Properties

Label 3234.2.a.bj.1.1
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 1.1
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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29066 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.29066 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.29066 q^{10} -1.00000 q^{11} -1.00000 q^{12} +3.23948 q^{13} +3.29066 q^{15} +1.00000 q^{16} -0.462230 q^{17} -1.00000 q^{18} -2.77725 q^{19} -3.29066 q^{20} +1.00000 q^{22} -2.90080 q^{23} +1.00000 q^{24} +5.82843 q^{25} -3.23948 q^{26} -1.00000 q^{27} +2.82843 q^{29} -3.29066 q^{30} +0.704871 q^{31} -1.00000 q^{32} +1.00000 q^{33} +0.462230 q^{34} +1.00000 q^{36} +0.996838 q^{37} +2.77725 q^{38} -3.23948 q^{39} +3.29066 q^{40} +0.462230 q^{41} +6.75605 q^{43} -1.00000 q^{44} -3.29066 q^{45} +2.90080 q^{46} -3.94882 q^{47} -1.00000 q^{48} -5.82843 q^{50} +0.462230 q^{51} +3.23948 q^{52} +12.3105 q^{53} +1.00000 q^{54} +3.29066 q^{55} +2.77725 q^{57} -2.82843 q^{58} -10.8284 q^{59} +3.29066 q^{60} +2.06791 q^{61} -0.704871 q^{62} +1.00000 q^{64} -10.6600 q^{65} -1.00000 q^{66} +15.3374 q^{67} -0.462230 q^{68} +2.90080 q^{69} +12.1358 q^{71} -1.00000 q^{72} -0.359873 q^{73} -0.996838 q^{74} -5.82843 q^{75} -2.77725 q^{76} +3.23948 q^{78} +0.168411 q^{79} -3.29066 q^{80} +1.00000 q^{81} -0.462230 q^{82} -7.53330 q^{83} +1.52104 q^{85} -6.75605 q^{86} -2.82843 q^{87} +1.00000 q^{88} +11.3753 q^{89} +3.29066 q^{90} -2.90080 q^{92} -0.704871 q^{93} +3.94882 q^{94} +9.13897 q^{95} +1.00000 q^{96} -3.89317 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 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 \(-0.763200\pi\)
−0.735813 + 0.677184i \(0.763200\pi\)
\(6\) 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
\(13\) 3.23948 0.898470 0.449235 0.893414i \(-0.351697\pi\)
0.449235 + 0.893414i \(0.351697\pi\)
\(14\) 0 0
\(15\) 3.29066 0.849644
\(16\) 1.00000 0.250000
\(17\) −0.462230 −0.112107 −0.0560537 0.998428i \(-0.517852\pi\)
−0.0560537 + 0.998428i \(0.517852\pi\)
\(18\) −1.00000 −0.235702
\(19\) −2.77725 −0.637145 −0.318572 0.947899i \(-0.603203\pi\)
−0.318572 + 0.947899i \(0.603203\pi\)
\(20\) −3.29066 −0.735813
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) −2.90080 −0.604860 −0.302430 0.953172i \(-0.597798\pi\)
−0.302430 + 0.953172i \(0.597798\pi\)
\(24\) 1.00000 0.204124
\(25\) 5.82843 1.16569
\(26\) −3.23948 −0.635314
\(27\) −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
\(31\) 0.704871 0.126599 0.0632993 0.997995i \(-0.479838\pi\)
0.0632993 + 0.997995i \(0.479838\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.00000 0.174078
\(34\) 0.462230 0.0792719
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.996838 0.163879 0.0819396 0.996637i \(-0.473889\pi\)
