Properties

Label 1078.2.a.q.1.1
Level $1078$
Weight $2$
Character 1078.1
Self dual yes
Analytic conductor $8.608$
Analytic rank $1$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.41421 q^{3} +1.00000 q^{4} +1.41421 q^{5} +1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} -1.41421 q^{10} -1.00000 q^{11} -1.41421 q^{12} +2.82843 q^{13} -2.00000 q^{15} +1.00000 q^{16} -5.65685 q^{17} +1.00000 q^{18} +2.82843 q^{19} +1.41421 q^{20} +1.00000 q^{22} -2.00000 q^{23} +1.41421 q^{24} -3.00000 q^{25} -2.82843 q^{26} +5.65685 q^{27} -6.00000 q^{29} +2.00000 q^{30} -1.41421 q^{31} -1.00000 q^{32} +1.41421 q^{33} +5.65685 q^{34} -1.00000 q^{36} -2.00000 q^{37} -2.82843 q^{38} -4.00000 q^{39} -1.41421 q^{40} +5.65685 q^{41} +4.00000 q^{43} -1.00000 q^{44} -1.41421 q^{45} +2.00000 q^{46} +4.24264 q^{47} -1.41421 q^{48} +3.00000 q^{50} +8.00000 q^{51} +2.82843 q^{52} -12.0000 q^{53} -5.65685 q^{54} -1.41421 q^{55} -4.00000 q^{57} +6.00000 q^{58} +4.24264 q^{59} -2.00000 q^{60} -5.65685 q^{61} +1.41421 q^{62} +1.00000 q^{64} +4.00000 q^{65} -1.41421 q^{66} -2.00000 q^{67} -5.65685 q^{68} +2.82843 q^{69} -10.0000 q^{71} +1.00000 q^{72} -14.1421 q^{73} +2.00000 q^{74} +4.24264 q^{75} +2.82843 q^{76} +4.00000 q^{78} -8.00000 q^{79} +1.41421 q^{80} -5.00000 q^{81} -5.65685 q^{82} -2.82843 q^{83} -8.00000 q^{85} -4.00000 q^{86} +8.48528 q^{87} +1.00000 q^{88} -7.07107 q^{89} +1.41421 q^{90} -2.00000 q^{92} +2.00000 q^{93} -4.24264 q^{94} +4.00000 q^{95} +1.41421 q^{96} -15.5563 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 2 q^{32} - 2 q^{36} - 4 q^{37} - 8 q^{39} + 8 q^{43} - 2 q^{44} + 4 q^{46} + 6 q^{50} + 16 q^{51} - 24 q^{53} - 8 q^{57} + 12 q^{58} - 4 q^{60} + 2 q^{64} + 8 q^{65} - 4 q^{67} - 20 q^{71} + 2 q^{72} + 4 q^{74} + 8 q^{78} - 16 q^{79} - 10 q^{81} - 16 q^{85} - 8 q^{86} + 2 q^{88} - 4 q^{92} + 4 q^{93} + 8 q^{95} + 2 q^{99}+O(q^{100}) \) 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
\(3\) −1.41421 −0.816497 −0.408248 0.912871i \(-0.633860\pi\)
−0.408248 + 0.912871i \(0.633860\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.41421 0.632456 0.316228 0.948683i \(-0.397584\pi\)
0.316228 + 0.948683i \(0.397584\pi\)
\(6\) 1.41421 0.577350
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) −1.41421 −0.447214
\(11\) −1.00000 −0.301511
\(12\) −1.41421 −0.408248
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −5.65685 −1.37199 −0.685994 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 1.00000 0.235702
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 1.41421 0.316228
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.00000 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 1.41421 0.288675
\(25\) −3.00000 −0.600000
\(26\) −2.82843 −0.554700
\(27\) 5.65685 1.08866
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 2.00000 0.365148
\(31\) −1.41421 −0.254000 −0.127000 0.991903i \(-0.540535\pi\)
−0.127000 + 0.991903i \(0.540535\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.41421 0.246183
\(34\) 5.65685 0.970143
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) −2.82843 −0.458831
\(39\) −4.00000 −0.640513
\(40\) −1.41421 −0.223607
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −1.00000 −0.150756
\(45\) −1.41421 −0.210819
\(46\) 2.00000 0.294884
\(47\) 4.24264 0.618853 0.309426 0.950923i \(-0.399863\pi\)
0.309426 + 0.950923i \(0.399863\pi\)
\(48\) −1.41421 −0.204124
\(49\) 0 0
\(50\) 3.00000 0.424264
\(51\) 8.00000 1.12022
\(52\) 2.82843 0.392232
\(53\) −12.0000 −1.64833 −0.824163 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) −5.65685 −0.769800
\(55\) −1.41421 −0.190693
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 6.00000 0.787839
\(59\) 4.24264 0.552345 0.276172 0.961108i \(-0.410934\pi\)
0.276172 + 0.961108i \(0.410934\pi\)
\(60\) −2.00000 −0.258199
\(61\) −5.65685 −0.724286 −0.362143 0.932123i \(-0.617955\pi\)
−0.362143 + 0.932123i \(0.617955\pi\)
\(62\) 1.41421 0.179605
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.00000 0.496139
\(66\) −1.41421 −0.174078
\(67\) −2.00000 −0.244339 −0.122169 0.992509i \(-0.538985\pi\)
−0.122169 + 0.992509i \(0.538985\pi\)
\(68\) −5.65685 −0.685994
\(69\) 2.82843 0.340503
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 1.00000 0.117851
\(73\) −14.1421 −1.65521 −0.827606 0.561310i \(-0.810298\pi\)
−0.827606 + 0.561310i \(0.810298\pi\)
\(74\) 2.00000 0.232495
\(75\) 4.24264 0.489898
\(76\) 2.82843 0.324443
\(77\) 0 0
\(78\) 4.00000 0.452911
