) −25.4558 −1.37649
\(343\) 0 0
\(344\) 4.00000 0.215666
\(345\) 0 0
\(346\) 5.65685 0.304114
\(347\) −4.00000 −0.214731 −0.107366 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −11.3137 −0.599625
\(357\) 0 0
\(358\) 12.0000 0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 8.48528 0.447214
\(361\) 53.0000 2.78947
\(362\) 8.48528 0.445976
\(363\) 0 0
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) −25.4558 −1.32878 −0.664392 0.747384i \(-0.731309\pi\)
−0.664392 + 0.747384i \(0.731309\pi\)
\(368\) −8.00000 −0.417029
\(369\) −25.4558 −1.32518
\(370\) 16.9706 0.882258
\(371\) 0 0
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 2.82843 0.146254
\(375\) 0 0
\(376\) −2.82843 −0.145865
\(377\) 33.9411 1.74806
\(378\) 0 0
\(379\) −28.0000 −1.43826 −0.719132 0.694874i \(-0.755460\pi\)
−0.719132 + 0.694874i \(0.755460\pi\)
\(380\) −24.0000 −1.23117
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) −14.1421 −0.722629 −0.361315 0.932444i \(-0.617672\pi\)
−0.361315 + 0.932444i \(0.617672\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 12.0000 0.609994
\(388\) −11.3137 −0.574367
\(389\) 10.0000 0.507020 0.253510 0.967333i \(-0.418415\pi\)
0.253510 + 0.967333i \(0.418415\pi\)
\(390\) 0 0
\(391\) 22.6274 1.14432
\(392\) 0 0
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 8.48528 0.425864 0.212932 0.977067i \(-0.431699\pi\)
0.212932 + 0.977067i \(0.431699\pi\)
\(398\) −14.1421 −0.708881
\(399\) 0 0
\(400\) 3.00000 0.150000
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) −48.0000 −2.39105
\(404\) −11.3137 −0.562878
\(405\) 25.4558 1.26491
\(406\) 0 0
\(407\) −6.00000 −0.297409
\(408\) 0 0
\(409\) −8.48528 −0.419570 −0.209785 0.977748i \(-0.567276\pi\)
−0.209785 + 0.977748i \(0.567276\pi\)
\(410\) −24.0000 −1.18528
\(411\) 0 0
\(412\) 14.1421 0.696733
\(413\) 0 0
\(414\) −24.0000 −1.17954
\(415\) 8.00000 0.392705
\(416\) 5.65685 0.277350
\(417\) 0 0
\(418\) 8.48528 0.415029
\(419\) −22.6274 −1.10542 −0.552711 0.833373i \(-0.686407\pi\)
−0.552711 + 0.833373i \(0.686407\pi\)
\(420\) 0 0
\(421\) −26.0000 −1.26716 −0.633581 0.773676i \(-0.718416\pi\)
−0.633581 + 0.773676i \(0.718416\pi\)
\(422\) −28.0000 −1.36302
\(423\) −8.48528 −0.412568
\(424\) −6.00000 −0.291386
\(425\) −8.48528 −0.411597
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000 0.193347
\(429\) 0 0
\(430\) 11.3137 0.545595
\(431\) 16.0000 0.770693 0.385346 0.922772i \(-0.374082\pi\)
0.385346 + 0.922772i \(0.374082\pi\)
\(432\) 0 0
\(433\) 33.9411 1.63111 0.815553 0.578682i \(-0.196433\pi\)
0.815553 + 0.578682i \(0.196433\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −14.0000 −0.670478
\(437\) 67.8823 3.24725
\(438\) 0 0
\(439\) 28.2843 1.34993 0.674967 0.737848i \(-0.264158\pi\)
0.674967 + 0.737848i \(0.264158\pi\)
\(440\) −2.82843 −0.134840
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −32.0000 −1.51695
\(446\) 2.82843 0.133930
\(447\) 0 0
\(448\) 0 0
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 9.00000 0.424264
\(451\) 8.48528 0.399556
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) 2.82843 0.132745
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) −14.1421 −0.660819
\(459\) 0 0
