Properties

Label 6034.2.a.i.1.2
Level $6034$
Weight $2$
Character 6034.1
Self dual yes
Analytic conductor $48.182$
Analytic rank $0$
Dimension $2$
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] = [6034,2,Mod(1,6034)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6034, 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("6034.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6034 = 2 \cdot 7 \cdot 431 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6034.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: \(48.1817325796\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6034.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +2.00000 q^{3} +1.00000 q^{4} +2.00000 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +5.46410 q^{11} +2.00000 q^{12} -4.73205 q^{13} +1.00000 q^{14} +1.00000 q^{16} +3.46410 q^{17} +1.00000 q^{18} +0.535898 q^{19} +2.00000 q^{21} +5.46410 q^{22} +6.00000 q^{23} +2.00000 q^{24} -5.00000 q^{25} -4.73205 q^{26} -4.00000 q^{27} +1.00000 q^{28} +7.46410 q^{29} -1.46410 q^{31} +1.00000 q^{32} +10.9282 q^{33} +3.46410 q^{34} +1.00000 q^{36} +5.26795 q^{37} +0.535898 q^{38} -9.46410 q^{39} -6.92820 q^{41} +2.00000 q^{42} +11.6603 q^{43} +5.46410 q^{44} +6.00000 q^{46} -8.39230 q^{47} +2.00000 q^{48} +1.00000 q^{49} -5.00000 q^{50} +6.92820 q^{51} -4.73205 q^{52} +2.00000 q^{53} -4.00000 q^{54} +1.00000 q^{56} +1.07180 q^{57} +7.46410 q^{58} -3.46410 q^{59} -1.46410 q^{61} -1.46410 q^{62} +1.00000 q^{63} +1.00000 q^{64} +10.9282 q^{66} +0.732051 q^{67} +3.46410 q^{68} +12.0000 q^{69} -2.92820 q^{71} +1.00000 q^{72} +2.00000 q^{73} +5.26795 q^{74} -10.0000 q^{75} +0.535898 q^{76} +5.46410 q^{77} -9.46410 q^{78} +12.0000 q^{79} -11.0000 q^{81} -6.92820 q^{82} +5.66025 q^{83} +2.00000 q^{84} +11.6603 q^{86} +14.9282 q^{87} +5.46410 q^{88} -2.00000 q^{89} -4.73205 q^{91} +6.00000 q^{92} -2.92820 q^{93} -8.39230 q^{94} +2.00000 q^{96} -8.00000 q^{97} +1.00000 q^{98} +5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 4 q^{3} + 2 q^{4} + 4 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{11} + 4 q^{12} - 6 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{18} + 8 q^{19} + 4 q^{21} + 4 q^{22} + 12 q^{23} + 4 q^{24} - 10 q^{25} - 6 q^{26} - 8 q^{27} + 2 q^{28} + 8 q^{29} + 4 q^{31} + 2 q^{32} + 8 q^{33} + 2 q^{36} + 14 q^{37} + 8 q^{38} - 12 q^{39} + 4 q^{42} + 6 q^{43} + 4 q^{44} + 12 q^{46} + 4 q^{47} + 4 q^{48} + 2 q^{49} - 10 q^{50} - 6 q^{52} + 4 q^{53} - 8 q^{54} + 2 q^{56} + 16 q^{57} + 8 q^{58} + 4 q^{61} + 4 q^{62} + 2 q^{63} + 2 q^{64} + 8 q^{66} - 2 q^{67} + 24 q^{69} + 8 q^{71} + 2 q^{72} + 4 q^{73} + 14 q^{74} - 20 q^{75} + 8 q^{76} + 4 q^{77} - 12 q^{78} + 24 q^{79} - 22 q^{81} - 6 q^{83} + 4 q^{84} + 6 q^{86} + 16 q^{87} + 4 q^{88} - 4 q^{89} - 6 q^{91} + 12 q^{92} + 8 q^{93} + 4 q^{94} + 4 q^{96} - 16 q^{97} + 2 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 2.00000 0.816497
\(7\) 1.00000 0.377964
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.46410 1.64749 0.823744 0.566961i \(-0.191881\pi\)
0.823744 + 0.566961i \(0.191881\pi\)
\(12\) 2.00000 0.577350
\(13\) −4.73205 −1.31243 −0.656217 0.754572i \(-0.727845\pi\)
−0.656217 + 0.754572i \(0.727845\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.46410 0.840168 0.420084 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.535898 0.122944 0.0614718 0.998109i \(-0.480421\pi\)
0.0614718 + 0.998109i \(0.480421\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 5.46410 1.16495
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 2.00000 0.408248
\(25\) −5.00000 −1.00000
\(26\) −4.73205 −0.928032
\(27\) −4.00000 −0.769800
\(28\) 1.00000 0.188982
\(29\) 7.46410 1.38605 0.693024 0.720914i \(-0.256278\pi\)
0.693024 + 0.720914i \(0.256278\pi\)
\(30\) 0 0
\(31\) −1.46410 −0.262960 −0.131480 0.991319i \(-0.541973\pi\)
−0.131480 + 0.991319i \(0.541973\pi\)
\(32\) 1.00000 0.176777
\(33\) 10.9282 1.90236
\(34\) 3.46410 0.594089
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.26795 0.866046 0.433023 0.901383i \(-0.357447\pi\)
0.433023 + 0.901383i \(0.357447\pi\)
\(38\) 0.535898 0.0869342
\(39\) −9.46410 −1.51547
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 2.00000 0.308607
\(43\) 11.6603 1.77817 0.889086 0.457740i \(-0.151341\pi\)
0.889086 + 0.457740i \(0.151341\pi\)
\(44\) 5.46410 0.823744
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) −8.39230 −1.22414 −0.612072 0.790802i \(-0.709664\pi\)
−0.612072 + 0.790802i \(0.709664\pi\)
\(48\) 2.00000 0.288675
\(49\) 1.00000 0.142857
\(50\) −5.00000 −0.707107
\(51\) 6.92820 0.970143
\(52\) −4.73205 −0.656217
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) −4.00000 −0.544331
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) 1.07180 0.141963
