Properties

Label 3234.2.a.bj.1.4
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $4$
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] = [3234,2,Mod(1,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.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: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.14336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 8x^{2} + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.32685\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.29066 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.29066 q^{10} -1.00000 q^{11} -1.00000 q^{12} -6.06791 q^{13} -3.29066 q^{15} +1.00000 q^{16} +6.11908 q^{17} -1.00000 q^{18} -0.0511786 q^{19} +3.29066 q^{20} +1.00000 q^{22} -6.75605 q^{23} +1.00000 q^{24} +5.82843 q^{25} +6.06791 q^{26} -1.00000 q^{27} +2.82843 q^{29} +3.29066 q^{30} -5.87644 q^{31} -1.00000 q^{32} +1.00000 q^{33} -6.11908 q^{34} +1.00000 q^{36} -8.31055 q^{37} +0.0511786 q^{38} +6.06791 q^{39} -3.29066 q^{40} -6.11908 q^{41} +2.90080 q^{43} -1.00000 q^{44} +3.29066 q^{45} +6.75605 q^{46} -1.22275 q^{47} -1.00000 q^{48} -5.82843 q^{50} -6.11908 q^{51} -6.06791 q^{52} +3.00316 q^{53} +1.00000 q^{54} -3.29066 q^{55} +0.0511786 q^{57} -2.82843 q^{58} -10.8284 q^{59} -3.29066 q^{60} -7.23948 q^{61} +5.87644 q^{62} +1.00000 q^{64} -19.9674 q^{65} -1.00000 q^{66} -1.68051 q^{67} +6.11908 q^{68} +6.75605 q^{69} -6.47896 q^{71} -1.00000 q^{72} +11.6736 q^{73} +8.31055 q^{74} -5.82843 q^{75} -0.0511786 q^{76} -6.06791 q^{78} -9.13897 q^{79} +3.29066 q^{80} +1.00000 q^{81} +6.11908 q^{82} -0.951983 q^{83} +20.1358 q^{85} -2.90080 q^{86} -2.82843 q^{87} +1.00000 q^{88} -16.5469 q^{89} -3.29066 q^{90} -6.75605 q^{92} +5.87644 q^{93} +1.22275 q^{94} -0.168411 q^{95} +1.00000 q^{96} +14.7216 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{11} - 4 q^{12} + 4 q^{16} - 4 q^{18} + 4 q^{22} - 8 q^{23} + 4 q^{24} + 12 q^{25} - 4 q^{27} - 16 q^{31} - 4 q^{32} + 4 q^{33} + 4 q^{36} + 8 q^{37} + 8 q^{43} - 4 q^{44} + 8 q^{46} - 16 q^{47} - 4 q^{48} - 12 q^{50} + 8 q^{53} + 4 q^{54} - 32 q^{59} - 16 q^{61} + 16 q^{62} + 4 q^{64} - 16 q^{65} - 4 q^{66} + 16 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{74} - 12 q^{75} + 16 q^{79} + 4 q^{81} + 32 q^{85} - 8 q^{86} + 4 q^{88} - 16 q^{89} - 8 q^{92} + 16 q^{93} + 16 q^{94} - 16 q^{95} + 4 q^{96} + 16 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 3.29066 1.47163 0.735813 0.677184i \(-0.236800\pi\)
0.735813 + 0.677184i \(0.236800\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.29066 −1.04060
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) −6.06791 −1.68293 −0.841467 0.540308i \(-0.818308\pi\)
−0.841467 + 0.540308i \(0.818308\pi\)
\(14\) 0 0
\(15\) −3.29066 −0.849644
\(16\) 1.00000 0.250000
\(17\) 6.11908 1.48410 0.742048 0.670347i \(-0.233855\pi\)
0.742048 + 0.670347i \(0.233855\pi\)
\(18\) −1.00000 −0.235702
\(19\) −0.0511786 −0.0117412 −0.00587059 0.999983i \(-0.501869\pi\)
−0.00587059 + 0.999983i \(0.501869\pi\)
\(20\) 3.29066 0.735813
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −6.75605 −1.40873 −0.704367 0.709836i \(-0.748769\pi\)
−0.704367 + 0.709836i \(0.748769\pi\)
\(24\) 1.00000 0.204124
\(25\) 5.82843 1.16569
\(26\) 6.06791 1.19001
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 3.29066 0.600789
\(31\) −5.87644 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) −6.11908 −1.04941
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −8.31055 −1.36625 −0.683123 0.730304i \(-0.739379\pi\)
−0.683123 + 0.730304i \(0.739379\pi\)
\(38\) 0.0511786 0.00830226
\(39\) 6.06791 0.971643
\(40\) −3.29066 −0.520299
\(41\) −6.11908 −0.955640 −0.477820 0.878458i \(-0.658573\pi\)
−0.477820 + 0.878458i \(0.658573\pi\)
\(42\) 0 0
\(43\) 2.90080 0.442369 0.221184 0.975232i \(-0.429008\pi\)
0.221184 + 0.975232i \(0.429008\pi\)
\(44\) −1.00000 −0.150756
\(45\) 3.29066 0.490542
\(46\) 6.75605 0.996125
\(47\) −1.22275 −0.178357 −0.0891783 0.996016i \(-0.528424\pi\)
−0.0891783 + 0.996016i \(0.528424\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −5.82843 −0.824264
\(51\) −6.11908 −0.856843
\(52\) −6.06791 −0.841467
\(53\) 3.00316 0.412516 0.206258 0.978498i \(-0.433871\pi\)
0.206258 + 0.978498i \(0.433871\pi\)
\(54\) 1.00000 0.136083
\(55\) −3.29066 −0.443712
\(56\) 0 0
\(57\) 0.0511786 0.00677877
\(58\) −2.82843 −0.371391
\(59\) −10.8284 −1.40974 −0.704871 0.709336i \(-0.748995\pi\)
−0.704871 + 0.709336i \(0.748995\pi\)
\(60\) −3.29066 −0.424822
\(61\) −7.23948 −0.926920 −0.463460 0.886118i \(-0.653392\pi\)
−0.463460 + 0.886118i \(0.653392\pi\)
\(62\) 5.87644 0.746309
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −19.9674 −2.47665
\(66\) −1.00000 −0.123091
\(67\) −1.68051 −0.205307 −0.102654 0.994717i \(-0.532733\pi\)
−0.102654 + 0.994717i \(0.532733\pi\)
\(68\) 6.11908 0.742048
\(69\) 6.75605 0.813333