0.0819396 + 0.996637i \(0.473889\pi\)
\(38\) 2.77725 0.450529
\(39\) −3.23948 −0.518732
\(40\) 3.29066 0.520299
\(41\) 0.462230 0.0721883 0.0360941 0.999348i \(-0.488508\pi\)
0.0360941 + 0.999348i \(0.488508\pi\)
\(42\) 0 0
\(43\) 6.75605 1.03029 0.515144 0.857104i \(-0.327738\pi\)
0.515144 + 0.857104i \(0.327738\pi\)
\(44\) −1.00000 −0.150756
\(45\) −3.29066 −0.490542
\(46\) 2.90080 0.427700
\(47\) −3.94882 −0.575995 −0.287997 0.957631i \(-0.592989\pi\)
−0.287997 + 0.957631i \(0.592989\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −5.82843 −0.824264
\(51\) 0.462230 0.0647252
\(52\) 3.23948 0.449235
\(53\) 12.3105 1.69098 0.845492 0.533988i \(-0.179307\pi\)
0.845492 + 0.533988i \(0.179307\pi\)
\(54\) 1.00000 0.136083
\(55\) 3.29066 0.443712
\(56\) 0 0
\(57\) 2.77725 0.367856
\(58\) −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
\(61\) 2.06791 0.264768 0.132384 0.991198i \(-0.457737\pi\)
0.132384 + 0.991198i \(0.457737\pi\)
\(62\) −0.704871 −0.0895187
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −10.6600 −1.32221
\(66\) −1.00000 −0.123091
\(67\) 15.3374 1.87376 0.936879 0.349655i \(-0.113701\pi\)
0.936879 + 0.349655i \(0.113701\pi\)
\(68\) −0.462230 −0.0560537
\(69\) 2.90080 0.349216
\(70\) 0 0
\(71\) 12.1358 1.44026 0.720128 0.693841i \(-0.244083\pi\)
0.720128 + 0.693841i \(0.244083\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.359873 −0.0421200 −0.0210600 0.999778i \(-0.506704\pi\)
−0.0210600 + 0.999778i \(0.506704\pi\)
\(74\) −0.996838 −0.115880
\(75\) −5.82843 −0.673009
\(76\) −2.77725 −0.318572
\(77\) 0 0
\(78\) 3.23948 0.366799
\(79\) 0.168411 0.0189477 0.00947387 0.999955i \(-0.496984\pi\)
0.00947387 + 0.999955i \(0.496984\pi\)
\(80\) −3.29066 −0.367907
\(81\) 1.00000 0.111111
\(82\) −0.462230 −0.0510448
\(83\) −7.53330 −0.826887 −0.413443 0.910530i \(-0.635674\pi\)
−0.413443 + 0.910530i \(0.635674\pi\)
\(84\) 0 0
\(85\) 1.52104 0.164980
\(86\) −6.75605 −0.728524
\(87\) −2.82843 −0.303239
\(88\) 1.00000 0.106600
\(89\) 11.3753 1.20578 0.602889 0.797825i \(-0.294016\pi\)
0.602889 + 0.797825i \(0.294016\pi\)
\(90\) 3.29066 0.346866
\(91\) 0 0
\(92\) −2.90080 −0.302430
\(93\) −0.704871 −0.0730917
\(94\) 3.94882 0.407290
\(95\) 9.13897 0.937639
\(96\) 1.00000 0.102062
\(97\) −3.89317 −0.395292 −0.197646 0.980273i \(-0.563330\pi\)
−0.197646 + 0.980273i \(0.563330\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 5.82843 0.582843
\(101\) −10.9687 −1.09143 −0.545714 0.837972i \(-0.683741\pi\)
−0.545714 + 0.837972i \(0.683741\pi\)
\(102\) −0.462230 −0.0457676
\(103\) −12.8072 −1.26193 −0.630967 0.775810i \(-0.717342\pi\)
−0.630967 + 0.775810i \(0.717342\pi\)
\(104\) −3.23948 −0.317657
\(105\) 0 0
\(106\) −12.3105 −1.19571