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 1.41421 0.158114
\(81\) −5.00000 −0.555556
\(82\) −5.65685 −0.624695
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) −4.00000 −0.431331
\(87\) 8.48528 0.909718
\(88\) 1.00000 0.106600
\(89\) −7.07107 −0.749532 −0.374766 0.927119i \(-0.622277\pi\)
−0.374766 + 0.927119i \(0.622277\pi\)
\(90\) 1.41421 0.149071
\(91\) 0 0
\(92\) −2.00000 −0.208514
\(93\) 2.00000 0.207390
\(94\) −4.24264 −0.437595
\(95\) 4.00000 0.410391
\(96\) 1.41421 0.144338
\(97\) −15.5563 −1.57951 −0.789754 0.613424i \(-0.789792\pi\)
−0.789754 + 0.613424i \(0.789792\pi\)
\(98\) 0 0
\(99\) 1.00000 0.100504
\(100\) −3.00000 −0.300000
\(101\) 2.82843 0.281439 0.140720 0.990050i \(-0.455058\pi\)
0.140720 + 0.990050i \(0.455058\pi\)
\(102\) −8.00000 −0.792118
\(103\) −4.24264 −0.418040 −0.209020 0.977911i \(-0.567027\pi\)
−0.209020 + 0.977911i \(0.567027\pi\)
\(104\) −2.82843 −0.277350
\(105\) 0 0
\(106\) 12.0000 1.16554
\(107\) −20.0000 −1.93347 −0.966736 0.255774i \(-0.917670\pi\)
−0.966736 + 0.255774i \(0.917670\pi\)
\(108\) 5.65685 0.544331
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 1.41421 0.134840
\(111\) 2.82843 0.268462
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 4.00000 0.374634
\(115\) −2.82843 −0.263752
\(116\) −6.00000 −0.557086
\(117\) −2.82843 −0.261488
\(118\) −4.24264 −0.390567
\(119\) 0 0
\(120\) 2.00000 0.182574
\(121\) 1.00000 0.0909091
\(122\) 5.65685 0.512148
\(123\) −8.00000 −0.721336
\(124\) −1.41421 −0.127000
\(125\) −11.3137 −1.01193
\(126\) 0 0
\(127\) 12.0000 1.06483 0.532414 0.846484i \(-0.321285\pi\)
0.532414 + 0.846484i \(0.321285\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.65685 −0.498058
\(130\) −4.00000 −0.350823
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 1.41421 0.123091
\(133\) 0 0
\(134\) 2.00000 0.172774
\(135\) 8.00000 0.688530
\(136\) 5.65685 0.485071
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) −2.82843 −0.240772
\(139\) −2.82843 −0.239904 −0.119952 0.992780i \(-0.538274\pi\)
−0.119952 + 0.992780i \(0.538274\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 10.0000 0.839181
\(143\) −2.82843 −0.236525
\(144\) −1.00000 −0.0833333
\(145\) −8.48528 −0.704664
\(146\) 14.1421 1.17041
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) −4.24264 −0.346410
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) −2.82843 −0.229416
\(153\) 5.65685 0.457330
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) −4.00000 −0.320256
\(157\) 18.3848 1.46726 0.733632 0.679546i \(-0.237823\pi\)
0.733632 + 0.679546i \(0.237823\pi\)
\(158\) 8.00000 0.636446
\(159\) 16.9706 1.34585
\(160\) −1.41421 −0.111803
\(161\) 0 0
\(162\) 5.00000 0.392837
\(163\) −22.0000 −1.72317 −0.861586 0.507611i \(-0.830529\pi\)
−0.861586 + 0.507611i \(0.830529\pi\)
\(164\) 5.65685 0.441726
\(165\) 2.00000 0.155700
\(166\) 2.82843 0.219529
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) −5.00000 −0.384615
\(170\) 8.00000 0.613572
\(171\) −2.82843 −0.216295
\(172\) 4.00000 0.304997
\(173\) 14.1421 1.07521 0.537603 0.843198i \(-0.319330\pi\)
0.537603 + 0.843198i \(0.319330\pi\)
\(174\) −8.48528 −0.643268
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −6.00000 −0.450988
\(178\) 7.07107 0.529999
\(179\) 20.0000 1.49487 0.747435 0.664335i \(-0.231285\pi\)
0.747435 + 0.664335i \(0.231285\pi\)
\(180\) −1.41421 −0.105409
\(181\) 12.7279 0.946059 0.473029 0.881047i \(-0.343160\pi\)
0.473029 + 0.881047i \(0.343160\pi\)
\(182\) 0 0
\(183\) 8.00000 0.591377
\(184\) 2.00000 0.147442
\(185\) −2.82843 −0.207950
\(186\) −2.00000 −0.146647
\(187\) 5.65685 0.413670
\(188\) 4.24264 0.309426
\(189\) 0 0
\(190\) −4.00000 −0.290191
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −1.41421 −0.102062
\(193\) 6.00000 0.431889 0.215945 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) 15.5563 1.11688
\(195\) −5.65685 −0.405096
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 21.2132 1.50376 0.751882 0.659298i \(-0.229146\pi\)
0.751882 + 0.659298i \(0.229146\pi\)
\(200\) 3.00000 0.212132
\(201\) 2.82843 0.199502
\(202\) −2.82843 −0.199007
\(203\) 0 0
\(204\) 8.00000 0.560112
\(205\) 8.00000 0.558744
\(206\) 4.24264 0.295599
\(207\) 2.00000 0.139010
\(208\) 2.82843 0.196116
\(209\) −2.82843 −0.195646
\(210\) 0 0
\(211\) 28.0000 1.92760 0.963800 0.266627i \(-0.0859092\pi\)
0.963800 + 0.266627i \(0.0859092\pi\)
\(212\) −12.0000 −0.824163
\(213\) 14.1421 0.969003
\(214\) 20.0000 1.36717