\(70\) 0 0
\(71\) −6.47896 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(72\) −1.00000 −0.117851
\(73\) 11.6736 1.36629 0.683145 0.730283i \(-0.260612\pi\)
0.683145 + 0.730283i \(0.260612\pi\)
\(74\) 8.31055 0.966081
\(75\) −5.82843 −0.673009
\(76\) −0.0511786 −0.00587059
\(77\) 0 0
\(78\) −6.06791 −0.687055
\(79\) −9.13897 −1.02821 −0.514107 0.857726i \(-0.671877\pi\)
−0.514107 + 0.857726i \(0.671877\pi\)
\(80\) 3.29066 0.367907
\(81\) 1.00000 0.111111
\(82\) 6.11908 0.675740
\(83\) −0.951983 −0.104494 −0.0522469 0.998634i \(-0.516638\pi\)
−0.0522469 + 0.998634i \(0.516638\pi\)
\(84\) 0 0
\(85\) 20.1358 2.18404
\(86\) −2.90080 −0.312802
\(87\) −2.82843 −0.303239
\(88\) 1.00000 0.106600
\(89\) −16.5469 −1.75396 −0.876982 0.480523i \(-0.840447\pi\)
−0.876982 + 0.480523i \(0.840447\pi\)
\(90\) −3.29066 −0.346866
\(91\) 0 0
\(92\) −6.75605 −0.704367
\(93\) 5.87644 0.609359
\(94\) 1.22275 0.126117
\(95\) −0.168411 −0.0172786
\(96\) 1.00000 0.102062
\(97\) 14.7216 1.49475 0.747376 0.664401i \(-0.231313\pi\)
0.747376 + 0.664401i \(0.231313\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 5.82843 0.582843
\(101\) −5.51657 −0.548919 −0.274460 0.961599i \(-0.588499\pi\)
−0.274460 + 0.961599i \(0.588499\pi\)
\(102\) 6.11908 0.605880
\(103\) −11.6781 −1.15067 −0.575336 0.817917i \(-0.695129\pi\)
−0.575336 + 0.817917i \(0.695129\pi\)
\(104\) 6.06791 0.595007
\(105\) 0 0
\(106\) −3.00316 −0.291693
\(107\) 15.9611 1.54302 0.771508 0.636220i \(-0.219503\pi\)
0.771508 + 0.636220i \(0.219503\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −5.65053 −0.541223 −0.270611 0.962689i \(-0.587226\pi\)
−0.270611 + 0.962689i \(0.587226\pi\)
\(110\) 3.29066 0.313752
\(111\) 8.31055 0.788802
\(112\) 0 0
\(113\) −0.247112 −0.0232463 −0.0116232 0.999932i \(-0.503700\pi\)
−0.0116232 + 0.999932i \(0.503700\pi\)
\(114\) −0.0511786 −0.00479331
\(115\) −22.2318 −2.07313
\(116\) 2.82843 0.262613
\(117\) −6.06791 −0.560978
\(118\) 10.8284 0.996838
\(119\) 0 0
\(120\) 3.29066 0.300395
\(121\) 1.00000 0.0909091
\(122\) 7.23948 0.655432
\(123\) 6.11908 0.551739
\(124\) −5.87644 −0.527720
\(125\) 2.72607 0.243827
\(126\) 0 0
\(127\) 13.3310 1.18294 0.591469 0.806328i \(-0.298548\pi\)
0.591469 + 0.806328i \(0.298548\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.90080 −0.255402
\(130\) 19.9674 1.75126
\(131\) −3.04802 −0.266306 −0.133153 0.991095i \(-0.542510\pi\)
−0.133153 + 0.991095i \(0.542510\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 1.68051 0.145174
\(135\) −3.29066 −0.283215
\(136\) −6.11908 −0.524707
\(137\) 8.47896 0.724406 0.362203 0.932099i \(-0.382025\pi\)
0.362203 + 0.932099i \(0.382025\pi\)
\(138\) −6.75605 −0.575113
\(139\) 3.80407 0.322657 0.161328 0.986901i \(-0.448422\pi\)
0.161328 + 0.986901i \(0.448422\pi\)
\(140\) 0 0
\(141\) 1.22275 0.102974
\(142\) 6.47896 0.543702
\(143\) 6.06791 0.507424
\(144\) 1.00000 0.0833333
\(145\) 9.30739 0.772936
\(146\) −11.6736 −0.966113
\(147\) 0 0
\(148\) −8.31055 −0.683123
\(149\) 24.1269 1.97655 0.988275 0.152684i \(-0.0487916\pi\)
0.988275 + 0.152684i \(0.0487916\pi\)
\(150\) 5.82843 0.475889
\(151\) 7.85525 0.639251 0.319625 0.947544i \(-0.396443\pi\)
0.319625 + 0.947544i \(0.396443\pi\)
\(152\) 0.0511786 0.00415113
\(153\) 6.11908 0.494699
\(154\) 0 0
\(155\) −19.3374 −1.55321
\(156\) 6.06791 0.485821
\(157\) 18.2549 1.45690 0.728450 0.685099i \(-0.240241\pi\)
0.728450 + 0.685099i \(0.240241\pi\)
\(158\) 9.13897 0.727058
\(159\) −3.00316 −0.238166
\(160\) −3.29066 −0.260149
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 2.34315 0.183529 0.0917647 0.995781i \(-0.470749\pi\)
0.0917647 + 0.995781i \(0.470749\pi\)
\(164\) −6.11908 −0.477820
\(165\) 3.29066 0.256177
\(166\) 0.951983 0.0738882
\(167\) −19.7863 −1.53111 −0.765557 0.643369i \(-0.777536\pi\)
−0.765557 + 0.643369i \(0.777536\pi\)
\(168\) 0 0
\(169\) 23.8195 1.83227
\(170\) −20.1358 −1.54435
\(171\) −0.0511786 −0.00391372
\(172\) 2.90080 0.221184
\(173\) −10.5768 −0.804143 −0.402071 0.915608i \(-0.631710\pi\)
−0.402071 + 0.915608i \(0.631710\pi\)
\(174\) 2.82843 0.214423
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 10.8284 0.813914
\(178\) 16.5469 1.24024
\(179\) 18.3016 1.36793 0.683963 0.729517i \(-0.260255\pi\)
0.683963 + 0.729517i \(0.260255\pi\)
\(180\) 3.29066 0.245271
\(181\) −25.0499 −1.86194 −0.930971 0.365093i \(-0.881037\pi\)
−0.930971 + 0.365093i \(0.881037\pi\)
\(182\) 0 0
\(183\) 7.23948 0.535158
\(184\) 6.75605 0.498063
\(185\) −27.3472 −2.01060
\(186\) −5.87644 −0.430882
\(187\) −6.11908 −0.447472
\(188\) −1.22275 −0.0891783
\(189\) 0 0
\(190\) 0.168411 0.0122178
\(191\) −21.8650 −1.58210 −0.791050 0.611752i \(-0.790465\pi\)
−0.791050 + 0.611752i \(0.790465\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 20.8195 1.49862 0.749310 0.662220i \(-0.230385\pi\)
0.749310 + 0.662220i \(0.230385\pi\)