\(107\) −11.9611 −1.15632 −0.578161 0.815923i \(-0.696229\pi\)
−0.578161 + 0.815923i \(0.696229\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 12.9642 1.24175 0.620874 0.783910i \(-0.286778\pi\)
0.620874 + 0.783910i \(0.286778\pi\)
\(110\) −3.29066 −0.313752
\(111\) −0.996838 −0.0946157
\(112\) 0 0
\(113\) −13.4097 −1.26148 −0.630741 0.775993i \(-0.717249\pi\)
−0.630741 + 0.775993i \(0.717249\pi\)
\(114\) −2.77725 −0.260113
\(115\) 9.54555 0.890128
\(116\) 2.82843 0.262613
\(117\) 3.23948 0.299490
\(118\) 10.8284 0.996838
\(119\) 0 0
\(120\) −3.29066 −0.300395
\(121\) 1.00000 0.0909091
\(122\) −2.06791 −0.187219
\(123\) −0.462230 −0.0416779
\(124\) 0.704871 0.0632993
\(125\) −2.72607 −0.243827
\(126\) 0 0
\(127\) −22.3016 −1.97895 −0.989474 0.144713i \(-0.953774\pi\)
−0.989474 + 0.144713i \(0.953774\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −6.75605 −0.594837
\(130\) 10.6600 0.934945
\(131\) 3.53330 0.308706 0.154353 0.988016i \(-0.450671\pi\)
0.154353 + 0.988016i \(0.450671\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −15.3374 −1.32495
\(135\) 3.29066 0.283215
\(136\) 0.462230 0.0396359
\(137\) −10.1358 −0.865961 −0.432980 0.901403i \(-0.642538\pi\)
−0.432980 + 0.901403i \(0.642538\pi\)
\(138\) −2.90080 −0.246933
\(139\) −6.63249 −0.562561 −0.281280 0.959626i \(-0.590759\pi\)
−0.281280 + 0.959626i \(0.590759\pi\)
\(140\) 0 0
\(141\) 3.94882 0.332551
\(142\) −12.1358 −1.01841
\(143\) −3.23948 −0.270899
\(144\) 1.00000 0.0833333
\(145\) −9.30739 −0.772936
\(146\) 0.359873 0.0297833
\(147\) 0 0
\(148\) 0.996838 0.0819396
\(149\) −20.8132 −1.70508 −0.852540 0.522662i \(-0.824939\pi\)
−0.852540 + 0.522662i \(0.824939\pi\)
\(150\) 5.82843 0.475889
\(151\) 0.144755 0.0117800 0.00588999 0.999983i \(-0.498125\pi\)
0.00588999 + 0.999983i \(0.498125\pi\)
\(152\) 2.77725 0.225265
\(153\) −0.462230 −0.0373691
\(154\) 0 0
\(155\) −2.31949 −0.186306
\(156\) −3.23948 −0.259366
\(157\) −6.94119 −0.553967 −0.276984 0.960875i \(-0.589335\pi\)
−0.276984 + 0.960875i \(0.589335\pi\)
\(158\) −0.168411 −0.0133981
\(159\) −12.3105 −0.976290
\(160\) 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 \(0.470749\pi\)
\(164\) 0.462230 0.0360941
\(165\) −3.29066 −0.256177
\(166\) 7.53330 0.584697
\(167\) 17.4432 1.34980 0.674898 0.737911i \(-0.264188\pi\)
0.674898 + 0.737911i \(0.264188\pi\)
\(168\) 0 0
\(169\) −2.50578 −0.192752
\(170\) −1.52104 −0.116659
\(171\) −2.77725 −0.212382
\(172\) 6.75605 0.515144
\(173\) 15.7484 1.19733 0.598665 0.801000i \(-0.295698\pi\)
0.598665 + 0.801000i \(0.295698\pi\)
\(174\) 2.82843 0.214423
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 10.8284 0.813914
\(178\) −11.3753 −0.852614
\(179\) −17.3310 −1.29538 −0.647691 0.761903i \(-0.724265\pi\)
−0.647691 + 0.761903i \(0.724265\pi\)