\(460\) −22.6274 −1.05501
\(461\) −11.3137 −0.526932 −0.263466 0.964669i \(-0.584866\pi\)
−0.263466 + 0.964669i \(0.584866\pi\)
\(462\) 0 0
\(463\) −24.0000 −1.11537 −0.557687 0.830051i \(-0.688311\pi\)
−0.557687 + 0.830051i \(0.688311\pi\)
\(464\) −6.00000 −0.278543
\(465\) 0 0
\(466\) 26.0000 1.20443
\(467\) −33.9411 −1.57061 −0.785304 0.619110i \(-0.787493\pi\)
−0.785304 + 0.619110i \(0.787493\pi\)
\(468\) 16.9706 0.784465
\(469\) 0 0
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −5.65685 −0.260378
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) −25.4558 −1.16799
\(476\) 0 0
\(477\) −18.0000 −0.824163
\(478\) −24.0000 −1.09773
\(479\) −11.3137 −0.516937 −0.258468 0.966020i \(-0.583218\pi\)
−0.258468 + 0.966020i \(0.583218\pi\)
\(480\) 0 0
\(481\) 33.9411 1.54758
\(482\) 2.82843 0.128831
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −32.0000 −1.45305
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) −5.65685 −0.256074
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 16.9706 0.764316
\(494\) −48.0000 −2.15962
\(495\) −8.48528 −0.381385
\(496\) 8.48528 0.381000
\(497\) 0 0
\(498\) 0 0
\(499\) 44.0000 1.96971 0.984855 0.173379i \(-0.0554684\pi\)
0.984855 + 0.173379i \(0.0554684\pi\)
\(500\) −5.65685 −0.252982
\(501\) 0 0
\(502\) −5.65685 −0.252478
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) −32.0000 −1.42398
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) −31.1127 −1.37905 −0.689523 0.724264i \(-0.742180\pi\)
−0.689523 + 0.724264i \(0.742180\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 11.3137 0.499026
\(515\) 40.0000 1.76261
\(516\) 0 0
\(517\) 2.82843 0.124394
\(518\) 0 0
\(519\) 0 0
\(520\) 16.0000 0.701646
\(521\) −11.3137 −0.495663 −0.247831 0.968803i \(-0.579718\pi\)
−0.247831 + 0.968803i \(0.579718\pi\)
\(522\) −18.0000 −0.787839
\(523\) 31.1127 1.36046 0.680232 0.732997i \(-0.261879\pi\)
0.680232 + 0.732997i \(0.261879\pi\)
\(524\) −2.82843 −0.123560
\(525\) 0 0
\(526\) 16.0000 0.697633
\(527\) −24.0000 −1.04546
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −16.9706 −0.737154
\(531\) −16.9706 −0.736460
\(532\) 0 0
\(533\) −48.0000 −2.07911
\(534\) 0 0
\(535\) 11.3137 0.489134
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −25.4558 −1.09748
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 28.2843 1.21491
\(543\) 0 0
\(544\) 2.82843 0.121268
\(545\) −39.5980 −1.69619
\(546\) 0 0
\(547\) −12.0000 −0.513083 −0.256541 0.966533i \(-0.582583\pi\)
−0.256541 + 0.966533i \(0.582583\pi\)
\(548\) −18.0000 −0.768922
\(549\) −16.9706 −0.724286
\(550\) −3.00000 −0.127920
\(551\) 50.9117 2.16891
\(552\) 0 0
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 0 0
\(556\) 8.48528 0.359856
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 25.4558 1.07763
\(559\) 22.6274 0.957038
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) 19.7990 0.834428 0.417214 0.908808i \(-0.363007\pi\)
0.417214 + 0.908808i \(0.363007\pi\)
\(564\) 0 0
\(565\) −16.9706 −0.713957
\(566\) 14.1421 0.594438
\(567\) 0 0
\(568\) 0 0
\(569\) 22.0000 0.922288 0.461144 0.887325i \(-0.347439\pi\)
0.461144 + 0.887325i \(0.347439\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) −5.65685 −0.236525