\(194\) −14.7216 −1.05695
\(195\) 19.9674 1.42990
\(196\) 0 0
\(197\) −20.4790 −1.45907 −0.729533 0.683946i \(-0.760262\pi\)
−0.729533 + 0.683946i \(0.760262\pi\)
\(198\) 1.00000 0.0710669
\(199\) −15.5333 −1.10113 −0.550563 0.834794i \(-0.685587\pi\)
−0.550563 + 0.834794i \(0.685587\pi\)
\(200\) −5.82843 −0.412132
\(201\) 1.68051 0.118534
\(202\) 5.51657 0.388145
\(203\) 0 0
\(204\) −6.11908 −0.428422
\(205\) −20.1358 −1.40635
\(206\) 11.6781 0.813649
\(207\) −6.75605 −0.469578
\(208\) −6.06791 −0.420734
\(209\) 0.0511786 0.00354010
\(210\) 0 0
\(211\) −6.40658 −0.441047 −0.220524 0.975382i \(-0.570777\pi\)
−0.220524 + 0.975382i \(0.570777\pi\)
\(212\) 3.00316 0.206258
\(213\) 6.47896 0.443931
\(214\) −15.9611 −1.09108
\(215\) 9.54555 0.651001
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 5.65053 0.382702
\(219\) −11.6736 −0.788828
\(220\) −3.29066 −0.221856
\(221\) −37.1300 −2.49764
\(222\) −8.31055 −0.557767
\(223\) −28.4912 −1.90791 −0.953956 0.299945i \(-0.903032\pi\)
−0.953956 + 0.299945i \(0.903032\pi\)
\(224\) 0 0
\(225\) 5.82843 0.388562
\(226\) 0.247112 0.0164376
\(227\) 2.90326 0.192696 0.0963481 0.995348i \(-0.469284\pi\)
0.0963481 + 0.995348i \(0.469284\pi\)
\(228\) 0.0511786 0.00338938
\(229\) −6.11908 −0.404360 −0.202180 0.979348i \(-0.564803\pi\)
−0.202180 + 0.979348i \(0.564803\pi\)
\(230\) 22.2318 1.46592
\(231\) 0 0
\(232\) −2.82843 −0.185695
\(233\) −11.6569 −0.763666 −0.381833 0.924231i \(-0.624707\pi\)
−0.381833 + 0.924231i \(0.624707\pi\)
\(234\) 6.06791 0.396671
\(235\) −4.02366 −0.262474
\(236\) −10.8284 −0.704871
\(237\) 9.13897 0.593640
\(238\) 0 0
\(239\) −21.3074 −1.37826 −0.689130 0.724638i \(-0.742007\pi\)
−0.689130 + 0.724638i \(0.742007\pi\)
\(240\) −3.29066 −0.212411
\(241\) 11.8783 0.765148 0.382574 0.923925i \(-0.375038\pi\)
0.382574 + 0.923925i \(0.375038\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) −7.23948 −0.463460
\(245\) 0 0
\(246\) −6.11908 −0.390139
\(247\) 0.310547 0.0197596
\(248\) 5.87644 0.373155
\(249\) 0.951983 0.0603295
\(250\) −2.72607 −0.172412
\(251\) 7.21135 0.455176 0.227588 0.973757i \(-0.426916\pi\)
0.227588 + 0.973757i \(0.426916\pi\)
\(252\) 0 0
\(253\) 6.75605 0.424749
\(254\) −13.3310 −0.836464
\(255\) −20.1358 −1.26095
\(256\) 1.00000 0.0625000
\(257\) −9.08840 −0.566919 −0.283459 0.958984i \(-0.591482\pi\)
−0.283459 + 0.958984i \(0.591482\pi\)
\(258\) 2.90080 0.180596
\(259\) 0 0
\(260\) −19.9674 −1.23833
\(261\) 2.82843 0.175075
\(262\) 3.04802 0.188307
\(263\) 5.28373 0.325809 0.162904 0.986642i \(-0.447914\pi\)
0.162904 + 0.986642i \(0.447914\pi\)
\(264\) −1.00000 −0.0615457
\(265\) 9.88238 0.607070
\(266\) 0 0
\(267\) 16.5469 1.01265
\(268\) −1.68051 −0.102654
\(269\) −20.2151 −1.23254 −0.616269 0.787536i \(-0.711356\pi\)
−0.616269 + 0.787536i \(0.711356\pi\)
\(270\) 3.29066 0.200263
\(271\) −11.1779 −0.679009 −0.339504 0.940604i \(-0.610259\pi\)
−0.339504 + 0.940604i \(0.610259\pi\)
\(272\) 6.11908 0.371024
\(273\) 0 0
\(274\) −8.47896 −0.512233
\(275\) −5.82843 −0.351467
\(276\) 6.75605 0.406666
\(277\) 25.7376 1.54642 0.773212 0.634148i \(-0.218649\pi\)
0.773212 + 0.634148i \(0.218649\pi\)
\(278\) −3.80407 −0.228153
\(279\) −5.87644 −0.351813
\(280\) 0 0
\(281\) −12.0063 −0.716237 −0.358119 0.933676i \(-0.616582\pi\)
−0.358119 + 0.933676i \(0.616582\pi\)
\(282\) −1.22275 −0.0728138
\(283\) 28.7746 1.71047 0.855237 0.518237i \(-0.173411\pi\)
0.855237 + 0.518237i \(0.173411\pi\)
\(284\) −6.47896 −0.384455
\(285\) 0.168411 0.00997582
\(286\) −6.06791 −0.358803
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 20.4432 1.20254
\(290\) −9.30739 −0.546548
\(291\) −14.7216 −0.862995
\(292\) 11.6736 0.683145
\(293\) −20.2337 −1.18207 −0.591033 0.806648i \(-0.701280\pi\)
−0.591033 + 0.806648i \(0.701280\pi\)
\(294\) 0 0
\(295\) −35.6326 −2.07461
\(296\) 8.31055 0.483041
\(297\) 1.00000 0.0580259
\(298\) −24.1269 −1.39763
\(299\) 40.9951 2.37081
\(300\) −5.82843 −0.336504
\(301\) 0 0
\(302\) −7.85525 −0.452019
\(303\) 5.51657 0.316919
\(304\) −0.0511786 −0.00293529
\(305\) −23.8226 −1.36408
\(306\) −6.11908 −0.349805
\(307\) 17.5033 0.998967 0.499484 0.866323i \(-0.333523\pi\)
0.499484 + 0.866323i \(0.333523\pi\)
\(308\) 0 0
\(309\) 11.6781 0.664341
\(310\) 19.3374 1.09829
\(311\) −30.0486 −1.70390 −0.851949 0.523625i \(-0.824579\pi\)
−0.851949 + 0.523625i \(0.824579\pi\)
\(312\) −6.06791 −0.343528
\(313\) −24.6256 −1.39192 −0.695960 0.718081i \(-0.745021\pi\)
−0.695960 + 0.718081i \(0.745021\pi\)
\(314\) −18.2549 −1.03018
\(315\) 0 0
\(316\) −9.13897 −0.514107
\(317\) 11.1479 0.626129 0.313065 0.949732i \(-0.398644\pi\)
0.313065 + 0.949732i \(0.398644\pi\)
\(318\) 3.00316 0.168409
\(319\) −2.82843 −0.158362
\(320\) 3.29066 0.183953
\(321\) −15.9611 −0.890860
\(322\) 0 0
\(323\) −0.313166 −0.0174250
\(324\) 1.00000 0.0555556
\(325\) −35.3663 −1.96177