\(215\) 5.65685 0.385794
\(216\) −5.65685 −0.384900
\(217\) 0 0
\(218\) −10.0000 −0.677285
\(219\) 20.0000 1.35147
\(220\) −1.41421 −0.0953463
\(221\) −16.0000 −1.07628
\(222\) −2.82843 −0.189832
\(223\) −15.5563 −1.04173 −0.520865 0.853639i \(-0.674391\pi\)
−0.520865 + 0.853639i \(0.674391\pi\)
\(224\) 0 0
\(225\) 3.00000 0.200000
\(226\) 18.0000 1.19734
\(227\) −28.2843 −1.87729 −0.938647 0.344881i \(-0.887919\pi\)
−0.938647 + 0.344881i \(0.887919\pi\)
\(228\) −4.00000 −0.264906
\(229\) −4.24264 −0.280362 −0.140181 0.990126i \(-0.544768\pi\)
−0.140181 + 0.990126i \(0.544768\pi\)
\(230\) 2.82843 0.186501
\(231\) 0 0
\(232\) 6.00000 0.393919
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 2.82843 0.184900
\(235\) 6.00000 0.391397
\(236\) 4.24264 0.276172
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −2.00000 −0.129099
\(241\) −5.65685 −0.364390 −0.182195 0.983262i \(-0.558320\pi\)
−0.182195 + 0.983262i \(0.558320\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −9.89949 −0.635053
\(244\) −5.65685 −0.362143
\(245\) 0 0
\(246\) 8.00000 0.510061
\(247\) 8.00000 0.509028
\(248\) 1.41421 0.0898027
\(249\) 4.00000 0.253490
\(250\) 11.3137 0.715542
\(251\) −24.0416 −1.51749 −0.758747 0.651385i \(-0.774188\pi\)
−0.758747 + 0.651385i \(0.774188\pi\)
\(252\) 0 0
\(253\) 2.00000 0.125739
\(254\) −12.0000 −0.752947
\(255\) 11.3137 0.708492
\(256\) 1.00000 0.0625000
\(257\) −9.89949 −0.617514 −0.308757 0.951141i \(-0.599913\pi\)
−0.308757 + 0.951141i \(0.599913\pi\)
\(258\) 5.65685 0.352180
\(259\) 0 0
\(260\) 4.00000 0.248069
\(261\) 6.00000 0.371391
\(262\) −11.3137 −0.698963
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) −1.41421 −0.0870388
\(265\) −16.9706 −1.04249
\(266\) 0 0
\(267\) 10.0000 0.611990
\(268\) −2.00000 −0.122169
\(269\) −24.0416 −1.46584 −0.732922 0.680313i \(-0.761844\pi\)
−0.732922 + 0.680313i \(0.761844\pi\)
\(270\) −8.00000 −0.486864
\(271\) 8.48528 0.515444 0.257722 0.966219i \(-0.417028\pi\)
0.257722 + 0.966219i \(0.417028\pi\)
\(272\) −5.65685 −0.342997
\(273\) 0 0
\(274\) 2.00000 0.120824
\(275\) 3.00000 0.180907
\(276\) 2.82843 0.170251
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 2.82843 0.169638
\(279\) 1.41421 0.0846668
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 6.00000 0.357295
\(283\) 16.9706 1.00880 0.504398 0.863472i \(-0.331715\pi\)
0.504398 + 0.863472i \(0.331715\pi\)
\(284\) −10.0000 −0.593391
\(285\) −5.65685 −0.335083
\(286\) 2.82843 0.167248
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 15.0000 0.882353
\(290\) 8.48528 0.498273
\(291\) 22.0000 1.28966
\(292\) −14.1421 −0.827606
\(293\) 16.9706 0.991431 0.495715 0.868485i \(-0.334906\pi\)
0.495715 + 0.868485i \(0.334906\pi\)
\(294\) 0 0
\(295\) 6.00000 0.349334
\(296\) 2.00000 0.116248
\(297\) −5.65685 −0.328244
\(298\) 10.0000 0.579284
\(299\) −5.65685 −0.327144
\(300\) 4.24264 0.244949
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) −4.00000 −0.229794
\(304\) 2.82843 0.162221
\(305\) −8.00000 −0.458079
\(306\) −5.65685 −0.323381
\(307\) −11.3137 −0.645707 −0.322854 0.946449i \(-0.604642\pi\)
−0.322854 + 0.946449i \(0.604642\pi\)
\(308\) 0 0
\(309\) 6.00000 0.341328
\(310\) 2.00000 0.113592
\(311\) −9.89949 −0.561349 −0.280674 0.959803i \(-0.590558\pi\)
−0.280674 + 0.959803i \(0.590558\pi\)
\(312\) 4.00000 0.226455
\(313\) 26.8701 1.51879 0.759393 0.650633i \(-0.225496\pi\)
0.759393 + 0.650633i \(0.225496\pi\)
\(314\) −18.3848 −1.03751
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −26.0000 −1.46031 −0.730153 0.683284i \(-0.760551\pi\)
−0.730153 + 0.683284i \(0.760551\pi\)
\(318\) −16.9706 −0.951662
\(319\) 6.00000 0.335936
\(320\) 1.41421 0.0790569
\(321\) 28.2843 1.57867
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) −5.00000 −0.277778
\(325\) −8.48528 −0.470679
\(326\) 22.0000 1.21847
\(327\) −14.1421 −0.782062
\(328\) −5.65685 −0.312348
\(329\) 0 0
\(330\) −2.00000 −0.110096
\(331\) 28.0000 1.53902 0.769510 0.638635i \(-0.220501\pi\)
0.769510 + 0.638635i \(0.220501\pi\)
\(332\) −2.82843 −0.155230
\(333\) 2.00000 0.109599
\(334\) 11.3137 0.619059
\(335\) −2.82843 −0.154533
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 5.00000 0.271964
\(339\) 25.4558 1.38257
\(340\) −8.00000 −0.433861
\(341\) 1.41421 0.0765840
\(342\) 2.82843 0.152944
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 4.00000 0.215353
\(346\) −14.1421 −0.760286
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 8.48528 0.454859