\(58\) 7.46410 0.980085
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) −1.46410 −0.187459 −0.0937295 0.995598i \(-0.529879\pi\)
−0.0937295 + 0.995598i \(0.529879\pi\)
\(62\) −1.46410 −0.185941
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.9282 1.34517
\(67\) 0.732051 0.0894342 0.0447171 0.999000i \(-0.485761\pi\)
0.0447171 + 0.999000i \(0.485761\pi\)
\(68\) 3.46410 0.420084
\(69\) 12.0000 1.44463
\(70\) 0 0
\(71\) −2.92820 −0.347514 −0.173757 0.984789i \(-0.555591\pi\)
−0.173757 + 0.984789i \(0.555591\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 5.26795 0.612387
\(75\) −10.0000 −1.15470
\(76\) 0.535898 0.0614718
\(77\) 5.46410 0.622692
\(78\) −9.46410 −1.07160
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) −6.92820 −0.765092
\(83\) 5.66025 0.621294 0.310647 0.950525i \(-0.399454\pi\)
0.310647 + 0.950525i \(0.399454\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) 11.6603 1.25736
\(87\) 14.9282 1.60047
\(88\) 5.46410 0.582475
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) −4.73205 −0.496054
\(92\) 6.00000 0.625543
\(93\) −2.92820 −0.303641
\(94\) −8.39230 −0.865600
\(95\) 0 0
\(96\) 2.00000 0.204124
\(97\) −8.00000 −0.812277 −0.406138 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 1.00000 0.101015
\(99\) 5.46410 0.549163
\(100\) −5.00000 −0.500000
\(101\) −4.73205 −0.470857 −0.235428 0.971892i \(-0.575649\pi\)
−0.235428 + 0.971892i \(0.575649\pi\)
\(102\) 6.92820 0.685994
\(103\) 2.53590 0.249869 0.124935 0.992165i \(-0.460128\pi\)
0.124935 + 0.992165i \(0.460128\pi\)
\(104\) −4.73205 −0.464016
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −2.19615 −0.212310 −0.106155 0.994350i \(-0.533854\pi\)
−0.106155 + 0.994350i \(0.533854\pi\)
\(108\) −4.00000 −0.384900
\(109\) 4.92820 0.472036 0.236018 0.971749i \(-0.424158\pi\)
0.236018 + 0.971749i \(0.424158\pi\)
\(110\) 0 0
\(111\) 10.5359 1.00002
\(112\) 1.00000 0.0944911
\(113\) −4.53590 −0.426701 −0.213351 0.976976i \(-0.568438\pi\)
−0.213351 + 0.976976i \(0.568438\pi\)
\(114\) 1.07180 0.100383
\(115\) 0 0
\(116\) 7.46410 0.693024
\(117\) −4.73205 −0.437478
\(118\) −3.46410 −0.318896
\(119\) 3.46410 0.317554
\(120\) 0 0
\(121\) 18.8564 1.71422
\(122\) −1.46410 −0.132554
\(123\) −13.8564 −1.24939
\(124\) −1.46410 −0.131480
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) 6.92820 0.614779 0.307389 0.951584i \(-0.400545\pi\)
0.307389 + 0.951584i \(0.400545\pi\)
\(128\) 1.00000 0.0883883
\(129\) 23.3205 2.05326
\(130\) 0 0
\(131\) 1.26795 0.110781 0.0553906 0.998465i \(-0.482360\pi\)
0.0553906 + 0.998465i \(0.482360\pi\)
\(132\) 10.9282 0.951178
\(133\) 0.535898 0.0464683
\(134\) 0.732051 0.0632396
\(135\) 0 0
\(136\) 3.46410 0.297044
\(137\) 10.3923 0.887875 0.443937 0.896058i \(-0.353581\pi\)
0.443937 + 0.896058i \(0.353581\pi\)
\(138\) 12.0000 1.02151
\(139\) 18.3923 1.56001 0.780007 0.625770i \(-0.215215\pi\)
0.780007 + 0.625770i \(0.215215\pi\)
\(140\) 0 0
\(141\) −16.7846 −1.41352
\(142\) −2.92820 −0.245729
\(143\) −25.8564 −2.16222
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.00000 0.165521
\(147\) 2.00000 0.164957
\(148\) 5.26795 0.433023
\(149\) −19.8564 −1.62670 −0.813350 0.581775i \(-0.802359\pi\)
−0.813350 + 0.581775i \(0.802359\pi\)
\(150\) −10.0000 −0.816497
\(151\) −4.92820 −0.401051 −0.200526 0.979688i \(-0.564265\pi\)
−0.200526 + 0.979688i \(0.564265\pi\)
\(152\) 0.535898 0.0434671
\(153\) 3.46410 0.280056
\(154\) 5.46410 0.440310
\(155\) 0 0
\(156\) −9.46410 −0.757735
\(157\) −3.60770 −0.287925 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(158\) 12.0000 0.954669
\(159\) 4.00000 0.317221
\(160\) 0 0
\(161\) 6.00000 0.472866
\(162\) −11.0000 −0.864242
\(163\) 6.53590 0.511931 0.255966 0.966686i \(-0.417607\pi\)
0.255966 + 0.966686i \(0.417607\pi\)
\(164\) −6.92820 −0.541002
\(165\) 0 0
\(166\) 5.66025 0.439321
\(167\) 15.3205 1.18554 0.592768 0.805373i \(-0.298035\pi\)
0.592768 + 0.805373i \(0.298035\pi\)
\(168\) 2.00000 0.154303
\(169\) 9.39230 0.722485
\(170\) 0 0
\(171\) 0.535898 0.0409812
\(172\) 11.6603 0.889086
\(173\) −21.8564 −1.66171 −0.830856 0.556488i \(-0.812149\pi\)
−0.830856 + 0.556488i \(0.812149\pi\)
\(174\) 14.9282 1.13170
\(175\) −5.00000 −0.377964
\(176\) 5.46410 0.411872
\(177\) −6.92820 −0.520756
\(178\) −2.00000 −0.149906
\(179\) −9.46410 −0.707380 −0.353690 0.935363i \(-0.615073\pi\)
−0.353690 + 0.935363i \(0.615073\pi\)
\(180\) 0 0
\(181\) −21.1244 −1.57016 −0.785080 0.619394i \(-0.787378\pi\)
−0.785080 + 0.619394i \(0.787378\pi\)
\(182\) −4.73205 −0.350763
\(183\) −2.92820 −0.216459
\(184\) 6.00000 0.442326
\(185\) 0 0
\(186\) −2.92820 −0.214706
\(187\) 18.9282 1.38417
\(188\) −8.39230 −0.612072
\(189\) −4.00000 −0.290957