\(326\) −2.34315 −0.129775
\(327\) 5.65053 0.312475
\(328\) 6.11908 0.337870
\(329\) 0 0
\(330\) −3.29066 −0.181145
\(331\) −4.16841 −0.229117 −0.114558 0.993417i \(-0.536545\pi\)
−0.114558 + 0.993417i \(0.536545\pi\)
\(332\) −0.951983 −0.0522469
\(333\) −8.31055 −0.455415
\(334\) 19.7863 1.08266
\(335\) −5.52998 −0.302135
\(336\) 0 0
\(337\) −15.8163 −0.861570 −0.430785 0.902455i \(-0.641763\pi\)
−0.430785 + 0.902455i \(0.641763\pi\)
\(338\) −23.8195 −1.29561
\(339\) 0.247112 0.0134213
\(340\) 20.1358 1.09202
\(341\) 5.87644 0.318227
\(342\) 0.0511786 0.00276742
\(343\) 0 0
\(344\) −2.90080 −0.156401
\(345\) 22.2318 1.19692
\(346\) 10.5768 0.568615
\(347\) −16.1058 −0.864606 −0.432303 0.901728i \(-0.642299\pi\)
−0.432303 + 0.901728i \(0.642299\pi\)
\(348\) −2.82843 −0.151620
\(349\) 9.19076 0.491970 0.245985 0.969274i \(-0.420889\pi\)
0.245985 + 0.969274i \(0.420889\pi\)
\(350\) 0 0
\(351\) 6.06791 0.323881
\(352\) 1.00000 0.0533002
\(353\) −24.8874 −1.32462 −0.662311 0.749229i \(-0.730424\pi\)
−0.662311 + 0.749229i \(0.730424\pi\)
\(354\) −10.8284 −0.575524
\(355\) −21.3200 −1.13155
\(356\) −16.5469 −0.876982
\(357\) 0 0
\(358\) −18.3016 −0.967270
\(359\) −6.37945 −0.336694 −0.168347 0.985728i \(-0.553843\pi\)
−0.168347 + 0.985728i \(0.553843\pi\)
\(360\) −3.29066 −0.173433
\(361\) −18.9974 −0.999862
\(362\) 25.0499 1.31659
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 38.4138 2.01067
\(366\) −7.23948 −0.378414
\(367\) −6.26831 −0.327203 −0.163602 0.986526i \(-0.552311\pi\)
−0.163602 + 0.986526i \(0.552311\pi\)
\(368\) −6.75605 −0.352183
\(369\) −6.11908 −0.318547
\(370\) 27.3472 1.42171
\(371\) 0 0
\(372\) 5.87644 0.304679
\(373\) −10.8348 −0.561002 −0.280501 0.959854i \(-0.590501\pi\)
−0.280501 + 0.959854i \(0.590501\pi\)
\(374\) 6.11908 0.316410
\(375\) −2.72607 −0.140774
\(376\) 1.22275 0.0630586
\(377\) −17.1626 −0.883920
\(378\) 0 0
\(379\) 12.7658 0.655738 0.327869 0.944723i \(-0.393670\pi\)
0.327869 + 0.944723i \(0.393670\pi\)
\(380\) −0.168411 −0.00863931
\(381\) −13.3310 −0.682970
\(382\) 21.8650 1.11871
\(383\) 32.6633 1.66902 0.834509 0.550994i \(-0.185751\pi\)
0.834509 + 0.550994i \(0.185751\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −20.8195 −1.05968
\(387\) 2.90080 0.147456
\(388\) 14.7216 0.747376
\(389\) 31.0179 1.57267 0.786334 0.617801i \(-0.211976\pi\)
0.786334 + 0.617801i \(0.211976\pi\)
\(390\) −19.9674 −1.01109
\(391\) −41.3408 −2.09070
\(392\) 0 0
\(393\) 3.04802 0.153752
\(394\) 20.4790 1.03171
\(395\) −30.0732 −1.51315
\(396\) −1.00000 −0.0502519
\(397\) 15.0986 0.757777 0.378888 0.925442i \(-0.376306\pi\)
0.378888 + 0.925442i \(0.376306\pi\)
\(398\) 15.5333 0.778614
\(399\) 0 0
\(400\) 5.82843 0.291421
\(401\) −9.16525 −0.457691 −0.228845 0.973463i \(-0.573495\pi\)
−0.228845 + 0.973463i \(0.573495\pi\)
\(402\) −1.68051 −0.0838162
\(403\) 35.6577 1.77624
\(404\) −5.51657 −0.274460
\(405\) 3.29066 0.163514
\(406\) 0 0
\(407\) 8.31055 0.411939
\(408\) 6.11908 0.302940
\(409\) 9.32149 0.460918 0.230459 0.973082i \(-0.425977\pi\)
0.230459 + 0.973082i \(0.425977\pi\)
\(410\) 20.1358 0.994437
\(411\) −8.47896 −0.418236
\(412\) −11.6781 −0.575336
\(413\) 0 0
\(414\) 6.75605 0.332042
\(415\) −3.13265 −0.153776
\(416\) 6.06791 0.297504
\(417\) −3.80407 −0.186286
\(418\) −0.0511786 −0.00250323
\(419\) 13.5456 0.661744 0.330872 0.943676i \(-0.392657\pi\)
0.330872 + 0.943676i \(0.392657\pi\)
\(420\) 0 0
\(421\) −9.10899 −0.443945 −0.221973 0.975053i \(-0.571250\pi\)
−0.221973 + 0.975053i \(0.571250\pi\)
\(422\) 6.40658 0.311867
\(423\) −1.22275 −0.0594522
\(424\) −3.00316 −0.145846
\(425\) 35.6646 1.72999
\(426\) −6.47896 −0.313907
\(427\) 0 0
\(428\) 15.9611 0.771508
\(429\) −6.06791 −0.292961
\(430\) −9.54555 −0.460328
\(431\) 4.02366 0.193813 0.0969064 0.995294i \(-0.469105\pi\)
0.0969064 + 0.995294i \(0.469105\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 8.09789 0.389160 0.194580 0.980887i \(-0.437666\pi\)
0.194580 + 0.980887i \(0.437666\pi\)
\(434\) 0 0
\(435\) −9.30739 −0.446255
\(436\) −5.65053 −0.270611
\(437\) 0.345765 0.0165402
\(438\) 11.6736 0.557785
\(439\) 6.82843 0.325903 0.162952 0.986634i \(-0.447899\pi\)
0.162952 + 0.986634i \(0.447899\pi\)
\(440\) 3.29066 0.156876
\(441\) 0 0
\(442\) 37.1300 1.76610
\(443\) −1.62055 −0.0769947 −0.0384974 0.999259i \(-0.512257\pi\)
−0.0384974 + 0.999259i \(0.512257\pi\)
\(444\) 8.31055 0.394401
\(445\) −54.4501 −2.58118
\(446\) 28.4912 1.34910
\(447\) −24.1269 −1.14116
\(448\) 0 0
\(449\) 30.1756 1.42407 0.712037 0.702142i \(-0.247773\pi\)
0.712037 + 0.702142i \(0.247773\pi\)
\(450\) −5.82843 −0.274755
\(451\) 6.11908 0.288136
\(452\) −0.247112 −0.0116232
\(453\) −7.85525 −0.369072
\(454\) −2.90326 −0.136257
\(455\) 0 0
\(456\) −0.0511786 −0.00239666
\(457\) 11.3200 0.529529 0.264764 0.964313i \(-0.414706\pi\)
0.264764 + 0.964313i \(0.414706\pi\)