\(349\) −5.65685 −0.302804 −0.151402 0.988472i \(-0.548379\pi\)
−0.151402 + 0.988472i \(0.548379\pi\)
\(350\) 0 0
\(351\) 16.0000 0.854017
\(352\) 1.00000 0.0533002
\(353\) 1.41421 0.0752710 0.0376355 0.999292i \(-0.488017\pi\)
0.0376355 + 0.999292i \(0.488017\pi\)
\(354\) 6.00000 0.318896
\(355\) −14.1421 −0.750587
\(356\) −7.07107 −0.374766
\(357\) 0 0
\(358\) −20.0000 −1.05703
\(359\) 32.0000 1.68890 0.844448 0.535638i \(-0.179929\pi\)
0.844448 + 0.535638i \(0.179929\pi\)
\(360\) 1.41421 0.0745356
\(361\) −11.0000 −0.578947
\(362\) −12.7279 −0.668965
\(363\) −1.41421 −0.0742270
\(364\) 0 0
\(365\) −20.0000 −1.04685
\(366\) −8.00000 −0.418167
\(367\) 29.6985 1.55025 0.775124 0.631809i \(-0.217687\pi\)
0.775124 + 0.631809i \(0.217687\pi\)
\(368\) −2.00000 −0.104257
\(369\) −5.65685 −0.294484
\(370\) 2.82843 0.147043
\(371\) 0 0
\(372\) 2.00000 0.103695
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) −5.65685 −0.292509
\(375\) 16.0000 0.826236
\(376\) −4.24264 −0.218797
\(377\) −16.9706 −0.874028
\(378\) 0 0
\(379\) 6.00000 0.308199 0.154100 0.988055i \(-0.450752\pi\)
0.154100 + 0.988055i \(0.450752\pi\)
\(380\) 4.00000 0.205196
\(381\) −16.9706 −0.869428
\(382\) 0 0
\(383\) 26.8701 1.37300 0.686498 0.727132i \(-0.259147\pi\)
0.686498 + 0.727132i \(0.259147\pi\)
\(384\) 1.41421 0.0721688
\(385\) 0 0
\(386\) −6.00000 −0.305392
\(387\) −4.00000 −0.203331
\(388\) −15.5563 −0.789754
\(389\) −28.0000 −1.41966 −0.709828 0.704375i \(-0.751227\pi\)
−0.709828 + 0.704375i \(0.751227\pi\)
\(390\) 5.65685 0.286446
\(391\) 11.3137 0.572159
\(392\) 0 0
\(393\) −16.0000 −0.807093
\(394\) −18.0000 −0.906827
\(395\) −11.3137 −0.569254
\(396\) 1.00000 0.0502519
\(397\) −24.0416 −1.20661 −0.603307 0.797509i \(-0.706151\pi\)
−0.603307 + 0.797509i \(0.706151\pi\)
\(398\) −21.2132 −1.06332
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) −2.82843 −0.141069
\(403\) −4.00000 −0.199254
\(404\) 2.82843 0.140720
\(405\) −7.07107 −0.351364
\(406\) 0 0
\(407\) 2.00000 0.0991363
\(408\) −8.00000 −0.396059
\(409\) 31.1127 1.53842 0.769212 0.638994i \(-0.220649\pi\)
0.769212 + 0.638994i \(0.220649\pi\)
\(410\) −8.00000 −0.395092
\(411\) 2.82843 0.139516
\(412\) −4.24264 −0.209020
\(413\) 0 0
\(414\) −2.00000 −0.0982946
\(415\) −4.00000 −0.196352
\(416\) −2.82843 −0.138675
\(417\) 4.00000 0.195881
\(418\) 2.82843 0.138343
\(419\) −12.7279 −0.621800 −0.310900 0.950443i \(-0.600630\pi\)
−0.310900 + 0.950443i \(0.600630\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) −28.0000 −1.36302
\(423\) −4.24264 −0.206284
\(424\) 12.0000 0.582772
\(425\) 16.9706 0.823193
\(426\) −14.1421 −0.685189
\(427\) 0 0
\(428\) −20.0000 −0.966736
\(429\) 4.00000 0.193122
\(430\) −5.65685 −0.272798
\(431\) −20.0000 −0.963366 −0.481683 0.876346i \(-0.659974\pi\)
−0.481683 + 0.876346i \(0.659974\pi\)
\(432\) 5.65685 0.272166
\(433\) 38.1838 1.83499 0.917497 0.397742i \(-0.130206\pi\)
0.917497 + 0.397742i \(0.130206\pi\)
\(434\) 0 0
\(435\) 12.0000 0.575356
\(436\) 10.0000 0.478913
\(437\) −5.65685 −0.270604
\(438\) −20.0000 −0.955637
\(439\) −25.4558 −1.21494 −0.607471 0.794342i \(-0.707816\pi\)
−0.607471 + 0.794342i \(0.707816\pi\)
\(440\) 1.41421 0.0674200
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) −12.0000 −0.570137 −0.285069 0.958507i \(-0.592016\pi\)
−0.285069 + 0.958507i \(0.592016\pi\)
\(444\) 2.82843 0.134231
\(445\) −10.0000 −0.474045
\(446\) 15.5563 0.736614
\(447\) 14.1421 0.668900
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) −3.00000 −0.141421
\(451\) −5.65685 −0.266371
\(452\) −18.0000 −0.846649
\(453\) −22.6274 −1.06313
\(454\) 28.2843 1.32745
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 4.24264 0.198246
\(459\) −32.0000 −1.49363
\(460\) −2.82843 −0.131876
\(461\) 28.2843 1.31733 0.658665 0.752436i \(-0.271121\pi\)
0.658665 + 0.752436i \(0.271121\pi\)
\(462\) 0 0
\(463\) −14.0000 −0.650635 −0.325318 0.945605i \(-0.605471\pi\)
−0.325318 + 0.945605i \(0.605471\pi\)
\(464\) −6.00000 −0.278543
\(465\) 2.82843 0.131165
\(466\) −6.00000 −0.277945
\(467\) 4.24264 0.196326 0.0981630 0.995170i \(-0.468703\pi\)
0.0981630 + 0.995170i \(0.468703\pi\)
\(468\) −2.82843 −0.130744
\(469\) 0 0
\(470\) −6.00000 −0.276759
\(471\) −26.0000 −1.19802
\(472\) −4.24264 −0.195283
\(473\) −4.00000 −0.183920
\(474\) −11.3137 −0.519656
\(475\) −8.48528 −0.389331
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 16.0000 0.731823
\(479\) 25.4558 1.16311 0.581554 0.813508i \(-0.302445\pi\)