\(180\) −3.29066 −0.245271
\(181\) −23.9207 −1.77801 −0.889006 0.457896i \(-0.848603\pi\)
−0.889006 + 0.457896i \(0.848603\pi\)
\(182\) 0 0
\(183\) −2.06791 −0.152864
\(184\) 2.90080 0.213850
\(185\) −3.28025 −0.241169
\(186\) 0.704871 0.0516837
\(187\) 0.462230 0.0338016
\(188\) −3.94882 −0.287997
\(189\) 0 0
\(190\) −9.13897 −0.663011
\(191\) −7.10552 −0.514137 −0.257069 0.966393i \(-0.582757\pi\)
−0.257069 + 0.966393i \(0.582757\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −5.50578 −0.396314 −0.198157 0.980170i \(-0.563496\pi\)
−0.198157 + 0.980170i \(0.563496\pi\)
\(194\) 3.89317 0.279513
\(195\) 10.6600 0.763380
\(196\) 0 0
\(197\) −1.86419 −0.132818 −0.0664089 0.997792i \(-0.521154\pi\)
−0.0664089 + 0.997792i \(0.521154\pi\)
\(198\) 1.00000 0.0710669
\(199\) −8.95198 −0.634589 −0.317295 0.948327i \(-0.602774\pi\)
−0.317295 + 0.948327i \(0.602774\pi\)
\(200\) −5.82843 −0.412132
\(201\) −15.3374 −1.08181
\(202\) 10.9687 0.771756
\(203\) 0 0
\(204\) 0.462230 0.0323626
\(205\) −1.52104 −0.106234
\(206\) 12.8072 0.892322
\(207\) −2.90080 −0.201620
\(208\) 3.23948 0.224617
\(209\) 2.77725 0.192106
\(210\) 0 0
\(211\) 16.0634 1.10585 0.552926 0.833230i \(-0.313511\pi\)
0.552926 + 0.833230i \(0.313511\pi\)
\(212\) 12.3105 0.845492
\(213\) −12.1358 −0.831532
\(214\) 11.9611 0.817642
\(215\) −22.2318 −1.51620
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) −12.9642 −0.878049
\(219\) 0.359873 0.0243180
\(220\) 3.29066 0.221856
\(221\) −1.49739 −0.100725
\(222\) 0.996838 0.0669034
\(223\) 15.3196 1.02588 0.512940 0.858425i \(-0.328557\pi\)
0.512940 + 0.858425i \(0.328557\pi\)
\(224\) 0 0
\(225\) 5.82843 0.388562
\(226\) 13.4097 0.892003
\(227\) −11.3885 −0.755884 −0.377942 0.925829i \(-0.623368\pi\)
−0.377942 + 0.925829i \(0.623368\pi\)
\(228\) 2.77725 0.183928
\(229\) 0.462230 0.0305450 0.0152725 0.999883i \(-0.495138\pi\)
0.0152725 + 0.999883i \(0.495138\pi\)
\(230\) −9.54555 −0.629415
\(231\) 0 0
\(232\) −2.82843 −0.185695
\(233\) −11.6569 −0.763666 −0.381833 0.924231i \(-0.624707\pi\)
−0.381833 + 0.924231i \(0.624707\pi\)
\(234\) −3.23948 −0.211771
\(235\) 12.9942 0.847649
\(236\) −10.8284 −0.704871
\(237\) −0.168411 −0.0109395
\(238\) 0 0
\(239\) −2.69261 −0.174171 −0.0870854 0.996201i \(-0.527755\pi\)
−0.0870854 + 0.996201i \(0.527755\pi\)
\(240\) 3.29066 0.212411
\(241\) 10.7491 0.692412 0.346206 0.938159i \(-0.387470\pi\)
0.346206 + 0.938159i \(0.387470\pi\)
\(242\) −1.00000 −0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 2.06791 0.132384
\(245\) 0 0
\(246\) 0.462230 0.0294707
\(247\) −8.99684 −0.572455
\(248\) −0.704871 −0.0447594
\(249\) 7.53330 0.477403
\(250\) 2.72607 0.172412
\(251\) 1.75921 0.111040 0.0555202 0.998458i \(-0.482318\pi\)