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) −3.00000 −0.125000
\(577\) −16.9706 −0.706494 −0.353247 0.935530i \(-0.614922\pi\)
−0.353247 + 0.935530i \(0.614922\pi\)
\(578\) 9.00000 0.374351
\(579\) 0 0
\(580\) −16.9706 −0.704664
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −8.48528 −0.351123
\(585\) 48.0000 1.98456
\(586\) 5.65685 0.233682
\(587\) 39.5980 1.63438 0.817192 0.576366i \(-0.195530\pi\)
0.817192 + 0.576366i \(0.195530\pi\)
\(588\) 0 0
\(589\) −72.0000 −2.96671
\(590\) −16.0000 −0.658710
\(591\) 0 0
\(592\) −6.00000 −0.246598
\(593\) −36.7696 −1.50994 −0.754972 0.655757i \(-0.772350\pi\)
−0.754972 + 0.655757i \(0.772350\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) −45.2548 −1.85061
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −19.7990 −0.807618 −0.403809 0.914843i \(-0.632314\pi\)
−0.403809 + 0.914843i \(0.632314\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) 8.00000 0.325515
\(605\) 2.82843 0.114992
\(606\) 0 0
\(607\) −33.9411 −1.37763 −0.688814 0.724938i \(-0.741868\pi\)
−0.688814 + 0.724938i \(0.741868\pi\)
\(608\) 8.48528 0.344124
\(609\) 0 0
\(610\) −16.0000 −0.647821
\(611\) −16.0000 −0.647291
\(612\) 8.48528 0.342997
\(613\) 2.00000 0.0807792 0.0403896 0.999184i \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 14.1421 0.570730
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) −11.3137 −0.454736 −0.227368 0.973809i \(-0.573012\pi\)
−0.227368 + 0.973809i \(0.573012\pi\)
\(620\) 24.0000 0.963863
\(621\) 0 0
\(622\) 14.1421 0.567048
\(623\) 0 0
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) −16.9706 −0.678280
\(627\) 0 0
\(628\) 19.7990 0.790066
\(629\) 16.9706 0.676661
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 2.00000 0.0794301
\(635\) 22.6274 0.897942
\(636\) 0 0
\(637\) 0 0
\(638\) 6.00000 0.237542
\(639\) 0 0
\(640\) −2.82843 −0.111803
\(641\) 46.0000 1.81689 0.908445 0.418004i \(-0.137270\pi\)
0.908445 + 0.418004i \(0.137270\pi\)
\(642\) 0 0
\(643\) 5.65685 0.223085 0.111542 0.993760i \(-0.464421\pi\)
0.111542 + 0.993760i \(0.464421\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −36.7696 −1.44556 −0.722780 0.691078i \(-0.757136\pi\)
−0.722780 + 0.691078i \(0.757136\pi\)
\(648\) −9.00000 −0.353553
\(649\) 5.65685 0.222051
\(650\) 16.9706 0.665640
\(651\) 0 0
\(652\) 12.0000 0.469956
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0 0
\(655\) −8.00000 −0.312586
\(656\) 8.48528 0.331295
\(657\) −25.4558 −0.993127
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) −8.48528 −0.330039 −0.165020 0.986290i \(-0.552769\pi\)
−0.165020 + 0.986290i \(0.552769\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −2.82843 −0.109764
\(665\) 0 0
\(666\) −18.0000 −0.697486
\(667\) 48.0000 1.85857
\(668\) 5.65685 0.218870
\(669\) 0 0
\(670\) 11.3137 0.437087
\(671\) 5.65685 0.218380
\(672\) 0 0
\(673\) 14.0000 0.539660 0.269830 0.962908i \(-0.413032\pi\)
0.269830 + 0.962908i \(0.413032\pi\)
\(674\) 18.0000 0.693334
\(675\) 0 0
\(676\) 19.0000 0.730769
\(677\) 28.2843 1.08705 0.543526 0.839392i \(-0.317089\pi\)
0.543526 + 0.839392i \(0.317089\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) 0 0