\(458\) 6.11908 0.285926
\(459\) −6.11908 −0.285614
\(460\) −22.2318 −1.03657
\(461\) 7.51025 0.349787 0.174894 0.984587i \(-0.444042\pi\)
0.174894 + 0.984587i \(0.444042\pi\)
\(462\) 0 0
\(463\) −3.10899 −0.144487 −0.0722436 0.997387i \(-0.523016\pi\)
−0.0722436 + 0.997387i \(0.523016\pi\)
\(464\) 2.82843 0.131306
\(465\) 19.3374 0.896749
\(466\) 11.6569 0.539993
\(467\) −36.3189 −1.68064 −0.840320 0.542091i \(-0.817633\pi\)
−0.840320 + 0.542091i \(0.817633\pi\)
\(468\) −6.06791 −0.280489
\(469\) 0 0
\(470\) 4.02366 0.185597
\(471\) −18.2549 −0.841141
\(472\) 10.8284 0.498419
\(473\) −2.90080 −0.133379
\(474\) −9.13897 −0.419767
\(475\) −0.298291 −0.0136865
\(476\) 0 0
\(477\) 3.00316 0.137505
\(478\) 21.3074 0.974577
\(479\) 30.5035 1.39374 0.696870 0.717198i \(-0.254576\pi\)
0.696870 + 0.717198i \(0.254576\pi\)
\(480\) 3.29066 0.150197
\(481\) 50.4276 2.29930
\(482\) −11.8783 −0.541042
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 48.4437 2.19972
\(486\) 1.00000 0.0453609
\(487\) −12.6001 −0.570963 −0.285482 0.958384i \(-0.592154\pi\)
−0.285482 + 0.958384i \(0.592154\pi\)
\(488\) 7.23948 0.327716
\(489\) −2.34315 −0.105961
\(490\) 0 0
\(491\) −31.3137 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(492\) 6.11908 0.275870
\(493\) 17.3074 0.779485
\(494\) −0.310547 −0.0139722
\(495\) −3.29066 −0.147904
\(496\) −5.87644 −0.263860
\(497\) 0 0
\(498\) −0.951983 −0.0426594
\(499\) −3.49477 −0.156447 −0.0782236 0.996936i \(-0.524925\pi\)
−0.0782236 + 0.996936i \(0.524925\pi\)
\(500\) 2.72607 0.121914
\(501\) 19.7863 0.883989
\(502\) −7.21135 −0.321858
\(503\) −20.2627 −0.903468 −0.451734 0.892153i \(-0.649194\pi\)
−0.451734 + 0.892153i \(0.649194\pi\)
\(504\) 0 0
\(505\) −18.1531 −0.807804
\(506\) −6.75605 −0.300343
\(507\) −23.8195 −1.05786
\(508\) 13.3310 0.591469
\(509\) 18.4123 0.816111 0.408055 0.912957i \(-0.366207\pi\)
0.408055 + 0.912957i \(0.366207\pi\)
\(510\) 20.1358 0.891629
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0.0511786 0.00225959
\(514\) 9.08840 0.400872
\(515\) −38.4285 −1.69336
\(516\) −2.90080 −0.127701
\(517\) 1.22275 0.0537765
\(518\) 0 0
\(519\) 10.5768 0.464272
\(520\) 19.9674 0.875628
\(521\) 24.0053 1.05169 0.525846 0.850580i \(-0.323749\pi\)
0.525846 + 0.850580i \(0.323749\pi\)
\(522\) −2.82843 −0.123797
\(523\) −39.3894 −1.72238 −0.861189 0.508285i \(-0.830280\pi\)
−0.861189 + 0.508285i \(0.830280\pi\)
\(524\) −3.04802 −0.133153
\(525\) 0 0
\(526\) −5.28373 −0.230382
\(527\) −35.9585 −1.56638
\(528\) 1.00000 0.0435194
\(529\) 22.6442 0.984531
\(530\) −9.88238 −0.429263
\(531\) −10.8284 −0.469914
\(532\) 0 0
\(533\) 37.1300 1.60828
\(534\) −16.5469 −0.716053
\(535\) 52.5224 2.27074
\(536\) 1.68051 0.0725870
\(537\) −18.3016 −0.789773
\(538\) 20.2151 0.871536
\(539\) 0 0
\(540\) −3.29066 −0.141607
\(541\) 35.1064 1.50934 0.754670 0.656104i \(-0.227797\pi\)
0.754670 + 0.656104i \(0.227797\pi\)
\(542\) 11.1779 0.480132
\(543\) 25.0499 1.07499
\(544\) −6.11908 −0.262354
\(545\) −18.5940 −0.796478
\(546\) 0 0
\(547\) 30.3342 1.29700 0.648498 0.761216i \(-0.275397\pi\)
0.648498 + 0.761216i \(0.275397\pi\)
\(548\) 8.47896 0.362203
\(549\) −7.23948 −0.308973
\(550\) 5.82843 0.248525
\(551\) −0.144755 −0.00616676
\(552\) −6.75605 −0.287557
\(553\) 0 0
\(554\) −25.7376 −1.09349
\(555\) 27.3472 1.16082
\(556\) 3.80407 0.161328
\(557\) −36.0643 −1.52809 −0.764047 0.645161i \(-0.776790\pi\)
−0.764047 + 0.645161i \(0.776790\pi\)
\(558\) 5.87644 0.248770
\(559\) −17.6018 −0.744477
\(560\) 0 0
\(561\) 6.11908 0.258348
\(562\) 12.0063 0.506456
\(563\) −38.7908 −1.63484 −0.817418 0.576046i \(-0.804595\pi\)
−0.817418 + 0.576046i \(0.804595\pi\)
\(564\) 1.22275 0.0514871
\(565\) −0.813161 −0.0342099
\(566\) −28.7746 −1.20949
\(567\) 0 0
\(568\) 6.47896 0.271851
\(569\) −12.4101 −0.520257 −0.260128 0.965574i \(-0.583765\pi\)
−0.260128 + 0.965574i \(0.583765\pi\)
\(570\) −0.168411 −0.00705397
\(571\) −35.7863 −1.49761 −0.748806 0.662789i \(-0.769372\pi\)
−0.748806 + 0.662789i \(0.769372\pi\)
\(572\) 6.06791 0.253712
\(573\) 21.8650 0.913425
\(574\) 0 0
\(575\) −39.3771 −1.64214
\(576\) 1.00000 0.0416667
\(577\) 24.1313 1.00460 0.502300 0.864693i \(-0.332487\pi\)
0.502300 + 0.864693i \(0.332487\pi\)
\(578\) −20.4432 −0.850325
\(579\) −20.8195 −0.865228
\(580\) 9.30739 0.386468
\(581\) 0 0
\(582\) 14.7216 0.610230
\(583\) −3.00316 −0.124378
\(584\) −11.6736 −0.483056
\(585\) −19.9674 −0.825550
\(586\) 20.2337 0.835846
\(587\) 31.0539 1.28173 0.640867 0.767652i \(-0.278575\pi\)
0.640867 + 0.767652i \(0.278575\pi\)
\(588\) 0 0
\(589\) 0.300748 0.0123921
\(590\) 35.6326 1.46697
\(591\) 20.4790 0.842392
\(592\) −8.31055 −0.341561
\(593\) 23.5378 0.966580 0.483290 0.875460i \(-0.339442\pi\)
0.483290 + 0.875460i \(0.339442\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 24.1269 0.988275
\(597\) 15.5333 0.635736