\(190\) 0 0
\(191\) −5.46410 −0.395369 −0.197684 0.980266i \(-0.563342\pi\)
−0.197684 + 0.980266i \(0.563342\pi\)
\(192\) 2.00000 0.144338
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 17.3205 1.23404 0.617018 0.786949i \(-0.288341\pi\)
0.617018 + 0.786949i \(0.288341\pi\)
\(198\) 5.46410 0.388317
\(199\) 9.46410 0.670892 0.335446 0.942059i \(-0.391113\pi\)
0.335446 + 0.942059i \(0.391113\pi\)
\(200\) −5.00000 −0.353553
\(201\) 1.46410 0.103270
\(202\) −4.73205 −0.332946
\(203\) 7.46410 0.523877
\(204\) 6.92820 0.485071
\(205\) 0 0
\(206\) 2.53590 0.176684
\(207\) 6.00000 0.417029
\(208\) −4.73205 −0.328109
\(209\) 2.92820 0.202548
\(210\) 0 0
\(211\) −10.5885 −0.728939 −0.364470 0.931215i \(-0.618750\pi\)
−0.364470 + 0.931215i \(0.618750\pi\)
\(212\) 2.00000 0.137361
\(213\) −5.85641 −0.401274
\(214\) −2.19615 −0.150126
\(215\) 0 0
\(216\) −4.00000 −0.272166
\(217\) −1.46410 −0.0993897
\(218\) 4.92820 0.333780
\(219\) 4.00000 0.270295
\(220\) 0 0
\(221\) −16.3923 −1.10267
\(222\) 10.5359 0.707123
\(223\) 24.2487 1.62381 0.811907 0.583787i \(-0.198430\pi\)
0.811907 + 0.583787i \(0.198430\pi\)
\(224\) 1.00000 0.0668153
\(225\) −5.00000 −0.333333
\(226\) −4.53590 −0.301723
\(227\) 8.53590 0.566547 0.283274 0.959039i \(-0.408580\pi\)
0.283274 + 0.959039i \(0.408580\pi\)
\(228\) 1.07180 0.0709815
\(229\) −28.3923 −1.87622 −0.938108 0.346342i \(-0.887424\pi\)
−0.938108 + 0.346342i \(0.887424\pi\)
\(230\) 0 0
\(231\) 10.9282 0.719023
\(232\) 7.46410 0.490042
\(233\) −14.0000 −0.917170 −0.458585 0.888650i \(-0.651644\pi\)
−0.458585 + 0.888650i \(0.651644\pi\)
\(234\) −4.73205 −0.309344
\(235\) 0 0
\(236\) −3.46410 −0.225494
\(237\) 24.0000 1.55897
\(238\) 3.46410 0.224544
\(239\) 17.4641 1.12966 0.564829 0.825208i \(-0.308942\pi\)
0.564829 + 0.825208i \(0.308942\pi\)
\(240\) 0 0
\(241\) −7.07180 −0.455534 −0.227767 0.973716i \(-0.573143\pi\)
−0.227767 + 0.973716i \(0.573143\pi\)
\(242\) 18.8564 1.21214
\(243\) −10.0000 −0.641500
\(244\) −1.46410 −0.0937295
\(245\) 0 0
\(246\) −13.8564 −0.883452
\(247\) −2.53590 −0.161355
\(248\) −1.46410 −0.0929705
\(249\) 11.3205 0.717408
\(250\) 0 0
\(251\) −13.6603 −0.862228 −0.431114 0.902298i \(-0.641879\pi\)
−0.431114 + 0.902298i \(0.641879\pi\)
\(252\) 1.00000 0.0629941
\(253\) 32.7846 2.06115
\(254\) 6.92820 0.434714
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 0.535898 0.0334284 0.0167142 0.999860i \(-0.494679\pi\)
0.0167142 + 0.999860i \(0.494679\pi\)
\(258\) 23.3205 1.45187
\(259\) 5.26795 0.327334
\(260\) 0 0
\(261\) 7.46410 0.462016
\(262\) 1.26795 0.0783342
\(263\) −7.85641 −0.484447 −0.242223 0.970221i \(-0.577877\pi\)
−0.242223 + 0.970221i \(0.577877\pi\)
\(264\) 10.9282 0.672584
\(265\) 0 0
\(266\) 0.535898 0.0328580
\(267\) −4.00000 −0.244796
\(268\) 0.732051 0.0447171
\(269\) −18.5885 −1.13336 −0.566679 0.823939i \(-0.691772\pi\)
−0.566679 + 0.823939i \(0.691772\pi\)
\(270\) 0 0
\(271\) 8.78461 0.533627 0.266814 0.963748i \(-0.414029\pi\)
0.266814 + 0.963748i \(0.414029\pi\)
\(272\) 3.46410 0.210042
\(273\) −9.46410 −0.572793
\(274\) 10.3923 0.627822
\(275\) −27.3205 −1.64749
\(276\) 12.0000 0.722315
\(277\) −9.60770 −0.577270 −0.288635 0.957439i \(-0.593201\pi\)
−0.288635 + 0.957439i \(0.593201\pi\)
\(278\) 18.3923 1.10310
\(279\) −1.46410 −0.0876535
\(280\) 0 0
\(281\) 11.8564 0.707294 0.353647 0.935379i \(-0.384941\pi\)
0.353647 + 0.935379i \(0.384941\pi\)
\(282\) −16.7846 −0.999509
\(283\) 20.9282 1.24405 0.622026 0.782996i \(-0.286310\pi\)
0.622026 + 0.782996i \(0.286310\pi\)
\(284\) −2.92820 −0.173757
\(285\) 0 0
\(286\) −25.8564 −1.52892
\(287\) −6.92820 −0.408959
\(288\) 1.00000 0.0589256
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 2.00000 0.117041
\(293\) −11.6603 −0.681199 −0.340600 0.940208i \(-0.610630\pi\)
−0.340600 + 0.940208i \(0.610630\pi\)
\(294\) 2.00000 0.116642
\(295\) 0 0
\(296\) 5.26795 0.306193
\(297\) −21.8564 −1.26824
\(298\) −19.8564 −1.15025
\(299\) −28.3923 −1.64197
\(300\) −10.0000 −0.577350
\(301\) 11.6603 0.672086
\(302\) −4.92820 −0.283586
\(303\) −9.46410 −0.543698
\(304\) 0.535898 0.0307359
\(305\) 0 0
\(306\) 3.46410 0.198030
\(307\) 7.46410 0.425999 0.212999 0.977052i \(-0.431677\pi\)
0.212999 + 0.977052i \(0.431677\pi\)
\(308\) 5.46410 0.311346
\(309\) 5.07180 0.288524
\(310\) 0 0
\(311\) 4.00000 0.226819 0.113410 0.993548i \(-0.463823\pi\)
0.113410 + 0.993548i \(0.463823\pi\)
\(312\) −9.46410 −0.535799
\(313\) −0.535898 −0.0302908 −0.0151454 0.999885i \(-0.504821\pi\)
−0.0151454 + 0.999885i \(0.504821\pi\)
\(314\) −3.60770 −0.203594
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) −22.7321 −1.27676 −0.638380 0.769722i \(-0.720395\pi\)
−0.638380 + 0.769722i \(0.720395\pi\)
\(318\) 4.00000 0.224309