\(682\) −8.48528 −0.324918
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 25.4558 0.973329
\(685\) −50.9117 −1.94524
\(686\) 0 0
\(687\) 0 0
\(688\) −4.00000 −0.152499
\(689\) −33.9411 −1.29305
\(690\) 0 0
\(691\) −50.9117 −1.93677 −0.968386 0.249457i \(-0.919748\pi\)
−0.968386 + 0.249457i \(0.919748\pi\)
\(692\) −5.65685 −0.215041
\(693\) 0 0
\(694\) 4.00000 0.151838
\(695\) 24.0000 0.910372
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 50.9117 1.92017
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 11.3137 0.423999
\(713\) −67.8823 −2.54221
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) −12.0000 −0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) −25.4558 −0.949343 −0.474671 0.880163i \(-0.657433\pi\)
−0.474671 + 0.880163i \(0.657433\pi\)
\(720\) −8.48528 −0.316228
\(721\) 0 0
\(722\) −53.0000 −1.97246
\(723\) 0 0
\(724\) −8.48528 −0.315353
\(725\) −18.0000 −0.668503
\(726\) 0 0
\(727\) −14.1421 −0.524503 −0.262251 0.965000i \(-0.584465\pi\)
−0.262251 + 0.965000i \(0.584465\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) −24.0000 −0.888280
\(731\) 11.3137 0.418453
\(732\) 0 0
\(733\) −16.9706 −0.626822 −0.313411 0.949618i \(-0.601472\pi\)
−0.313411 + 0.949618i \(0.601472\pi\)
\(734\) 25.4558 0.939592
\(735\) 0 0
\(736\) 8.00000 0.294884
\(737\) −4.00000 −0.147342
\(738\) 25.4558 0.937043
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −16.9706 −0.623850
\(741\) 0 0
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) −28.2843 −1.03626
\(746\) 2.00000 0.0732252
\(747\) −8.48528 −0.310460
\(748\) −2.82843 −0.103418
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 2.82843 0.103142
\(753\) 0 0
\(754\) −33.9411 −1.23606
\(755\) 22.6274 0.823496
\(756\) 0 0
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 28.0000 1.01701
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 2.82843 0.102530 0.0512652 0.998685i \(-0.483675\pi\)
0.0512652 + 0.998685i \(0.483675\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −8.00000 −0.289430
\(765\) 24.0000 0.867722
\(766\) 14.1421 0.510976
\(767\) −32.0000 −1.15545
\(768\) 0 0
\(769\) 36.7696 1.32594 0.662972 0.748644i \(-0.269295\pi\)
0.662972 + 0.748644i \(0.269295\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.0000 −0.503871
\(773\) −2.82843 −0.101731 −0.0508657 0.998706i \(-0.516198\pi\)
−0.0508657 + 0.998706i \(0.516198\pi\)
\(774\) −12.0000 −0.431331
\(775\) 25.4558 0.914401
\(776\) 11.3137 0.406138
\(777\) 0 0
\(778\) −10.0000 −0.358517
\(779\) −72.0000 −2.57967
\(780\) 0 0
\(781\) 0 0
\(782\) −22.6274 −0.809155
\(783\) 0 0
\(784\) 0 0
\(785\) 56.0000 1.99873
\(786\) 0 0
\(787\) 25.4558 0.907403 0.453701 0.891154i \(-0.350103\pi\)
0.453701 + 0.891154i \(0.350103\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 3.00000 0.106600
\(793\) −32.0000 −1.13635
\(794\) −8.48528 −0.301131
\(795\) 0 0
\(796\) 14.1421 0.501255
\(797\) 19.7990 0.701316 0.350658 0.936504i \(-0.385958\pi\)
0.350658 + 0.936504i \(0.385958\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) −3.00000 −0.106066
\(801\) 33.9411 1.19925
\(802\) 10.0000 0.353112
\(803\) 8.48528 0.299439
\(804\) 0 0
\(805\) 0 0