\(598\) −40.9951 −1.67641
\(599\) −20.0037 −0.817329 −0.408665 0.912685i \(-0.634005\pi\)
−0.408665 + 0.912685i \(0.634005\pi\)
\(600\) 5.82843 0.237945
\(601\) 0.799053 0.0325940 0.0162970 0.999867i \(-0.494812\pi\)
0.0162970 + 0.999867i \(0.494812\pi\)
\(602\) 0 0
\(603\) −1.68051 −0.0684357
\(604\) 7.85525 0.319625
\(605\) 3.29066 0.133784
\(606\) −5.51657 −0.224095
\(607\) −21.3200 −0.865353 −0.432677 0.901549i \(-0.642431\pi\)
−0.432677 + 0.901549i \(0.642431\pi\)
\(608\) 0.0511786 0.00207557
\(609\) 0 0
\(610\) 23.8226 0.964551
\(611\) 7.41954 0.300162
\(612\) 6.11908 0.247349
\(613\) −8.37398 −0.338222 −0.169111 0.985597i \(-0.554090\pi\)
−0.169111 + 0.985597i \(0.554090\pi\)
\(614\) −17.5033 −0.706376
\(615\) 20.1358 0.811954
\(616\) 0 0
\(617\) 3.74395 0.150726 0.0753628 0.997156i \(-0.475989\pi\)
0.0753628 + 0.997156i \(0.475989\pi\)
\(618\) −11.6781 −0.469760
\(619\) −0.924461 −0.0371572 −0.0185786 0.999827i \(-0.505914\pi\)
−0.0185786 + 0.999827i \(0.505914\pi\)
\(620\) −19.3374 −0.776607
\(621\) 6.75605 0.271111
\(622\) 30.0486 1.20484
\(623\) 0 0
\(624\) 6.06791 0.242911
\(625\) −20.1716 −0.806863
\(626\) 24.6256 0.984236
\(627\) −0.0511786 −0.00204388
\(628\) 18.2549 0.728450
\(629\) −50.8529 −2.02764
\(630\) 0 0
\(631\) 40.4374 1.60979 0.804894 0.593419i \(-0.202222\pi\)
0.804894 + 0.593419i \(0.202222\pi\)
\(632\) 9.13897 0.363529
\(633\) 6.40658 0.254639
\(634\) −11.1479 −0.442740
\(635\) 43.8679 1.74084
\(636\) −3.00316 −0.119083
\(637\) 0 0
\(638\) 2.82843 0.111979
\(639\) −6.47896 −0.256304
\(640\) −3.29066 −0.130075
\(641\) 13.6777 0.540235 0.270118 0.962827i \(-0.412937\pi\)
0.270118 + 0.962827i \(0.412937\pi\)
\(642\) 15.9611 0.629933
\(643\) 40.2627 1.58781 0.793903 0.608045i \(-0.208046\pi\)
0.793903 + 0.608045i \(0.208046\pi\)
\(644\) 0 0
\(645\) −9.54555 −0.375856
\(646\) 0.313166 0.0123214
\(647\) 6.61985 0.260253 0.130127 0.991497i \(-0.458462\pi\)
0.130127 + 0.991497i \(0.458462\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 10.8284 0.425053
\(650\) 35.3663 1.38718
\(651\) 0 0
\(652\) 2.34315 0.0917647
\(653\) −15.3200 −0.599519 −0.299760 0.954015i \(-0.596906\pi\)
−0.299760 + 0.954015i \(0.596906\pi\)
\(654\) −5.65053 −0.220953
\(655\) −10.0300 −0.391904
\(656\) −6.11908 −0.238910
\(657\) 11.6736 0.455430
\(658\) 0 0
\(659\) 31.7690 1.23754 0.618772 0.785570i \(-0.287630\pi\)
0.618772 + 0.785570i \(0.287630\pi\)
\(660\) 3.29066 0.128089
\(661\) 29.8296 1.16024 0.580118 0.814532i \(-0.303006\pi\)
0.580118 + 0.814532i \(0.303006\pi\)
\(662\) 4.16841 0.162010
\(663\) 37.1300 1.44201
\(664\) 0.951983 0.0369441
\(665\) 0 0
\(666\) 8.31055 0.322027
\(667\) −19.1090 −0.739903
\(668\) −19.7863 −0.765557
\(669\) 28.4912 1.10153
\(670\) 5.52998 0.213642
\(671\) 7.23948 0.279477
\(672\) 0 0
\(673\) 23.7690 0.916228 0.458114 0.888893i \(-0.348525\pi\)
0.458114 + 0.888893i \(0.348525\pi\)
\(674\) 15.8163 0.609222
\(675\) −5.82843 −0.224336
\(676\) 23.8195 0.916134
\(677\) 22.6368 0.870003 0.435002 0.900430i \(-0.356748\pi\)
0.435002 + 0.900430i \(0.356748\pi\)
\(678\) −0.247112 −0.00949028
\(679\) 0 0
\(680\) −20.1358 −0.772173
\(681\) −2.90326 −0.111253
\(682\) −5.87644 −0.225021
\(683\) 6.79583 0.260035 0.130018 0.991512i \(-0.458497\pi\)
0.130018 + 0.991512i \(0.458497\pi\)
\(684\) −0.0511786 −0.00195686
\(685\) 27.9013 1.06606
\(686\) 0 0
\(687\) 6.11908 0.233458
\(688\) 2.90080 0.110592
\(689\) −18.2229 −0.694237
\(690\) −22.2318 −0.846352
\(691\) −16.2292 −0.617389 −0.308694 0.951161i \(-0.599892\pi\)
−0.308694 + 0.951161i \(0.599892\pi\)
\(692\) −10.5768 −0.402071
\(693\) 0 0
\(694\) 16.1058 0.611369
\(695\) 12.5179 0.474830
\(696\) 2.82843 0.107211
\(697\) −37.4432 −1.41826
\(698\) −9.19076 −0.347875
\(699\) 11.6569 0.440903
\(700\) 0 0
\(701\) −32.3253 −1.22091 −0.610454 0.792052i \(-0.709013\pi\)
−0.610454 + 0.792052i \(0.709013\pi\)
\(702\) −6.06791 −0.229018
\(703\) 0.425322 0.0160413
\(704\) −1.00000 −0.0376889
\(705\) 4.02366 0.151540
\(706\) 24.8874 0.936649
\(707\) 0 0
\(708\) 10.8284 0.406957
\(709\) −10.8458 −0.407321 −0.203661 0.979042i \(-0.565284\pi\)
−0.203661 + 0.979042i \(0.565284\pi\)
\(710\) 21.3200 0.800127
\(711\) −9.13897 −0.342738
\(712\) 16.5469 0.620120
\(713\) 39.7015 1.48683
\(714\) 0 0
\(715\) 19.9674 0.746738
\(716\) 18.3016 0.683963
\(717\) 21.3074 0.795739
\(718\) 6.37945 0.238079
\(719\) −3.69277 −0.137717 −0.0688585 0.997626i \(-0.521936\pi\)
−0.0688585 + 0.997626i \(0.521936\pi\)
\(720\) 3.29066 0.122636
\(721\) 0 0
\(722\) 18.9974 0.707009
\(723\) −11.8783 −0.441759
\(724\) −25.0499 −0.930971
\(725\) 16.4853 0.612248
\(726\) 1.00000 0.0371135
\(727\) −34.8470 −1.29240 −0.646202 0.763166i \(-0.723644\pi\)
−0.646202 + 0.763166i \(0.723644\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −38.4138 −1.42176
\(731\) 17.7503 0.656517
\(732\) 7.23948 0.267579