\(319\) 40.7846 2.28350
\(320\) 0 0
\(321\) −4.39230 −0.245155
\(322\) 6.00000 0.334367
\(323\) 1.85641 0.103293
\(324\) −11.0000 −0.611111
\(325\) 23.6603 1.31243
\(326\) 6.53590 0.361990
\(327\) 9.85641 0.545061
\(328\) −6.92820 −0.382546
\(329\) −8.39230 −0.462683
\(330\) 0 0
\(331\) 18.9808 1.04328 0.521639 0.853167i \(-0.325321\pi\)
0.521639 + 0.853167i \(0.325321\pi\)
\(332\) 5.66025 0.310647
\(333\) 5.26795 0.288682
\(334\) 15.3205 0.838301
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) 16.3923 0.892946 0.446473 0.894797i \(-0.352680\pi\)
0.446473 + 0.894797i \(0.352680\pi\)
\(338\) 9.39230 0.510874
\(339\) −9.07180 −0.492712
\(340\) 0 0
\(341\) −8.00000 −0.433224
\(342\) 0.535898 0.0289781
\(343\) 1.00000 0.0539949
\(344\) 11.6603 0.628679
\(345\) 0 0
\(346\) −21.8564 −1.17501
\(347\) 5.07180 0.272268 0.136134 0.990690i \(-0.456532\pi\)
0.136134 + 0.990690i \(0.456532\pi\)
\(348\) 14.9282 0.800236
\(349\) −23.6603 −1.26650 −0.633252 0.773946i \(-0.718280\pi\)
−0.633252 + 0.773946i \(0.718280\pi\)
\(350\) −5.00000 −0.267261
\(351\) 18.9282 1.01031
\(352\) 5.46410 0.291238
\(353\) −17.0718 −0.908640 −0.454320 0.890839i \(-0.650118\pi\)
−0.454320 + 0.890839i \(0.650118\pi\)
\(354\) −6.92820 −0.368230
\(355\) 0 0
\(356\) −2.00000 −0.106000
\(357\) 6.92820 0.366679
\(358\) −9.46410 −0.500193
\(359\) −18.9282 −0.998992 −0.499496 0.866316i \(-0.666482\pi\)
−0.499496 + 0.866316i \(0.666482\pi\)
\(360\) 0 0
\(361\) −18.7128 −0.984885
\(362\) −21.1244 −1.11027
\(363\) 37.7128 1.97941
\(364\) −4.73205 −0.248027
\(365\) 0 0
\(366\) −2.92820 −0.153060
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 6.00000 0.312772
\(369\) −6.92820 −0.360668
\(370\) 0 0
\(371\) 2.00000 0.103835
\(372\) −2.92820 −0.151820
\(373\) −25.6603 −1.32864 −0.664319 0.747449i \(-0.731278\pi\)
−0.664319 + 0.747449i \(0.731278\pi\)
\(374\) 18.9282 0.978754
\(375\) 0 0
\(376\) −8.39230 −0.432800
\(377\) −35.3205 −1.81910
\(378\) −4.00000 −0.205738
\(379\) −24.7846 −1.27310 −0.636550 0.771235i \(-0.719639\pi\)
−0.636550 + 0.771235i \(0.719639\pi\)
\(380\) 0 0
\(381\) 13.8564 0.709885
\(382\) −5.46410 −0.279568
\(383\) 5.46410 0.279203 0.139601 0.990208i \(-0.455418\pi\)
0.139601 + 0.990208i \(0.455418\pi\)
\(384\) 2.00000 0.102062
\(385\) 0 0
\(386\) −10.0000 −0.508987
\(387\) 11.6603 0.592724
\(388\) −8.00000 −0.406138
\(389\) −32.2487 −1.63507 −0.817537 0.575876i \(-0.804661\pi\)
−0.817537 + 0.575876i \(0.804661\pi\)
\(390\) 0 0
\(391\) 20.7846 1.05112
\(392\) 1.00000 0.0505076
\(393\) 2.53590 0.127919
\(394\) 17.3205 0.872595
\(395\) 0 0
\(396\) 5.46410 0.274581
\(397\) 18.9282 0.949979 0.474990 0.879991i \(-0.342452\pi\)
0.474990 + 0.879991i \(0.342452\pi\)
\(398\) 9.46410 0.474393
\(399\) 1.07180 0.0536570
\(400\) −5.00000 −0.250000
\(401\) 16.9282 0.845354 0.422677 0.906280i \(-0.361090\pi\)
0.422677 + 0.906280i \(0.361090\pi\)
\(402\) 1.46410 0.0730228
\(403\) 6.92820 0.345118
\(404\) −4.73205 −0.235428
\(405\) 0 0
\(406\) 7.46410 0.370437
\(407\) 28.7846 1.42680
\(408\) 6.92820 0.342997
\(409\) 8.53590 0.422073 0.211037 0.977478i \(-0.432316\pi\)
0.211037 + 0.977478i \(0.432316\pi\)
\(410\) 0 0
\(411\) 20.7846 1.02523
\(412\) 2.53590 0.124935
\(413\) −3.46410 −0.170457
\(414\) 6.00000 0.294884
\(415\) 0 0
\(416\) −4.73205 −0.232008
\(417\) 36.7846 1.80135
\(418\) 2.92820 0.143223
\(419\) 8.19615 0.400408 0.200204 0.979754i \(-0.435839\pi\)
0.200204 + 0.979754i \(0.435839\pi\)
\(420\) 0 0
\(421\) −14.7321 −0.717996 −0.358998 0.933338i \(-0.616882\pi\)
−0.358998 + 0.933338i \(0.616882\pi\)
\(422\) −10.5885 −0.515438
\(423\) −8.39230 −0.408048
\(424\) 2.00000 0.0971286
\(425\) −17.3205 −0.840168
\(426\) −5.85641 −0.283744
\(427\) −1.46410 −0.0708528
\(428\) −2.19615 −0.106155
\(429\) −51.7128 −2.49672
\(430\) 0 0
\(431\) 1.00000 0.0481683
\(432\) −4.00000 −0.192450
\(433\) 18.0000 0.865025 0.432512 0.901628i \(-0.357627\pi\)
0.432512 + 0.901628i \(0.357627\pi\)
\(434\) −1.46410 −0.0702791
\(435\) 0 0
\(436\) 4.92820 0.236018
\(437\) 3.21539 0.153813
\(438\) 4.00000 0.191127
\(439\) −24.0000 −1.14546 −0.572729 0.819745i \(-0.694115\pi\)
−0.572729 + 0.819745i \(0.694115\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) −16.3923 −0.779702
\(443\) −12.3923 −0.588776 −0.294388 0.955686i \(-0.595116\pi\)
−0.294388 + 0.955686i \(0.595116\pi\)
\(444\) 10.5359 0.500012
\(445\) 0 0
\(446\) 24.2487 1.14821
\(447\) −39.7128 −1.87835
\(448\) 1.00000 0.0472456
\(449\) 0.143594 0.00677660 0.00338830 0.999994i \(-0.498921\pi\)
0.00338830 + 0.999994i \(0.498921\pi\)
\(450\) −5.00000 −0.235702
\(451\) −37.8564 −1.78259
\(452\) −4.53590 −0.213351
\(453\) −9.85641 −0.463094
\(454\) 8.53590 0.400610
\(455\) 0 0
\(456\) 1.07180 0.0501915