0.581554 + 0.813508i \(0.302445\pi\)
\(480\) 2.00000 0.0912871
\(481\) −5.65685 −0.257930
\(482\) 5.65685 0.257663
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −22.0000 −0.998969
\(486\) 9.89949 0.449050
\(487\) 10.0000 0.453143 0.226572 0.973995i \(-0.427248\pi\)
0.226572 + 0.973995i \(0.427248\pi\)
\(488\) 5.65685 0.256074
\(489\) 31.1127 1.40696
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) −8.00000 −0.360668
\(493\) 33.9411 1.52863
\(494\) −8.00000 −0.359937
\(495\) 1.41421 0.0635642
\(496\) −1.41421 −0.0635001
\(497\) 0 0
\(498\) −4.00000 −0.179244
\(499\) −30.0000 −1.34298 −0.671492 0.741012i \(-0.734346\pi\)
−0.671492 + 0.741012i \(0.734346\pi\)
\(500\) −11.3137 −0.505964
\(501\) 16.0000 0.714827
\(502\) 24.0416 1.07303
\(503\) 19.7990 0.882793 0.441397 0.897312i \(-0.354483\pi\)
0.441397 + 0.897312i \(0.354483\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) −2.00000 −0.0889108
\(507\) 7.07107 0.314037
\(508\) 12.0000 0.532414
\(509\) 4.24264 0.188052 0.0940259 0.995570i \(-0.470026\pi\)
0.0940259 + 0.995570i \(0.470026\pi\)
\(510\) −11.3137 −0.500979
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 16.0000 0.706417
\(514\) 9.89949 0.436648
\(515\) −6.00000 −0.264392
\(516\) −5.65685 −0.249029
\(517\) −4.24264 −0.186591
\(518\) 0 0
\(519\) −20.0000 −0.877903
\(520\) −4.00000 −0.175412
\(521\) −15.5563 −0.681536 −0.340768 0.940147i \(-0.610687\pi\)
−0.340768 + 0.940147i \(0.610687\pi\)
\(522\) −6.00000 −0.262613
\(523\) −16.9706 −0.742071 −0.371035 0.928619i \(-0.620997\pi\)
−0.371035 + 0.928619i \(0.620997\pi\)
\(524\) 11.3137 0.494242
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) 8.00000 0.348485
\(528\) 1.41421 0.0615457
\(529\) −19.0000 −0.826087
\(530\) 16.9706 0.737154
\(531\) −4.24264 −0.184115
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) −10.0000 −0.432742
\(535\) −28.2843 −1.22284
\(536\) 2.00000 0.0863868
\(537\) −28.2843 −1.22056
\(538\) 24.0416 1.03651
\(539\) 0 0
\(540\) 8.00000 0.344265
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) −8.48528 −0.364474
\(543\) −18.0000 −0.772454
\(544\) 5.65685 0.242536
\(545\) 14.1421 0.605783
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 5.65685 0.241429
\(550\) −3.00000 −0.127920
\(551\) −16.9706 −0.722970
\(552\) −2.82843 −0.120386
\(553\) 0 0
\(554\) 10.0000 0.424859
\(555\) 4.00000 0.169791
\(556\) −2.82843 −0.119952
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) −1.41421 −0.0598684
\(559\) 11.3137 0.478519
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) −18.0000 −0.759284
\(563\) −14.1421 −0.596020 −0.298010 0.954563i \(-0.596323\pi\)
−0.298010 + 0.954563i \(0.596323\pi\)
\(564\) −6.00000 −0.252646
\(565\) −25.4558 −1.07094
\(566\) −16.9706 −0.713326
\(567\) 0 0
\(568\) 10.0000 0.419591
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 5.65685 0.236940
\(571\) 8.00000 0.334790 0.167395 0.985890i \(-0.446465\pi\)
0.167395 + 0.985890i \(0.446465\pi\)
\(572\) −2.82843 −0.118262
\(573\) 0 0
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) −1.00000 −0.0416667
\(577\) −29.6985 −1.23636 −0.618182 0.786035i \(-0.712131\pi\)
−0.618182 + 0.786035i \(0.712131\pi\)
\(578\) −15.0000 −0.623918
\(579\) −8.48528 −0.352636
\(580\) −8.48528 −0.352332
\(581\) 0 0
\(582\) −22.0000 −0.911929
\(583\) 12.0000 0.496989
\(584\) 14.1421 0.585206
\(585\) −4.00000 −0.165380
\(586\) −16.9706 −0.701047
\(587\) −1.41421 −0.0583708 −0.0291854 0.999574i \(-0.509291\pi\)
−0.0291854 + 0.999574i \(0.509291\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) −6.00000 −0.247016
\(591\) −25.4558 −1.04711
\(592\) −2.00000 −0.0821995
\(593\) 19.7990 0.813047 0.406524 0.913640i \(-0.366741\pi\)
0.406524 + 0.913640i \(0.366741\pi\)
\(594\) 5.65685 0.232104
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) −30.0000 −1.22782
\(598\) 5.65685 0.231326
\(599\) 18.0000 0.735460 0.367730 0.929933i \(-0.380135\pi\)
0.367730 + 0.929933i \(0.380135\pi\)
\(600\) −4.24264 −0.173205
\(601\) 45.2548 1.84598 0.922992 0.384820i \(-0.125737\pi\)
0.922992 + 0.384820i \(0.125737\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 16.0000 0.651031
\(605\) 1.41421 0.0574960
\(606\) 4.00000 0.162489
\(607\) −33.9411 −1.37763 −0.688814 0.724938i \(-0.741868\pi\)
−0.688814 + 0.724938i \(0.741868\pi\)
\(608\) −2.82843 −0.114708
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 12.0000 0.485468
\(612\) 5.65685 0.228665