\(733\) 39.9168 1.47436 0.737181 0.675696i \(-0.236157\pi\)
0.737181 + 0.675696i \(0.236157\pi\)
\(734\) 6.26831 0.231368
\(735\) 0 0
\(736\) 6.75605 0.249031
\(737\) 1.68051 0.0619024
\(738\) 6.11908 0.225247
\(739\) 4.77479 0.175644 0.0878218 0.996136i \(-0.472009\pi\)
0.0878218 + 0.996136i \(0.472009\pi\)
\(740\) −27.3472 −1.00530
\(741\) −0.310547 −0.0114082
\(742\) 0 0
\(743\) −25.1153 −0.921392 −0.460696 0.887558i \(-0.652400\pi\)
−0.460696 + 0.887558i \(0.652400\pi\)
\(744\) −5.87644 −0.215441
\(745\) 79.3933 2.90874
\(746\) 10.8348 0.396688
\(747\) −0.951983 −0.0348312
\(748\) −6.11908 −0.223736
\(749\) 0 0
\(750\) 2.72607 0.0995420
\(751\) 54.7154 1.99659 0.998296 0.0583532i \(-0.0185850\pi\)
0.998296 + 0.0583532i \(0.0185850\pi\)
\(752\) −1.22275 −0.0445892
\(753\) −7.21135 −0.262796
\(754\) 17.1626 0.625026
\(755\) 25.8489 0.940739
\(756\) 0 0
\(757\) 16.9316 0.615391 0.307695 0.951485i \(-0.400442\pi\)
0.307695 + 0.951485i \(0.400442\pi\)
\(758\) −12.7658 −0.463676
\(759\) −6.75605 −0.245229
\(760\) 0.168411 0.00610892
\(761\) 4.16781 0.151083 0.0755414 0.997143i \(-0.475931\pi\)
0.0755414 + 0.997143i \(0.475931\pi\)
\(762\) 13.3310 0.482933
\(763\) 0 0
\(764\) −21.8650 −0.791050
\(765\) 20.1358 0.728012
\(766\) −32.6633 −1.18017
\(767\) 65.7059 2.37250
\(768\) −1.00000 −0.0360844
\(769\) −16.8273 −0.606807 −0.303403 0.952862i \(-0.598123\pi\)
−0.303403 + 0.952862i \(0.598123\pi\)
\(770\) 0 0
\(771\) 9.08840 0.327311
\(772\) 20.8195 0.749310
\(773\) 1.60929 0.0578822 0.0289411 0.999581i \(-0.490786\pi\)
0.0289411 + 0.999581i \(0.490786\pi\)
\(774\) −2.90080 −0.104267
\(775\) −34.2504 −1.23031
\(776\) −14.7216 −0.528475
\(777\) 0 0
\(778\) −31.0179 −1.11204
\(779\) 0.313166 0.0112203
\(780\) 19.9674 0.714948
\(781\) 6.47896 0.231835
\(782\) 41.3408 1.47835
\(783\) −2.82843 −0.101080
\(784\) 0 0
\(785\) 60.0706 2.14401
\(786\) −3.04802 −0.108719
\(787\) 32.6123 1.16250 0.581252 0.813724i \(-0.302563\pi\)
0.581252 + 0.813724i \(0.302563\pi\)
\(788\) −20.4790 −0.729533
\(789\) −5.28373 −0.188106
\(790\) 30.0732 1.06996
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 43.9285 1.55995
\(794\) −15.0986 −0.535829
\(795\) −9.88238 −0.350492
\(796\) −15.5333 −0.550563
\(797\) −35.4599 −1.25605 −0.628027 0.778191i \(-0.716137\pi\)
−0.628027 + 0.778191i \(0.716137\pi\)
\(798\) 0 0
\(799\) −7.48212 −0.264698
\(800\) −5.82843 −0.206066
\(801\) −16.5469 −0.584655
\(802\) 9.16525 0.323636
\(803\) −11.6736 −0.411952
\(804\) 1.68051 0.0592670
\(805\) 0 0
\(806\) −35.6577 −1.25599
\(807\) 20.2151 0.711606
\(808\) 5.51657 0.194072
\(809\) −53.3953 −1.87728 −0.938640 0.344899i \(-0.887913\pi\)
−0.938640 + 0.344899i \(0.887913\pi\)
\(810\) −3.29066 −0.115622
\(811\) −0.178048 −0.00625211 −0.00312606 0.999995i \(-0.500995\pi\)
−0.00312606 + 0.999995i \(0.500995\pi\)
\(812\) 0 0
\(813\) 11.1779 0.392026
\(814\) −8.31055 −0.291285
\(815\) 7.71049 0.270087
\(816\) −6.11908 −0.214211
\(817\) −0.148459 −0.00519392
\(818\) −9.32149 −0.325918
\(819\) 0 0
\(820\) −20.1358 −0.703173
\(821\) 4.15740 0.145094 0.0725472 0.997365i \(-0.476887\pi\)
0.0725472 + 0.997365i \(0.476887\pi\)
\(822\) 8.47896 0.295738
\(823\) 0.0210370 0.000733305 0 0.000366653 1.00000i \(-0.499883\pi\)
0.000366653 1.00000i \(0.499883\pi\)
\(824\) 11.6781 0.406824
\(825\) 5.82843 0.202920
\(826\) 0 0
\(827\) −43.7774 −1.52229 −0.761145 0.648582i \(-0.775362\pi\)
−0.761145 + 0.648582i \(0.775362\pi\)
\(828\) −6.75605 −0.234789
\(829\) −32.0602 −1.11350 −0.556749 0.830681i \(-0.687951\pi\)
−0.556749 + 0.830681i \(0.687951\pi\)
\(830\) 3.13265 0.108736
\(831\) −25.7376 −0.892828
\(832\) −6.06791 −0.210367
\(833\) 0 0
\(834\) 3.80407 0.131724
\(835\) −65.1101 −2.25323
\(836\) 0.0511786 0.00177005
\(837\) 5.87644 0.203120
\(838\) −13.5456 −0.467923
\(839\) −6.92200 −0.238974 −0.119487 0.992836i \(-0.538125\pi\)
−0.119487 + 0.992836i \(0.538125\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 9.10899 0.313917
\(843\) 12.0063 0.413520
\(844\) −6.40658 −0.220524
\(845\) 78.3818 2.69641
\(846\) 1.22275 0.0420391
\(847\) 0 0
\(848\) 3.00316 0.103129
\(849\) −28.7746 −0.987543
\(850\) −35.6646 −1.22329
\(851\) 56.1465 1.92468
\(852\) 6.47896 0.221965
\(853\) 27.4727 0.940648 0.470324 0.882494i \(-0.344137\pi\)
0.470324 + 0.882494i \(0.344137\pi\)
\(854\) 0 0
\(855\) −0.168411 −0.00575954
\(856\) −15.9611 −0.545538
\(857\) 30.1949 1.03144 0.515720 0.856757i \(-0.327525\pi\)
0.515720 + 0.856757i \(0.327525\pi\)
\(858\) 6.06791 0.207155
\(859\) −42.2869 −1.44281 −0.721405 0.692513i \(-0.756503\pi\)
−0.721405 + 0.692513i \(0.756503\pi\)
\(860\) 9.54555 0.325501
\(861\) 0 0
\(862\) −4.02366 −0.137046
\(863\) 45.0340 1.53298 0.766488 0.642259i \(-0.222003\pi\)
0.766488 + 0.642259i \(0.222003\pi\)
\(864\) 1.00000 0.0340207
\(865\) −34.8048 −1.18340
\(866\) −8.09789 −0.275177
\(867\) −20.4432 −0.694287
\(868\) 0 0