\(457\) 19.0718 0.892141 0.446071 0.894998i \(-0.352823\pi\)
0.446071 + 0.894998i \(0.352823\pi\)
\(458\) −28.3923 −1.32669
\(459\) −13.8564 −0.646762
\(460\) 0 0
\(461\) 20.3923 0.949764 0.474882 0.880049i \(-0.342491\pi\)
0.474882 + 0.880049i \(0.342491\pi\)
\(462\) 10.9282 0.508426
\(463\) −26.9282 −1.25146 −0.625730 0.780040i \(-0.715199\pi\)
−0.625730 + 0.780040i \(0.715199\pi\)
\(464\) 7.46410 0.346512
\(465\) 0 0
\(466\) −14.0000 −0.648537
\(467\) 24.9282 1.15354 0.576770 0.816907i \(-0.304313\pi\)
0.576770 + 0.816907i \(0.304313\pi\)
\(468\) −4.73205 −0.218739
\(469\) 0.732051 0.0338030
\(470\) 0 0
\(471\) −7.21539 −0.332468
\(472\) −3.46410 −0.159448
\(473\) 63.7128 2.92952
\(474\) 24.0000 1.10236
\(475\) −2.67949 −0.122944
\(476\) 3.46410 0.158777
\(477\) 2.00000 0.0915737
\(478\) 17.4641 0.798789
\(479\) 28.5359 1.30384 0.651919 0.758288i \(-0.273964\pi\)
0.651919 + 0.758288i \(0.273964\pi\)
\(480\) 0 0
\(481\) −24.9282 −1.13663
\(482\) −7.07180 −0.322112
\(483\) 12.0000 0.546019
\(484\) 18.8564 0.857109
\(485\) 0 0
\(486\) −10.0000 −0.453609
\(487\) −2.92820 −0.132690 −0.0663448 0.997797i \(-0.521134\pi\)
−0.0663448 + 0.997797i \(0.521134\pi\)
\(488\) −1.46410 −0.0662768
\(489\) 13.0718 0.591127
\(490\) 0 0
\(491\) 6.53590 0.294961 0.147480 0.989065i \(-0.452884\pi\)
0.147480 + 0.989065i \(0.452884\pi\)
\(492\) −13.8564 −0.624695
\(493\) 25.8564 1.16451
\(494\) −2.53590 −0.114095
\(495\) 0 0
\(496\) −1.46410 −0.0657401
\(497\) −2.92820 −0.131348
\(498\) 11.3205 0.507284
\(499\) −3.66025 −0.163855 −0.0819277 0.996638i \(-0.526108\pi\)
−0.0819277 + 0.996638i \(0.526108\pi\)
\(500\) 0 0
\(501\) 30.6410 1.36894
\(502\) −13.6603 −0.609687
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0 0
\(506\) 32.7846 1.45745
\(507\) 18.7846 0.834254
\(508\) 6.92820 0.307389
\(509\) 16.7321 0.741635 0.370818 0.928706i \(-0.379078\pi\)
0.370818 + 0.928706i \(0.379078\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 1.00000 0.0441942
\(513\) −2.14359 −0.0946420
\(514\) 0.535898 0.0236375
\(515\) 0 0
\(516\) 23.3205 1.02663
\(517\) −45.8564 −2.01676
\(518\) 5.26795 0.231460
\(519\) −43.7128 −1.91878
\(520\) 0 0
\(521\) 21.0718 0.923172 0.461586 0.887095i \(-0.347281\pi\)
0.461586 + 0.887095i \(0.347281\pi\)
\(522\) 7.46410 0.326695
\(523\) −15.4641 −0.676198 −0.338099 0.941111i \(-0.609784\pi\)
−0.338099 + 0.941111i \(0.609784\pi\)
\(524\) 1.26795 0.0553906
\(525\) −10.0000 −0.436436
\(526\) −7.85641 −0.342556
\(527\) −5.07180 −0.220931
\(528\) 10.9282 0.475589
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) −3.46410 −0.150329
\(532\) 0.535898 0.0232341
\(533\) 32.7846 1.42006
\(534\) −4.00000 −0.173097
\(535\) 0 0
\(536\) 0.732051 0.0316198
\(537\) −18.9282 −0.816812
\(538\) −18.5885 −0.801405
\(539\) 5.46410 0.235356
\(540\) 0 0
\(541\) −28.9282 −1.24372 −0.621860 0.783128i \(-0.713623\pi\)
−0.621860 + 0.783128i \(0.713623\pi\)
\(542\) 8.78461 0.377331
\(543\) −42.2487 −1.81307
\(544\) 3.46410 0.148522
\(545\) 0 0
\(546\) −9.46410 −0.405026
\(547\) −33.1769 −1.41854 −0.709271 0.704936i \(-0.750976\pi\)
−0.709271 + 0.704936i \(0.750976\pi\)
\(548\) 10.3923 0.443937
\(549\) −1.46410 −0.0624863
\(550\) −27.3205 −1.16495
\(551\) 4.00000 0.170406
\(552\) 12.0000 0.510754
\(553\) 12.0000 0.510292
\(554\) −9.60770 −0.408192
\(555\) 0 0
\(556\) 18.3923 0.780007
\(557\) −18.3397 −0.777080 −0.388540 0.921432i \(-0.627020\pi\)
−0.388540 + 0.921432i \(0.627020\pi\)
\(558\) −1.46410 −0.0619804
\(559\) −55.1769 −2.33373
\(560\) 0 0
\(561\) 37.8564 1.59830
\(562\) 11.8564 0.500132
\(563\) 7.85641 0.331108 0.165554 0.986201i \(-0.447059\pi\)
0.165554 + 0.986201i \(0.447059\pi\)
\(564\) −16.7846 −0.706760
\(565\) 0 0
\(566\) 20.9282 0.879678
\(567\) −11.0000 −0.461957
\(568\) −2.92820 −0.122865
\(569\) −9.46410 −0.396756 −0.198378 0.980126i \(-0.563567\pi\)
−0.198378 + 0.980126i \(0.563567\pi\)
\(570\) 0 0
\(571\) 38.9808 1.63129 0.815647 0.578550i \(-0.196381\pi\)
0.815647 + 0.578550i \(0.196381\pi\)
\(572\) −25.8564 −1.08111
\(573\) −10.9282 −0.456532
\(574\) −6.92820 −0.289178
\(575\) −30.0000 −1.25109
\(576\) 1.00000 0.0416667
\(577\) −28.2487 −1.17601 −0.588005 0.808858i \(-0.700086\pi\)
−0.588005 + 0.808858i \(0.700086\pi\)
\(578\) −5.00000 −0.207973
\(579\) −20.0000 −0.831172
\(580\) 0 0
\(581\) 5.66025 0.234827
\(582\) −16.0000 −0.663221
\(583\) 10.9282 0.452600
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) −11.6603 −0.481681
\(587\) 21.2679 0.877822 0.438911 0.898530i \(-0.355364\pi\)
0.438911 + 0.898530i \(0.355364\pi\)
\(588\) 2.00000 0.0824786
\(589\) −0.784610 −0.0323293
\(590\) 0 0
\(591\) 34.6410 1.42494
\(592\) 5.26795 0.216511
\(593\) 9.71281 0.398857 0.199429 0.979912i \(-0.436091\pi\)
0.199429 + 0.979912i \(0.436091\pi\)