\(613\) −34.0000 −1.37325 −0.686624 0.727013i \(-0.740908\pi\)
−0.686624 + 0.727013i \(0.740908\pi\)
\(614\) 11.3137 0.456584
\(615\) −11.3137 −0.456213
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) −6.00000 −0.241355
\(619\) 12.7279 0.511578 0.255789 0.966733i \(-0.417665\pi\)
0.255789 + 0.966733i \(0.417665\pi\)
\(620\) −2.00000 −0.0803219
\(621\) −11.3137 −0.454003
\(622\) 9.89949 0.396934
\(623\) 0 0
\(624\) −4.00000 −0.160128
\(625\) −1.00000 −0.0400000
\(626\) −26.8701 −1.07394
\(627\) 4.00000 0.159745
\(628\) 18.3848 0.733632
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 8.00000 0.318223
\(633\) −39.5980 −1.57388
\(634\) 26.0000 1.03259
\(635\) 16.9706 0.673456
\(636\) 16.9706 0.672927
\(637\) 0 0
\(638\) −6.00000 −0.237542
\(639\) 10.0000 0.395594
\(640\) −1.41421 −0.0559017
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) −28.2843 −1.11629
\(643\) −12.7279 −0.501940 −0.250970 0.967995i \(-0.580750\pi\)
−0.250970 + 0.967995i \(0.580750\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 16.0000 0.629512
\(647\) 41.0122 1.61236 0.806178 0.591673i \(-0.201532\pi\)
0.806178 + 0.591673i \(0.201532\pi\)
\(648\) 5.00000 0.196419
\(649\) −4.24264 −0.166538
\(650\) 8.48528 0.332820
\(651\) 0 0
\(652\) −22.0000 −0.861586
\(653\) −48.0000 −1.87839 −0.939193 0.343391i \(-0.888424\pi\)
−0.939193 + 0.343391i \(0.888424\pi\)
\(654\) 14.1421 0.553001
\(655\) 16.0000 0.625172
\(656\) 5.65685 0.220863
\(657\) 14.1421 0.551737
\(658\) 0 0
\(659\) −4.00000 −0.155818 −0.0779089 0.996960i \(-0.524824\pi\)
−0.0779089 + 0.996960i \(0.524824\pi\)
\(660\) 2.00000 0.0778499
\(661\) 38.1838 1.48518 0.742588 0.669748i \(-0.233598\pi\)
0.742588 + 0.669748i \(0.233598\pi\)
\(662\) −28.0000 −1.08825
\(663\) 22.6274 0.878776
\(664\) 2.82843 0.109764
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 12.0000 0.464642
\(668\) −11.3137 −0.437741
\(669\) 22.0000 0.850569
\(670\) 2.82843 0.109272
\(671\) 5.65685 0.218380
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 2.00000 0.0770371
\(675\) −16.9706 −0.653197
\(676\) −5.00000 −0.192308
\(677\) 31.1127 1.19576 0.597879 0.801586i \(-0.296010\pi\)
0.597879 + 0.801586i \(0.296010\pi\)
\(678\) −25.4558 −0.977626
\(679\) 0 0
\(680\) 8.00000 0.306786
\(681\) 40.0000 1.53280
\(682\) −1.41421 −0.0541530
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) −2.82843 −0.108148
\(685\) −2.82843 −0.108069
\(686\) 0 0
\(687\) 6.00000 0.228914
\(688\) 4.00000 0.152499
\(689\) −33.9411 −1.29305
\(690\) −4.00000 −0.152277
\(691\) −32.5269 −1.23738 −0.618691 0.785634i \(-0.712337\pi\)
−0.618691 + 0.785634i \(0.712337\pi\)
\(692\) 14.1421 0.537603
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) −8.48528 −0.321634
\(697\) −32.0000 −1.21209
\(698\) 5.65685 0.214115
\(699\) −8.48528 −0.320943
\(700\) 0 0
\(701\) −10.0000 −0.377695 −0.188847 0.982006i \(-0.560475\pi\)
−0.188847 + 0.982006i \(0.560475\pi\)
\(702\) −16.0000 −0.603881
\(703\) −5.65685 −0.213352
\(704\) −1.00000 −0.0376889
\(705\) −8.48528 −0.319574
\(706\) −1.41421 −0.0532246
\(707\) 0 0
\(708\) −6.00000 −0.225494
\(709\) −8.00000 −0.300446 −0.150223 0.988652i \(-0.547999\pi\)
−0.150223 + 0.988652i \(0.547999\pi\)
\(710\) 14.1421 0.530745
\(711\) 8.00000 0.300023
\(712\) 7.07107 0.264999
\(713\) 2.82843 0.105925
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 20.0000 0.747435
\(717\) 22.6274 0.845036
\(718\) −32.0000 −1.19423
\(719\) −15.5563 −0.580154 −0.290077 0.957003i \(-0.593681\pi\)
−0.290077 + 0.957003i \(0.593681\pi\)
\(720\) −1.41421 −0.0527046
\(721\) 0 0
\(722\) 11.0000 0.409378
\(723\) 8.00000 0.297523
\(724\) 12.7279 0.473029
\(725\) 18.0000 0.668503
\(726\) 1.41421 0.0524864
\(727\) −32.5269 −1.20636 −0.603178 0.797606i \(-0.706099\pi\)
−0.603178 + 0.797606i \(0.706099\pi\)
\(728\) 0 0
\(729\) 29.0000 1.07407
\(730\) 20.0000 0.740233
\(731\) −22.6274 −0.836905
\(732\) 8.00000 0.295689
\(733\) −16.9706 −0.626822 −0.313411 0.949618i \(-0.601472\pi\)
−0.313411 + 0.949618i \(0.601472\pi\)
\(734\) −29.6985 −1.09619
\(735\) 0 0
\(736\) 2.00000 0.0737210
\(737\) 2.00000 0.0736709
\(738\) 5.65685 0.208232
\(739\) −24.0000 −0.882854 −0.441427 0.897297i \(-0.645528\pi\)
−0.441427 + 0.897297i \(0.645528\pi\)
\(740\) −2.82843 −0.103975
\(741\) −11.3137 −0.415619
\(742\) 0 0
\(743\) −48.0000 −1.76095 −0.880475 0.474093i \(-0.842776\pi\)
−0.880475 + 0.474093i \(0.842776\pi\)
\(744\) −2.00000 −0.0733236
\(745\) −14.1421 −0.518128
\(746\) 14.0000 0.512576
\(747\) 2.82843 0.103487