0.0555202 + 0.998458i \(0.482318\pi\)
\(252\) 0 0
\(253\) 2.90080 0.182372
\(254\) 22.3016 1.39933
\(255\) −1.52104 −0.0952513
\(256\) 1.00000 0.0625000
\(257\) 26.5442 1.65578 0.827892 0.560887i \(-0.189540\pi\)
0.827892 + 0.560887i \(0.189540\pi\)
\(258\) 6.75605 0.420613
\(259\) 0 0
\(260\) −10.6600 −0.661106
\(261\) 2.82843 0.175075
\(262\) −3.53330 −0.218288
\(263\) 3.68683 0.227340 0.113670 0.993519i \(-0.463739\pi\)
0.113670 + 0.993519i \(0.463739\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −40.5098 −2.48850
\(266\) 0 0
\(267\) −11.3753 −0.696157
\(268\) 15.3374 0.936879
\(269\) −0.471173 −0.0287279 −0.0143640 0.999897i \(-0.504572\pi\)
−0.0143640 + 0.999897i \(0.504572\pi\)
\(270\) −3.29066 −0.200263
\(271\) −29.7927 −1.80978 −0.904888 0.425650i \(-0.860045\pi\)
−0.904888 + 0.425650i \(0.860045\pi\)
\(272\) −0.462230 −0.0280268
\(273\) 0 0
\(274\) 10.1358 0.612327
\(275\) −5.82843 −0.351467
\(276\) 2.90080 0.174608
\(277\) −32.3650 −1.94463 −0.972313 0.233681i \(-0.924923\pi\)
−0.972313 + 0.233681i \(0.924923\pi\)
\(278\) 6.63249 0.397791
\(279\) 0.704871 0.0421995
\(280\) 0 0
\(281\) −30.6211 −1.82670 −0.913351 0.407174i \(-0.866514\pi\)
−0.913351 + 0.407174i \(0.866514\pi\)
\(282\) −3.94882 −0.235149
\(283\) 18.3381 1.09009 0.545043 0.838408i \(-0.316514\pi\)
0.545043 + 0.838408i \(0.316514\pi\)
\(284\) 12.1358 0.720128
\(285\) −9.13897 −0.541346
\(286\) 3.23948 0.191554
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −16.7863 −0.987432
\(290\) 9.30739 0.546548
\(291\) 3.89317 0.228222
\(292\) −0.359873 −0.0210600
\(293\) 6.09156 0.355873 0.177936 0.984042i \(-0.443058\pi\)
0.177936 + 0.984042i \(0.443058\pi\)
\(294\) 0 0
\(295\) 35.6326 2.07461
\(296\) −0.996838 −0.0579400
\(297\) 1.00000 0.0580259
\(298\) 20.8132 1.20567
\(299\) −9.39710 −0.543448
\(300\) −5.82843 −0.336504
\(301\) 0 0
\(302\) −0.144755 −0.00832971
\(303\) 10.9687 0.630136
\(304\) −2.77725 −0.159286
\(305\) −6.80477 −0.389640
\(306\) 0.462230 0.0264240
\(307\) 9.32511 0.532212 0.266106 0.963944i \(-0.414263\pi\)
0.266106 + 0.963944i \(0.414263\pi\)
\(308\) 0 0
\(309\) 12.8072 0.728578
\(310\) 2.31949 0.131738
\(311\) −25.0641 −1.42126 −0.710628 0.703567i \(-0.751589\pi\)
−0.710628 + 0.703567i \(0.751589\pi\)
\(312\) 3.23948 0.183399
\(313\) −19.1734 −1.08375 −0.541873 0.840460i \(-0.682285\pi\)
−0.541873 + 0.840460i \(0.682285\pi\)
\(314\) 6.94119 0.391714
\(315\) 0 0
\(316\) 0.168411 0.00947387
\(317\) 28.1658 1.58195 0.790974 0.611849i \(-0.209574\pi\)
0.790974 + 0.611849i \(0.209574\pi\)
\(318\) 12.3105 0.690341
\(319\) −2.82843 −0.158362
\(320\) −3.29066 −0.183953
\(321\) 11.9611 0.667602
\(322\) 0 0
\(323\) 1.28373 0.0714286
\(324\) 1.00000 0.0555556
\(325\) 18.8811 1.04733
\(326\) −2.34315 −0.129775