\(594\) −21.8564 −0.896779
\(595\) 0 0
\(596\) −19.8564 −0.813350
\(597\) 18.9282 0.774680
\(598\) −28.3923 −1.16105
\(599\) −25.4641 −1.04043 −0.520217 0.854034i \(-0.674149\pi\)
−0.520217 + 0.854034i \(0.674149\pi\)
\(600\) −10.0000 −0.408248
\(601\) 8.53590 0.348187 0.174093 0.984729i \(-0.444301\pi\)
0.174093 + 0.984729i \(0.444301\pi\)
\(602\) 11.6603 0.475236
\(603\) 0.732051 0.0298114
\(604\) −4.92820 −0.200526
\(605\) 0 0
\(606\) −9.46410 −0.384453
\(607\) −30.3923 −1.23359 −0.616793 0.787126i \(-0.711568\pi\)
−0.616793 + 0.787126i \(0.711568\pi\)
\(608\) 0.535898 0.0217335
\(609\) 14.9282 0.604921
\(610\) 0 0
\(611\) 39.7128 1.60661
\(612\) 3.46410 0.140028
\(613\) −43.4641 −1.75550 −0.877749 0.479120i \(-0.840956\pi\)
−0.877749 + 0.479120i \(0.840956\pi\)
\(614\) 7.46410 0.301227
\(615\) 0 0
\(616\) 5.46410 0.220155
\(617\) −4.53590 −0.182608 −0.0913042 0.995823i \(-0.529104\pi\)
−0.0913042 + 0.995823i \(0.529104\pi\)
\(618\) 5.07180 0.204018
\(619\) −1.66025 −0.0667312 −0.0333656 0.999443i \(-0.510623\pi\)
−0.0333656 + 0.999443i \(0.510623\pi\)
\(620\) 0 0
\(621\) −24.0000 −0.963087
\(622\) 4.00000 0.160385
\(623\) −2.00000 −0.0801283
\(624\) −9.46410 −0.378867
\(625\) 25.0000 1.00000
\(626\) −0.535898 −0.0214188
\(627\) 5.85641 0.233882
\(628\) −3.60770 −0.143963
\(629\) 18.2487 0.727624
\(630\) 0 0
\(631\) −0.784610 −0.0312348 −0.0156174 0.999878i \(-0.504971\pi\)
−0.0156174 + 0.999878i \(0.504971\pi\)
\(632\) 12.0000 0.477334
\(633\) −21.1769 −0.841707
\(634\) −22.7321 −0.902805
\(635\) 0 0
\(636\) 4.00000 0.158610
\(637\) −4.73205 −0.187491
\(638\) 40.7846 1.61468
\(639\) −2.92820 −0.115838
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) −4.39230 −0.173350
\(643\) −19.4641 −0.767589 −0.383795 0.923418i \(-0.625383\pi\)
−0.383795 + 0.923418i \(0.625383\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 1.85641 0.0730393
\(647\) −13.3205 −0.523683 −0.261842 0.965111i \(-0.584330\pi\)
−0.261842 + 0.965111i \(0.584330\pi\)
\(648\) −11.0000 −0.432121
\(649\) −18.9282 −0.742997
\(650\) 23.6603 0.928032
\(651\) −2.92820 −0.114765
\(652\) 6.53590 0.255966
\(653\) −28.1962 −1.10340 −0.551700 0.834042i \(-0.686021\pi\)
−0.551700 + 0.834042i \(0.686021\pi\)
\(654\) 9.85641 0.385416
\(655\) 0 0
\(656\) −6.92820 −0.270501
\(657\) 2.00000 0.0780274
\(658\) −8.39230 −0.327166
\(659\) 10.5359 0.410420 0.205210 0.978718i \(-0.434212\pi\)
0.205210 + 0.978718i \(0.434212\pi\)
\(660\) 0 0
\(661\) −3.60770 −0.140323 −0.0701615 0.997536i \(-0.522351\pi\)
−0.0701615 + 0.997536i \(0.522351\pi\)
\(662\) 18.9808 0.737708
\(663\) −32.7846 −1.27325
\(664\) 5.66025 0.219660
\(665\) 0 0
\(666\) 5.26795 0.204129
\(667\) 44.7846 1.73407
\(668\) 15.3205 0.592768
\(669\) 48.4974 1.87502
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 2.00000 0.0771517
\(673\) −47.8564 −1.84473 −0.922364 0.386321i \(-0.873746\pi\)
−0.922364 + 0.386321i \(0.873746\pi\)
\(674\) 16.3923 0.631408
\(675\) 20.0000 0.769800
\(676\) 9.39230 0.361242
\(677\) −7.60770 −0.292387 −0.146194 0.989256i \(-0.546702\pi\)
−0.146194 + 0.989256i \(0.546702\pi\)
\(678\) −9.07180 −0.348400
\(679\) −8.00000 −0.307012
\(680\) 0 0
\(681\) 17.0718 0.654193
\(682\) −8.00000 −0.306336
\(683\) 21.5167 0.823312 0.411656 0.911339i \(-0.364951\pi\)
0.411656 + 0.911339i \(0.364951\pi\)
\(684\) 0.535898 0.0204906
\(685\) 0 0
\(686\) 1.00000 0.0381802
\(687\) −56.7846 −2.16647
\(688\) 11.6603 0.444543
\(689\) −9.46410 −0.360554
\(690\) 0 0
\(691\) 18.0526 0.686752 0.343376 0.939198i \(-0.388430\pi\)
0.343376 + 0.939198i \(0.388430\pi\)
\(692\) −21.8564 −0.830856
\(693\) 5.46410 0.207564
\(694\) 5.07180 0.192523
\(695\) 0 0
\(696\) 14.9282 0.565852
\(697\) −24.0000 −0.909065
\(698\) −23.6603 −0.895554
\(699\) −28.0000 −1.05906
\(700\) −5.00000 −0.188982
\(701\) 19.4641 0.735149 0.367574 0.929994i \(-0.380188\pi\)
0.367574 + 0.929994i \(0.380188\pi\)
\(702\) 18.9282 0.714399
\(703\) 2.82309 0.106475
\(704\) 5.46410 0.205936
\(705\) 0 0
\(706\) −17.0718 −0.642506
\(707\) −4.73205 −0.177967
\(708\) −6.92820 −0.260378
\(709\) −47.1769 −1.77177 −0.885883 0.463909i \(-0.846447\pi\)
−0.885883 + 0.463909i \(0.846447\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) −2.00000 −0.0749532
\(713\) −8.78461 −0.328986
\(714\) 6.92820 0.259281
\(715\) 0 0
\(716\) −9.46410 −0.353690
\(717\) 34.9282 1.30442
\(718\) −18.9282 −0.706394
\(719\) −48.4974 −1.80865 −0.904324 0.426846i \(-0.859625\pi\)
−0.904324 + 0.426846i \(0.859625\pi\)
\(720\) 0 0
\(721\) 2.53590 0.0944418
\(722\) −18.7128 −0.696419
\(723\) −14.1436 −0.526006
\(724\) −21.1244 −0.785080
\(725\) −37.3205 −1.38605
\(726\) 37.7128 1.39965
\(727\) −34.5359 −1.28087 −0.640433 0.768014i \(-0.721245\pi\)