\(806\) 48.0000 1.69073
\(807\) 0 0
\(808\) 11.3137 0.398015
\(809\) 26.0000 0.914111 0.457056 0.889438i \(-0.348904\pi\)
0.457056 + 0.889438i \(0.348904\pi\)
\(810\) −25.4558 −0.894427
\(811\) 8.48528 0.297959 0.148979 0.988840i \(-0.452401\pi\)
0.148979 + 0.988840i \(0.452401\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 6.00000 0.210300
\(815\) 33.9411 1.18891
\(816\) 0 0
\(817\) 33.9411 1.18745
\(818\) 8.48528 0.296681
\(819\) 0 0
\(820\) 24.0000 0.838116
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) −14.1421 −0.492665
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 24.0000 0.834058
\(829\) 25.4558 0.884118 0.442059 0.896986i \(-0.354248\pi\)
0.442059 + 0.896986i \(0.354248\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −5.65685 −0.196116
\(833\) 0 0
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) −8.48528 −0.293470
\(837\) 0 0
\(838\) 22.6274 0.781651
\(839\) 53.7401 1.85531 0.927657 0.373432i \(-0.121819\pi\)
0.927657 + 0.373432i \(0.121819\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 26.0000 0.896019
\(843\) 0 0
\(844\) 28.0000 0.963800
\(845\) 53.7401 1.84872
\(846\) 8.48528 0.291730
\(847\) 0 0
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 8.48528 0.291043
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) 50.9117 1.74318 0.871592 0.490233i \(-0.163088\pi\)
0.871592 + 0.490233i \(0.163088\pi\)
\(854\) 0 0
\(855\) 72.0000 2.46235
\(856\) −4.00000 −0.136717
\(857\) 2.82843 0.0966172 0.0483086 0.998832i \(-0.484617\pi\)
0.0483086 + 0.998832i \(0.484617\pi\)
\(858\) 0 0
\(859\) −39.5980 −1.35107 −0.675533 0.737330i \(-0.736086\pi\)
−0.675533 + 0.737330i \(0.736086\pi\)
\(860\) −11.3137 −0.385794
\(861\) 0 0
\(862\) −16.0000 −0.544962
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −16.0000 −0.544016
\(866\) −33.9411 −1.15337
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 22.6274 0.766701
\(872\) 14.0000 0.474100
\(873\) 33.9411 1.14873
\(874\) −67.8823 −2.29615
\(875\) 0 0
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) −28.2843 −0.954548
\(879\) 0 0
\(880\) 2.82843 0.0953463
\(881\) −5.65685 −0.190584 −0.0952921 0.995449i \(-0.530379\pi\)
−0.0952921 + 0.995449i \(0.530379\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 16.0000 0.538138
\(885\) 0 0
\(886\) 20.0000 0.671913
\(887\) 45.2548 1.51951 0.759754 0.650210i \(-0.225319\pi\)
0.759754 + 0.650210i \(0.225319\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 32.0000 1.07264
\(891\) 9.00000 0.301511
\(892\) −2.82843 −0.0947027
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) −33.9411 −1.13453
\(896\) 0 0
\(897\) 0 0
\(898\) 6.00000 0.200223
\(899\) −50.9117 −1.69800
\(900\) −9.00000 −0.300000
\(901\) −16.9706 −0.565371
\(902\) −8.48528 −0.282529
\(903\) 0 0
\(904\) 6.00000 0.199557
\(905\) −24.0000 −0.797787
\(906\) 0 0
\(907\) −12.0000 −0.398453 −0.199227 0.979953i \(-0.563843\pi\)
−0.199227 + 0.979953i \(0.563843\pi\)
\(908\) −2.82843 −0.0938647
\(909\) 33.9411 1.12576
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) 0 0
\(913\) 2.82843 0.0936073
\(914\) −10.0000 −0.330771
\(915\) 0 0
\(916\) 14.1421 0.467269
\(917\) 0 0