\(327\) −12.9642 −0.716924
\(328\) −0.462230 −0.0255224
\(329\) 0 0
\(330\) 3.29066 0.181145
\(331\) 5.13897 0.282464 0.141232 0.989977i \(-0.454894\pi\)
0.141232 + 0.989977i \(0.454894\pi\)
\(332\) −7.53330 −0.413443
\(333\) 0.996838 0.0546264
\(334\) −17.4432 −0.954449
\(335\) −50.4700 −2.75747
\(336\) 0 0
\(337\) 19.8163 1.07946 0.539732 0.841837i \(-0.318526\pi\)
0.539732 + 0.841837i \(0.318526\pi\)
\(338\) 2.50578 0.136296
\(339\) 13.4097 0.728317
\(340\) 1.52104 0.0824901
\(341\) −0.704871 −0.0381709
\(342\) 2.77725 0.150176
\(343\) 0 0
\(344\) −6.75605 −0.364262
\(345\) −9.54555 −0.513915
\(346\) −15.7484 −0.846640
\(347\) 4.10583 0.220413 0.110206 0.993909i \(-0.464849\pi\)
0.110206 + 0.993909i \(0.464849\pi\)
\(348\) −2.82843 −0.151620
\(349\) −20.9897 −1.12356 −0.561778 0.827288i \(-0.689882\pi\)
−0.561778 + 0.827288i \(0.689882\pi\)
\(350\) 0 0
\(351\) −3.23948 −0.172911
\(352\) 1.00000 0.0533002
\(353\) 10.7453 0.571912 0.285956 0.958243i \(-0.407689\pi\)
0.285956 + 0.958243i \(0.407689\pi\)
\(354\) −10.8284 −0.575524
\(355\) −39.9348 −2.11952
\(356\) 11.3753 0.602889
\(357\) 0 0
\(358\) 17.3310 0.915974
\(359\) −26.5911 −1.40343 −0.701713 0.712460i \(-0.747581\pi\)
−0.701713 + 0.712460i \(0.747581\pi\)
\(360\) 3.29066 0.173433
\(361\) −11.2869 −0.594047
\(362\) 23.9207 1.25724
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 1.18422 0.0619849
\(366\) 2.06791 0.108091
\(367\) −20.5601 −1.07323 −0.536615 0.843827i \(-0.680297\pi\)
−0.536615 + 0.843827i \(0.680297\pi\)
\(368\) −2.90080 −0.151215
\(369\) 0.462230 0.0240628
\(370\) 3.28025 0.170532
\(371\) 0 0
\(372\) −0.704871 −0.0365459
\(373\) −29.4495 −1.52484 −0.762419 0.647083i \(-0.775989\pi\)
−0.762419 + 0.647083i \(0.775989\pi\)
\(374\) −0.462230 −0.0239014
\(375\) 2.72607 0.140774
\(376\) 3.94882 0.203645
\(377\) 9.16263 0.471899
\(378\) 0 0
\(379\) 1.86157 0.0956224 0.0478112 0.998856i \(-0.484775\pi\)
0.0478112 + 0.998856i \(0.484775\pi\)
\(380\) 9.13897 0.468819
\(381\) 22.3016 1.14255
\(382\) 7.10552 0.363550
\(383\) −9.55063 −0.488014 −0.244007 0.969773i \(-0.578462\pi\)
−0.244007 + 0.969773i \(0.578462\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 5.50578 0.280237
\(387\) 6.75605 0.343429
\(388\) −3.89317 −0.197646
\(389\) −3.01788 −0.153012 −0.0765062 0.997069i \(-0.524376\pi\)
−0.0765062 + 0.997069i \(0.524376\pi\)
\(390\) −10.6600 −0.539791
\(391\) 1.34084 0.0678092
\(392\) 0 0
\(393\) −3.53330 −0.178231
\(394\) 1.86419 0.0939164
\(395\) −0.554183 −0.0278840
\(396\) −1.00000 −0.0502519
\(397\) 34.8425 1.74870 0.874348 0.485299i \(-0.161289\pi\)
0.874348 + 0.485299i \(0.161289\pi\)
\(398\) 8.95198 0.448722
\(399\) 0 0
\(400\) 5.82843 0.291421
\(401\) 9.44952 0.471887 0.235943 0.971767i \(-0.424182\pi\)