−0.640433 + 0.768014i \(0.721245\pi\)
\(728\) −4.73205 −0.175381
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 40.3923 1.49396
\(732\) −2.92820 −0.108230
\(733\) 24.7846 0.915440 0.457720 0.889096i \(-0.348666\pi\)
0.457720 + 0.889096i \(0.348666\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 6.00000 0.221163
\(737\) 4.00000 0.147342
\(738\) −6.92820 −0.255031
\(739\) −12.7321 −0.468356 −0.234178 0.972194i \(-0.575240\pi\)
−0.234178 + 0.972194i \(0.575240\pi\)
\(740\) 0 0
\(741\) −5.07180 −0.186317
\(742\) 2.00000 0.0734223
\(743\) −31.7128 −1.16343 −0.581715 0.813393i \(-0.697618\pi\)
−0.581715 + 0.813393i \(0.697618\pi\)
\(744\) −2.92820 −0.107353
\(745\) 0 0
\(746\) −25.6603 −0.939489
\(747\) 5.66025 0.207098
\(748\) 18.9282 0.692084
\(749\) −2.19615 −0.0802457
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −8.39230 −0.306036
\(753\) −27.3205 −0.995615
\(754\) −35.3205 −1.28630
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) −16.6410 −0.604828 −0.302414 0.953177i \(-0.597793\pi\)
−0.302414 + 0.953177i \(0.597793\pi\)
\(758\) −24.7846 −0.900218
\(759\) 65.5692 2.38001
\(760\) 0 0
\(761\) 12.7846 0.463442 0.231721 0.972782i \(-0.425564\pi\)
0.231721 + 0.972782i \(0.425564\pi\)
\(762\) 13.8564 0.501965
\(763\) 4.92820 0.178413
\(764\) −5.46410 −0.197684
\(765\) 0 0
\(766\) 5.46410 0.197426
\(767\) 16.3923 0.591892
\(768\) 2.00000 0.0721688
\(769\) 18.7846 0.677390 0.338695 0.940896i \(-0.390014\pi\)
0.338695 + 0.940896i \(0.390014\pi\)
\(770\) 0 0
\(771\) 1.07180 0.0385998
\(772\) −10.0000 −0.359908
\(773\) −18.9282 −0.680800 −0.340400 0.940281i \(-0.610563\pi\)
−0.340400 + 0.940281i \(0.610563\pi\)
\(774\) 11.6603 0.419119
\(775\) 7.32051 0.262960
\(776\) −8.00000 −0.287183
\(777\) 10.5359 0.377973
\(778\) −32.2487 −1.15617
\(779\) −3.71281 −0.133025
\(780\) 0 0
\(781\) −16.0000 −0.572525
\(782\) 20.7846 0.743256
\(783\) −29.8564 −1.06698
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) 2.53590 0.0904525
\(787\) −20.5885 −0.733899 −0.366950 0.930241i \(-0.619598\pi\)
−0.366950 + 0.930241i \(0.619598\pi\)
\(788\) 17.3205 0.617018
\(789\) −15.7128 −0.559391
\(790\) 0 0
\(791\) −4.53590 −0.161278
\(792\) 5.46410 0.194158
\(793\) 6.92820 0.246028
\(794\) 18.9282 0.671737
\(795\) 0 0
\(796\) 9.46410 0.335446
\(797\) 43.0333 1.52432 0.762159 0.647390i \(-0.224139\pi\)
0.762159 + 0.647390i \(0.224139\pi\)
\(798\) 1.07180 0.0379412
\(799\) −29.0718 −1.02849
\(800\) −5.00000 −0.176777
\(801\) −2.00000 −0.0706665
\(802\) 16.9282 0.597756
\(803\) 10.9282 0.385648
\(804\) 1.46410 0.0516349
\(805\) 0 0
\(806\) 6.92820 0.244036
\(807\) −37.1769 −1.30869
\(808\) −4.73205 −0.166473
\(809\) 22.0000 0.773479 0.386739 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) 54.7846 1.92375 0.961874 0.273493i \(-0.0881791\pi\)
0.961874 + 0.273493i \(0.0881791\pi\)
\(812\) 7.46410 0.261939
\(813\) 17.5692 0.616179
\(814\) 28.7846 1.00890
\(815\) 0 0
\(816\) 6.92820 0.242536
\(817\) 6.24871 0.218615
\(818\) 8.53590 0.298451
\(819\) −4.73205 −0.165351
\(820\) 0 0
\(821\) −49.7654 −1.73682 −0.868412 0.495844i \(-0.834859\pi\)
−0.868412 + 0.495844i \(0.834859\pi\)
\(822\) 20.7846 0.724947
\(823\) −43.7128 −1.52373 −0.761866 0.647734i \(-0.775717\pi\)
−0.761866 + 0.647734i \(0.775717\pi\)
\(824\) 2.53590 0.0883422
\(825\) −54.6410 −1.90236
\(826\) −3.46410 −0.120532
\(827\) −46.6410 −1.62187 −0.810934 0.585138i \(-0.801040\pi\)
−0.810934 + 0.585138i \(0.801040\pi\)
\(828\) 6.00000 0.208514
\(829\) 44.0526 1.53001 0.765004 0.644025i \(-0.222737\pi\)
0.765004 + 0.644025i \(0.222737\pi\)
\(830\) 0 0
\(831\) −19.2154 −0.666575
\(832\) −4.73205 −0.164054
\(833\) 3.46410 0.120024
\(834\) 36.7846 1.27375
\(835\) 0 0
\(836\) 2.92820 0.101274
\(837\) 5.85641 0.202427
\(838\) 8.19615 0.283131
\(839\) −15.6077 −0.538837 −0.269419 0.963023i \(-0.586832\pi\)
−0.269419 + 0.963023i \(0.586832\pi\)
\(840\) 0 0
\(841\) 26.7128 0.921131
\(842\) −14.7321 −0.507700
\(843\) 23.7128 0.816713
\(844\) −10.5885 −0.364470
\(845\) 0 0
\(846\) −8.39230 −0.288533
\(847\) 18.8564 0.647914
\(848\) 2.00000 0.0686803
\(849\) 41.8564 1.43651
\(850\) −17.3205 −0.594089
\(851\) 31.6077 1.08350
\(852\) −5.85641 −0.200637
\(853\) 9.41154 0.322245 0.161123 0.986934i \(-0.448489\pi\)
0.161123 + 0.986934i \(0.448489\pi\)
\(854\) −1.46410 −0.0501005
\(855\) 0 0
\(856\) −2.19615 −0.0750629
\(857\) 50.1051 1.71156 0.855779 0.517341i \(-0.173078\pi\)
0.855779 + 0.517341i \(0.173078\pi\)
\(858\) −51.7128 −1.76545
\(859\) −29.6603 −1.01199 −0.505997 0.862535i \(-0.668875\pi\)
−0.505997 + 0.862535i \(0.668875\pi\)
\(860\) 0 0
\(861\) −13.8564 −0.472225
\(862\) 1.00000 0.0340601
\(863\) 46.6410 1.58768 0.793839 0.608128i \(-0.208079\pi\)
0.793839 + 0.608128i \(0.208079\pi\)