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 22.6274 0.746004
\(921\) 0 0
\(922\) 11.3137 0.372597
\(923\) 0 0
\(924\) 0 0
\(925\) −18.0000 −0.591836
\(926\) 24.0000 0.788689
\(927\) −42.4264 −1.39347
\(928\) 6.00000 0.196960
\(929\) 11.3137 0.371191 0.185595 0.982626i \(-0.440579\pi\)
0.185595 + 0.982626i \(0.440579\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −26.0000 −0.851658
\(933\) 0 0
\(934\) 33.9411 1.11059
\(935\) −8.00000 −0.261628
\(936\) −16.9706 −0.554700
\(937\) 19.7990 0.646805 0.323402 0.946262i \(-0.395173\pi\)
0.323402 + 0.946262i \(0.395173\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) −50.9117 −1.65967 −0.829837 0.558006i \(-0.811567\pi\)
−0.829837 + 0.558006i \(0.811567\pi\)
\(942\) 0 0
\(943\) −67.8823 −2.21055
\(944\) 5.65685 0.184115
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 52.0000 1.68977 0.844886 0.534946i \(-0.179668\pi\)
0.844886 + 0.534946i \(0.179668\pi\)
\(948\) 0 0
\(949\) −48.0000 −1.55815
\(950\) 25.4558 0.825897
\(951\) 0 0
\(952\) 0 0
\(953\) 42.0000 1.36051 0.680257 0.732974i \(-0.261868\pi\)
0.680257 + 0.732974i \(0.261868\pi\)
\(954\) 18.0000 0.582772
\(955\) −22.6274 −0.732206
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) 11.3137 0.365529
\(959\) 0 0
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) −33.9411 −1.09431
\(963\) −12.0000 −0.386695
\(964\) −2.82843 −0.0910975
\(965\) −39.5980 −1.27470
\(966\) 0 0
\(967\) −40.0000 −1.28631 −0.643157 0.765735i \(-0.722376\pi\)
−0.643157 + 0.765735i \(0.722376\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 32.0000 1.02746
\(971\) −11.3137 −0.363074 −0.181537 0.983384i \(-0.558107\pi\)
−0.181537 + 0.983384i \(0.558107\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −8.00000 −0.256337
\(975\) 0 0
\(976\) 5.65685 0.181071
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) 0 0
\(979\) −11.3137 −0.361588
\(980\) 0 0
\(981\) 42.0000 1.34096
\(982\) 12.0000 0.382935
\(983\) 48.0833 1.53362 0.766809 0.641875i \(-0.221843\pi\)
0.766809 + 0.641875i \(0.221843\pi\)
\(984\) 0 0
\(985\) −50.9117 −1.62218
\(986\) −16.9706 −0.540453
\(987\) 0 0
\(988\) 48.0000 1.52708
\(989\) 32.0000 1.01754
\(990\) 8.48528 0.269680
\(991\) 24.0000 0.762385 0.381193 0.924496i \(-0.375513\pi\)
0.381193 + 0.924496i \(0.375513\pi\)
\(992\) −8.48528 −0.269408
\(993\) 0 0
\(994\) 0 0
\(995\) 40.0000 1.26809
\(996\) 0 0
\(997\) −11.3137 −0.358309 −0.179154 0.983821i \(-0.557336\pi\)
−0.179154 + 0.983821i \(0.557336\pi\)
\(998\) −44.0000 −1.39280
\(999\) 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 1078.2.a.o.1.2 yes 2
3.2 odd 2 9702.2.a.dk.1.1 2
4.3 odd 2 8624.2.a.bo.1.2 2
7.2 even 3 1078.2.e.u.67.1 4
7.3 odd 6 1078.2.e.u.177.2 4
7.4 even 3 1078.2.e.u.177.1 4
7.5 odd 6 1078.2.e.u.67.2 4
7.6 odd 2 inner 1078.2.a.o.1.1 2
21.20 even 2 9702.2.a.dk.1.2 2
28.27 even 2 8624.2.a.bo.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.o.1.1 2 7.6 odd 2 inner
1078.2.a.o.1.2 yes 2 1.1 even 1 trivial
1078.2.e.u.67.1 4 7.2 even 3
1078.2.e.u.67.2 4 7.5 odd 6
1078.2.e.u.177.1 4 7.4 even 3
1078.2.e.u.177.2 4 7.3 odd 6
8624.2.a.bo.1.1 2 28.27 even 2
8624.2.a.bo.1.2 2 4.3 odd 2
9702.2.a.dk.1.1 2 3.2 odd 2
9702.2.a.dk.1.2 2 21.20 even 2