\(869\) 9.13897 0.310018
\(870\) 9.30739 0.315550
\(871\) 10.1972 0.345518
\(872\) 5.65053 0.191351
\(873\) 14.7216 0.498251
\(874\) −0.345765 −0.0116957
\(875\) 0 0
\(876\) −11.6736 −0.394414
\(877\) −42.6684 −1.44081 −0.720405 0.693554i \(-0.756044\pi\)
−0.720405 + 0.693554i \(0.756044\pi\)
\(878\) −6.82843 −0.230448
\(879\) 20.2337 0.682466
\(880\) −3.29066 −0.110928
\(881\) −11.1844 −0.376813 −0.188407 0.982091i \(-0.560332\pi\)
−0.188407 + 0.982091i \(0.560332\pi\)
\(882\) 0 0
\(883\) −24.2843 −0.817231 −0.408615 0.912707i \(-0.633988\pi\)
−0.408615 + 0.912707i \(0.633988\pi\)
\(884\) −37.1300 −1.24882
\(885\) 35.6326 1.19778
\(886\) 1.62055 0.0544435
\(887\) 26.2779 0.882327 0.441164 0.897427i \(-0.354566\pi\)
0.441164 + 0.897427i \(0.354566\pi\)
\(888\) −8.31055 −0.278884
\(889\) 0 0
\(890\) 54.4501 1.82517
\(891\) −1.00000 −0.0335013
\(892\) −28.4912 −0.953956
\(893\) 0.0625787 0.00209412
\(894\) 24.1269 0.806923
\(895\) 60.2243 2.01308
\(896\) 0 0
\(897\) −40.9951 −1.36879
\(898\) −30.1756 −1.00697
\(899\) −16.6211 −0.554345
\(900\) 5.82843 0.194281
\(901\) 18.3766 0.612213
\(902\) −6.11908 −0.203743
\(903\) 0 0
\(904\) 0.247112 0.00821882
\(905\) −82.4305 −2.74008
\(906\) 7.85525 0.260973
\(907\) −48.9164 −1.62424 −0.812121 0.583489i \(-0.801687\pi\)
−0.812121 + 0.583489i \(0.801687\pi\)
\(908\) 2.90326 0.0963481
\(909\) −5.51657 −0.182973
\(910\) 0 0
\(911\) 8.95444 0.296674 0.148337 0.988937i \(-0.452608\pi\)
0.148337 + 0.988937i \(0.452608\pi\)
\(912\) 0.0511786 0.00169469
\(913\) 0.951983 0.0315060
\(914\) −11.3200 −0.374433
\(915\) 23.8226 0.787552
\(916\) −6.11908 −0.202180
\(917\) 0 0
\(918\) 6.11908 0.201960
\(919\) −11.9222 −0.393276 −0.196638 0.980476i \(-0.563002\pi\)
−0.196638 + 0.980476i \(0.563002\pi\)
\(920\) 22.2318 0.732962
\(921\) −17.5033 −0.576754
\(922\) −7.51025 −0.247337
\(923\) 39.3137 1.29403
\(924\) 0 0
\(925\) −48.4374 −1.59261
\(926\) 3.10899 0.102168
\(927\) −11.6781 −0.383558
\(928\) −2.82843 −0.0928477
\(929\) 13.9105 0.456389 0.228194 0.973616i \(-0.426718\pi\)
0.228194 + 0.973616i \(0.426718\pi\)
\(930\) −19.3374 −0.634097
\(931\) 0 0
\(932\) −11.6569 −0.381833
\(933\) 30.0486 0.983746
\(934\) 36.3189 1.18839
\(935\) −20.1358 −0.658511
\(936\) 6.06791 0.198336
\(937\) −20.0104 −0.653711 −0.326856 0.945074i \(-0.605989\pi\)
−0.326856 + 0.945074i \(0.605989\pi\)
\(938\) 0 0
\(939\) 24.6256 0.803625
\(940\) −4.02366 −0.131237
\(941\) 54.9084 1.78996 0.894982 0.446103i \(-0.147188\pi\)
0.894982 + 0.446103i \(0.147188\pi\)
\(942\) 18.2549 0.594777
\(943\) 41.3408 1.34624
\(944\) −10.8284 −0.352435
\(945\) 0 0
\(946\) 2.90080 0.0943133
\(947\) −7.39850 −0.240419 −0.120210 0.992749i \(-0.538357\pi\)
−0.120210 + 0.992749i \(0.538357\pi\)
\(948\) 9.13897 0.296820
\(949\) −70.8342 −2.29938
\(950\) 0.298291 0.00967782
\(951\) −11.1479 −0.361496
\(952\) 0 0
\(953\) 21.1416 0.684843 0.342422 0.939546i \(-0.388753\pi\)
0.342422 + 0.939546i \(0.388753\pi\)
\(954\) −3.00316 −0.0972310
\(955\) −71.9504 −2.32826
\(956\) −21.3074 −0.689130
\(957\) 2.82843 0.0914301
\(958\) −30.5035 −0.985522
\(959\) 0 0
\(960\) −3.29066 −0.106206
\(961\) 3.53259 0.113955
\(962\) −50.4276 −1.62585
\(963\) 15.9611 0.514339
\(964\) 11.8783 0.382574
\(965\) 68.5098 2.20541
\(966\) 0 0
\(967\) 11.7174 0.376807 0.188404 0.982092i \(-0.439669\pi\)
0.188404 + 0.982092i \(0.439669\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0.313166 0.0100603
\(970\) −48.4437 −1.55543
\(971\) −35.9487 −1.15365 −0.576824 0.816869i \(-0.695708\pi\)
−0.576824 + 0.816869i \(0.695708\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 12.6001 0.403732
\(975\) 35.3663 1.13263
\(976\) −7.23948 −0.231730
\(977\) −4.33561 −0.138709 −0.0693543 0.997592i \(-0.522094\pi\)
−0.0693543 + 0.997592i \(0.522094\pi\)
\(978\) 2.34315 0.0749255
\(979\) 16.5469 0.528840
\(980\) 0 0
\(981\) −5.65053 −0.180408
\(982\) 31.3137 0.999261
\(983\) 16.3444 0.521305 0.260653 0.965433i \(-0.416062\pi\)
0.260653 + 0.965433i \(0.416062\pi\)
\(984\) −6.11908 −0.195069
\(985\) −67.3892 −2.14720
\(986\) −17.3074 −0.551179
\(987\) 0 0
\(988\) 0.310547 0.00987981
\(989\) −19.5980 −0.623180
\(990\) 3.29066 0.104584
\(991\) 11.5657 0.367398 0.183699 0.982983i \(-0.441193\pi\)
0.183699 + 0.982983i \(0.441193\pi\)
\(992\) 5.87644 0.186577
\(993\) 4.16841 0.132281
\(994\) 0 0
\(995\) −51.1148 −1.62045
\(996\) 0.951983 0.0301647
\(997\) −41.6956 −1.32051 −0.660257 0.751040i \(-0.729553\pi\)
−0.660257 + 0.751040i \(0.729553\pi\)
\(998\) 3.49477 0.110625
\(999\) 8.31055 0.262934
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 3234.2.a.bj.1.4 4
3.2 odd 2 9702.2.a.eb.1.1 4
7.6 odd 2 3234.2.a.bk.1.1 yes 4
21.20 even 2 9702.2.a.ec.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bj.1.4 4 1.1 even 1 trivial
3234.2.a.bk.1.1 yes 4 7.6 odd 2
9702.2.a.eb.1.1 4 3.2 odd 2
9702.2.a.ec.1.4 4 21.20 even 2