\(864\) −4.00000 −0.136083
\(865\) 0 0
\(866\) 18.0000 0.611665
\(867\) −10.0000 −0.339618
\(868\) −1.46410 −0.0496948
\(869\) 65.5692 2.22428
\(870\) 0 0
\(871\) −3.46410 −0.117377
\(872\) 4.92820 0.166890
\(873\) −8.00000 −0.270759
\(874\) 3.21539 0.108762
\(875\) 0 0
\(876\) 4.00000 0.135147
\(877\) 13.3205 0.449802 0.224901 0.974382i \(-0.427794\pi\)
0.224901 + 0.974382i \(0.427794\pi\)
\(878\) −24.0000 −0.809961
\(879\) −23.3205 −0.786581
\(880\) 0 0
\(881\) 0.928203 0.0312720 0.0156360 0.999878i \(-0.495023\pi\)
0.0156360 + 0.999878i \(0.495023\pi\)
\(882\) 1.00000 0.0336718
\(883\) 37.1244 1.24933 0.624667 0.780891i \(-0.285235\pi\)
0.624667 + 0.780891i \(0.285235\pi\)
\(884\) −16.3923 −0.551333
\(885\) 0 0
\(886\) −12.3923 −0.416328
\(887\) 23.7128 0.796198 0.398099 0.917342i \(-0.369670\pi\)
0.398099 + 0.917342i \(0.369670\pi\)
\(888\) 10.5359 0.353562
\(889\) 6.92820 0.232364
\(890\) 0 0
\(891\) −60.1051 −2.01360
\(892\) 24.2487 0.811907
\(893\) −4.49742 −0.150501
\(894\) −39.7128 −1.32820
\(895\) 0 0
\(896\) 1.00000 0.0334077
\(897\) −56.7846 −1.89598
\(898\) 0.143594 0.00479178
\(899\) −10.9282 −0.364476
\(900\) −5.00000 −0.166667
\(901\) 6.92820 0.230812
\(902\) −37.8564 −1.26048
\(903\) 23.3205 0.776058
\(904\) −4.53590 −0.150862
\(905\) 0 0
\(906\) −9.85641 −0.327457
\(907\) 50.6410 1.68151 0.840754 0.541418i \(-0.182112\pi\)
0.840754 + 0.541418i \(0.182112\pi\)
\(908\) 8.53590 0.283274
\(909\) −4.73205 −0.156952
\(910\) 0 0
\(911\) 45.0718 1.49330 0.746648 0.665220i \(-0.231662\pi\)
0.746648 + 0.665220i \(0.231662\pi\)
\(912\) 1.07180 0.0354907
\(913\) 30.9282 1.02357
\(914\) 19.0718 0.630839
\(915\) 0 0
\(916\) −28.3923 −0.938108
\(917\) 1.26795 0.0418714
\(918\) −13.8564 −0.457330
\(919\) 10.0000 0.329870 0.164935 0.986304i \(-0.447259\pi\)
0.164935 + 0.986304i \(0.447259\pi\)
\(920\) 0 0
\(921\) 14.9282 0.491901
\(922\) 20.3923 0.671585
\(923\) 13.8564 0.456089
\(924\) 10.9282 0.359511
\(925\) −26.3397 −0.866046
\(926\) −26.9282 −0.884916
\(927\) 2.53590 0.0832898
\(928\) 7.46410 0.245021
\(929\) 7.85641 0.257760 0.128880 0.991660i \(-0.458862\pi\)
0.128880 + 0.991660i \(0.458862\pi\)
\(930\) 0 0
\(931\) 0.535898 0.0175634
\(932\) −14.0000 −0.458585
\(933\) 8.00000 0.261908
\(934\) 24.9282 0.815676
\(935\) 0 0
\(936\) −4.73205 −0.154672
\(937\) −14.9282 −0.487683 −0.243842 0.969815i \(-0.578408\pi\)
−0.243842 + 0.969815i \(0.578408\pi\)
\(938\) 0.732051 0.0239023
\(939\) −1.07180 −0.0349768
\(940\) 0 0
\(941\) 42.5885 1.38834 0.694172 0.719809i \(-0.255771\pi\)
0.694172 + 0.719809i \(0.255771\pi\)
\(942\) −7.21539 −0.235090
\(943\) −41.5692 −1.35368
\(944\) −3.46410 −0.112747
\(945\) 0 0
\(946\) 63.7128 2.07148
\(947\) −40.4449 −1.31428 −0.657141 0.753768i \(-0.728234\pi\)
−0.657141 + 0.753768i \(0.728234\pi\)
\(948\) 24.0000 0.779484
\(949\) −9.46410 −0.307218
\(950\) −2.67949 −0.0869342
\(951\) −45.4641 −1.47427
\(952\) 3.46410 0.112272
\(953\) −24.1051 −0.780841 −0.390421 0.920637i \(-0.627670\pi\)
−0.390421 + 0.920637i \(0.627670\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 17.4641 0.564829
\(957\) 81.5692 2.63676
\(958\) 28.5359 0.921953
\(959\) 10.3923 0.335585
\(960\) 0 0
\(961\) −28.8564 −0.930852
\(962\) −24.9282 −0.803718
\(963\) −2.19615 −0.0707700
\(964\) −7.07180 −0.227767
\(965\) 0 0
\(966\) 12.0000 0.386094
\(967\) 29.1769 0.938266 0.469133 0.883127i \(-0.344566\pi\)
0.469133 + 0.883127i \(0.344566\pi\)
\(968\) 18.8564 0.606068
\(969\) 3.71281 0.119273
\(970\) 0 0
\(971\) 42.1051 1.35122 0.675609 0.737260i \(-0.263881\pi\)
0.675609 + 0.737260i \(0.263881\pi\)
\(972\) −10.0000 −0.320750
\(973\) 18.3923 0.589630
\(974\) −2.92820 −0.0938257
\(975\) 47.3205 1.51547
\(976\) −1.46410 −0.0468648
\(977\) 4.39230 0.140522 0.0702611 0.997529i \(-0.477617\pi\)
0.0702611 + 0.997529i \(0.477617\pi\)
\(978\) 13.0718 0.417990
\(979\) −10.9282 −0.349267
\(980\) 0 0
\(981\) 4.92820 0.157345
\(982\) 6.53590 0.208569
\(983\) 25.0718 0.799666 0.399833 0.916588i \(-0.369068\pi\)
0.399833 + 0.916588i \(0.369068\pi\)
\(984\) −13.8564 −0.441726
\(985\) 0 0
\(986\) 25.8564 0.823436
\(987\) −16.7846 −0.534260
\(988\) −2.53590 −0.0806777
\(989\) 69.9615 2.22465
\(990\) 0 0
\(991\) 3.21539 0.102140 0.0510701 0.998695i \(-0.483737\pi\)
0.0510701 + 0.998695i \(0.483737\pi\)
\(992\) −1.46410 −0.0464853
\(993\) 37.9615 1.20467
\(994\) −2.92820 −0.0928770
\(995\) 0 0
\(996\) 11.3205 0.358704
\(997\) −39.7128 −1.25772 −0.628859 0.777520i \(-0.716478\pi\)
−0.628859 + 0.777520i \(0.716478\pi\)
\(998\) −3.66025 −0.115863
\(999\) −21.0718 −0.666682
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 6034.2.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6034.2.a.i.1.2 2 1.1 even 1 trivial