\(748\) 5.65685 0.206835
\(749\) 0 0
\(750\) −16.0000 −0.584237
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) 4.24264 0.154713
\(753\) 34.0000 1.23903
\(754\) 16.9706 0.618031
\(755\) 22.6274 0.823496
\(756\) 0 0
\(757\) −44.0000 −1.59921 −0.799604 0.600528i \(-0.794957\pi\)
−0.799604 + 0.600528i \(0.794957\pi\)
\(758\) −6.00000 −0.217930
\(759\) −2.82843 −0.102665
\(760\) −4.00000 −0.145095
\(761\) 25.4558 0.922774 0.461387 0.887199i \(-0.347352\pi\)
0.461387 + 0.887199i \(0.347352\pi\)
\(762\) 16.9706 0.614779
\(763\) 0 0
\(764\) 0 0
\(765\) 8.00000 0.289241
\(766\) −26.8701 −0.970855
\(767\) 12.0000 0.433295
\(768\) −1.41421 −0.0510310
\(769\) 8.48528 0.305987 0.152994 0.988227i \(-0.451109\pi\)
0.152994 + 0.988227i \(0.451109\pi\)
\(770\) 0 0
\(771\) 14.0000 0.504198
\(772\) 6.00000 0.215945
\(773\) −1.41421 −0.0508657 −0.0254329 0.999677i \(-0.508096\pi\)
−0.0254329 + 0.999677i \(0.508096\pi\)
\(774\) 4.00000 0.143777
\(775\) 4.24264 0.152400
\(776\) 15.5563 0.558440
\(777\) 0 0
\(778\) 28.0000 1.00385
\(779\) 16.0000 0.573259
\(780\) −5.65685 −0.202548
\(781\) 10.0000 0.357828
\(782\) −11.3137 −0.404577
\(783\) −33.9411 −1.21296
\(784\) 0 0
\(785\) 26.0000 0.927980
\(786\) 16.0000 0.570701
\(787\) 42.4264 1.51234 0.756169 0.654376i \(-0.227069\pi\)
0.756169 + 0.654376i \(0.227069\pi\)
\(788\) 18.0000 0.641223
\(789\) −33.9411 −1.20834
\(790\) 11.3137 0.402524
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) −16.0000 −0.568177
\(794\) 24.0416 0.853206
\(795\) 24.0000 0.851192
\(796\) 21.2132 0.751882
\(797\) −18.3848 −0.651222 −0.325611 0.945504i \(-0.605570\pi\)
−0.325611 + 0.945504i \(0.605570\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 3.00000 0.106066
\(801\) 7.07107 0.249844
\(802\) −36.0000 −1.27120
\(803\) 14.1421 0.499065
\(804\) 2.82843 0.0997509
\(805\) 0 0
\(806\) 4.00000 0.140894
\(807\) 34.0000 1.19686
\(808\) −2.82843 −0.0995037
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 7.07107 0.248452
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) −12.0000 −0.420858
\(814\) −2.00000 −0.0701000
\(815\) −31.1127 −1.08983
\(816\) 8.00000 0.280056
\(817\) 11.3137 0.395817
\(818\) −31.1127 −1.08783
\(819\) 0 0
\(820\) 8.00000 0.279372
\(821\) 34.0000 1.18661 0.593304 0.804978i \(-0.297823\pi\)
0.593304 + 0.804978i \(0.297823\pi\)
\(822\) −2.82843 −0.0986527
\(823\) −32.0000 −1.11545 −0.557725 0.830026i \(-0.688326\pi\)
−0.557725 + 0.830026i \(0.688326\pi\)
\(824\) 4.24264 0.147799
\(825\) −4.24264 −0.147710
\(826\) 0 0
\(827\) −8.00000 −0.278187 −0.139094 0.990279i \(-0.544419\pi\)
−0.139094 + 0.990279i \(0.544419\pi\)
\(828\) 2.00000 0.0695048
\(829\) −35.3553 −1.22794 −0.613971 0.789329i \(-0.710429\pi\)
−0.613971 + 0.789329i \(0.710429\pi\)
\(830\) 4.00000 0.138842
\(831\) 14.1421 0.490585
\(832\) 2.82843 0.0980581
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) −16.0000 −0.553703
\(836\) −2.82843 −0.0978232
\(837\) −8.00000 −0.276520
\(838\) 12.7279 0.439679
\(839\) −43.8406 −1.51355 −0.756773 0.653678i \(-0.773225\pi\)
−0.756773 + 0.653678i \(0.773225\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −8.00000 −0.275698
\(843\) −25.4558 −0.876746
\(844\) 28.0000 0.963800
\(845\) −7.07107 −0.243252
\(846\) 4.24264 0.145865
\(847\) 0 0
\(848\) −12.0000 −0.412082
\(849\) −24.0000 −0.823678
\(850\) −16.9706 −0.582086
\(851\) 4.00000 0.137118
\(852\) 14.1421 0.484502
\(853\) −2.82843 −0.0968435 −0.0484218 0.998827i \(-0.515419\pi\)
−0.0484218 + 0.998827i \(0.515419\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 20.0000 0.683586
\(857\) −8.48528 −0.289852 −0.144926 0.989443i \(-0.546294\pi\)
−0.144926 + 0.989443i \(0.546294\pi\)
\(858\) −4.00000 −0.136558
\(859\) 38.1838 1.30281 0.651407 0.758729i \(-0.274179\pi\)
0.651407 + 0.758729i \(0.274179\pi\)
\(860\) 5.65685 0.192897
\(861\) 0 0
\(862\) 20.0000 0.681203
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) −5.65685 −0.192450
\(865\) 20.0000 0.680020
\(866\) −38.1838 −1.29754
\(867\) −21.2132 −0.720438
\(868\) 0 0
\(869\) 8.00000 0.271381
\(870\) −12.0000 −0.406838
\(871\) −5.65685 −0.191675
\(872\) −10.0000 −0.338643
\(873\) 15.5563 0.526503
\(874\) 5.65685 0.191346
\(875\) 0 0
\(876\) 20.0000 0.675737
\(877\) −10.0000 −0.337676 −0.168838 0.985644i \(-0.554001\pi\)
−0.168838 + 0.985644i \(0.554001\pi\)
\(878\) 25.4558 0.859093
\(879\) −24.0000 −0.809500
\(880\) −1.41421 −0.0476731
\(881\) 4.24264 0.142938 0.0714691 0.997443i \(-0.477231\pi\)
0.0714691 + 0.997443i \(0.477231\pi\)
\(882\) 0 0
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) −16.0000 −0.538138
\(885\) −8.48528 −0.285230
\(886\) 12.0000 0.403148
\(887\) 48.0833 1.61448 0.807239 0.590225i \(-0.200961\pi\)
0.807239 + 0.590225i \(0.200961\pi\)
\(888\) −2.82843 −0.0949158
\(889\) 0 0
\(890\) 10.0000 0.335201
\(891\) 5.00000 0.167506
\(892\) −15.5563 −0.520865
\(893\) 12.0000 0.401565
\(894\) −14.1421 −0.472984
\(895\) 28.2843 0.945439
\(896\) 0 0
\(897\) 8.00000 0.267112
\(898\) 12.0000 0.400445
\(899\) 8.48528 0.283000
\(900\) 3.00000 0.100000
\(901\) 67.8823 2.26149
\(902\) 5.65685 0.188353
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 18.0000 0.598340
\(906\) 22.6274 0.751746
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) −28.2843 −0.938647
\(909\) −2.82843 −0.0938130
\(910\) 0 0
\(911\) 30.0000 0.993944 0.496972 0.867766i \(-0.334445\pi\)
0.496972 + 0.867766i \(0.334445\pi\)
\(912\) −4.00000 −0.132453
\(913\) 2.82843 0.0936073
\(914\) −2.00000 −0.0661541
\(915\) 11.3137 0.374020
\(916\) −4.24264 −0.140181
\(917\) 0 0
\(918\) 32.0000 1.05616
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 2.82843 0.0932505
\(921\) 16.0000 0.527218
\(922\) −28.2843 −0.931493
\(923\) −28.2843 −0.930988
\(924\) 0 0
\(925\) 6.00000 0.197279
\(926\) 14.0000 0.460069
\(927\) 4.24264 0.139347
\(928\) 6.00000 0.196960
\(929\) −15.5563 −0.510387 −0.255194 0.966890i \(-0.582139\pi\)
−0.255194 + 0.966890i \(0.582139\pi\)
\(930\) −2.82843 −0.0927478
\(931\) 0 0
\(932\) 6.00000 0.196537
\(933\) 14.0000 0.458339
\(934\) −4.24264 −0.138823
\(935\) 8.00000 0.261628
\(936\) 2.82843 0.0924500
\(937\) −25.4558 −0.831606 −0.415803 0.909455i \(-0.636499\pi\)
−0.415803 + 0.909455i \(0.636499\pi\)
\(938\) 0 0
\(939\) −38.0000 −1.24008
\(940\) 6.00000 0.195698
\(941\) 22.6274 0.737633 0.368816 0.929502i \(-0.379763\pi\)
0.368816 + 0.929502i \(0.379763\pi\)
\(942\) 26.0000 0.847126
\(943\) −11.3137 −0.368425
\(944\) 4.24264 0.138086
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 11.3137 0.367452
\(949\) −40.0000 −1.29845
\(950\) 8.48528 0.275299
\(951\) 36.7696 1.19233
\(952\) 0 0
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) −12.0000 −0.388514
\(955\) 0 0
\(956\) −16.0000 −0.517477
\(957\) −8.48528 −0.274290
\(958\) −25.4558 −0.822441
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −29.0000 −0.935484
\(962\) 5.65685 0.182384
\(963\) 20.0000 0.644491
\(964\) −5.65685 −0.182195
\(965\) 8.48528 0.273151
\(966\) 0 0
\(967\) 28.0000 0.900419 0.450210 0.892923i \(-0.351349\pi\)
0.450210 + 0.892923i \(0.351349\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 22.6274 0.726897
\(970\) 22.0000 0.706377
\(971\) 26.8701 0.862301 0.431151 0.902280i \(-0.358108\pi\)
0.431151 + 0.902280i \(0.358108\pi\)
\(972\) −9.89949 −0.317526
\(973\) 0 0
\(974\) −10.0000 −0.320421
\(975\) 12.0000 0.384308
\(976\) −5.65685 −0.181071
\(977\) 28.0000 0.895799 0.447900 0.894084i \(-0.352172\pi\)
0.447900 + 0.894084i \(0.352172\pi\)
\(978\) −31.1127 −0.994874
\(979\) 7.07107 0.225992
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) −4.00000 −0.127645
\(983\) 57.9828 1.84936 0.924681 0.380742i \(-0.124331\pi\)
0.924681 + 0.380742i \(0.124331\pi\)
\(984\) 8.00000 0.255031
\(985\) 25.4558 0.811091
\(986\) −33.9411 −1.08091
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) −8.00000 −0.254385
\(990\) −1.41421 −0.0449467
\(991\) 50.0000 1.58830 0.794151 0.607720i \(-0.207916\pi\)
0.794151 + 0.607720i \(0.207916\pi\)
\(992\) 1.41421 0.0449013
\(993\) −39.5980 −1.25660
\(994\) 0 0
\(995\) 30.0000 0.951064
\(996\) 4.00000 0.126745
\(997\) 59.3970 1.88112 0.940560 0.339626i \(-0.110301\pi\)
0.940560 + 0.339626i \(0.110301\pi\)
\(998\) 30.0000 0.949633
\(999\) −11.3137 −0.357950
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.q.1.1 2
3.2 odd 2 9702.2.a.dq.1.1 2
4.3 odd 2 8624.2.a.bw.1.2 2
7.2 even 3 1078.2.e.s.67.2 4
7.3 odd 6 1078.2.e.s.177.1 4
7.4 even 3 1078.2.e.s.177.2 4
7.5 odd 6 1078.2.e.s.67.1 4
7.6 odd 2 inner 1078.2.a.q.1.2 yes 2
21.20 even 2 9702.2.a.dq.1.2 2
28.27 even 2 8624.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1078.2.a.q.1.1 2 1.1 even 1 trivial
1078.2.a.q.1.2 yes 2 7.6 odd 2 inner
1078.2.e.s.67.1 4 7.5 odd 6
1078.2.e.s.67.2 4 7.2 even 3
1078.2.e.s.177.1 4 7.3 odd 6
1078.2.e.s.177.2 4 7.4 even 3
8624.2.a.bw.1.1 2 28.27 even 2
8624.2.a.bw.1.2 2 4.3 odd 2
9702.2.a.dq.1.1 2 3.2 odd 2
9702.2.a.dq.1.2 2 21.20 even 2