0.235943 + 0.971767i \(0.424182\pi\)
\(402\) 15.3374 0.764958
\(403\) 2.28342 0.113745
\(404\) −10.9687 −0.545714
\(405\) −3.29066 −0.163514
\(406\) 0 0
\(407\) −0.996838 −0.0494114
\(408\) −0.462230 −0.0228838
\(409\) −29.0372 −1.43580 −0.717899 0.696147i \(-0.754896\pi\)
−0.717899 + 0.696147i \(0.754896\pi\)
\(410\) 1.52104 0.0751189
\(411\) 10.1358 0.499963
\(412\) −12.8072 −0.630967
\(413\) 0 0
\(414\) 2.90080 0.142567
\(415\) 24.7895 1.21687
\(416\) −3.23948 −0.158829
\(417\) 6.63249 0.324795
\(418\) −2.77725 −0.135840
\(419\) −18.2318 −0.890684 −0.445342 0.895361i \(-0.646918\pi\)
−0.445342 + 0.895361i \(0.646918\pi\)
\(420\) 0 0
\(421\) 1.79529 0.0874969 0.0437484 0.999043i \(-0.486070\pi\)
0.0437484 + 0.999043i \(0.486070\pi\)
\(422\) −16.0634 −0.781956
\(423\) −3.94882 −0.191998
\(424\) −12.3105 −0.597853
\(425\) −2.69408 −0.130682
\(426\) 12.1358 0.587982
\(427\) 0 0
\(428\) −11.9611 −0.578161
\(429\) 3.23948 0.156404
\(430\) 22.2318 1.07211
\(431\) −12.9942 −0.625910 −0.312955 0.949768i \(-0.601319\pi\)
−0.312955 + 0.949768i \(0.601319\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 0.387396 0.0186170 0.00930852 0.999957i \(-0.497037\pi\)
0.00930852 + 0.999957i \(0.497037\pi\)
\(434\) 0 0
\(435\) 9.30739 0.446255
\(436\) 12.9642 0.620874
\(437\) 8.05626 0.385383
\(438\) −0.359873 −0.0171954
\(439\) 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
\(442\) 1.49739 0.0712234
\(443\) 18.5911 0.883290 0.441645 0.897190i \(-0.354395\pi\)
0.441645 + 0.897190i \(0.354395\pi\)
\(444\) −0.996838 −0.0473079
\(445\) −37.4322 −1.77446
\(446\) −15.3196 −0.725406
\(447\) 20.8132 0.984429
\(448\) 0 0
\(449\) 6.10868 0.288286 0.144143 0.989557i \(-0.453957\pi\)
0.144143 + 0.989557i \(0.453957\pi\)
\(450\) −5.82843 −0.274755
\(451\) −0.462230 −0.0217656
\(452\) −13.4097 −0.630741
\(453\) −0.144755 −0.00680118
\(454\) 11.3885 0.534491
\(455\) 0 0
\(456\) −2.77725 −0.130057
\(457\) 29.9348 1.40029 0.700145 0.714000i \(-0.253118\pi\)
0.700145 + 0.714000i \(0.253118\pi\)
\(458\) −0.462230 −0.0215986
\(459\) 0.462230 0.0215751
\(460\) 9.54555 0.445064
\(461\) −5.65238 −0.263258 −0.131629 0.991299i \(-0.542021\pi\)
−0.131629 + 0.991299i \(0.542021\pi\)
\(462\) 0 0
\(463\) 7.79529 0.362278 0.181139 0.983458i \(-0.442022\pi\)
0.181139 + 0.983458i \(0.442022\pi\)
\(464\) 2.82843 0.131306
\(465\) 2.31949 0.107564
\(466\) 11.6569 0.539993
\(467\) 34.9464 1.61712 0.808562 0.588411i \(-0.200246\pi\)
0.808562 + 0.588411i \(0.200246\pi\)
\(468\) 3.23948 0.149745
\(469\) 0 0
\(470\) −12.9942 −0.599379
\(471\) 6.94119 0.319833
\(472\) 10.8284 0.498419
\(473\) −6.75605 −0.310643
\(474\) 0.168411 0.00773538
\(475\) −16.1870 −0.742710
\(476\) 0 0
\(477\) 12.3105 0.563661
\(478\) 2.69261 0.123157
\(479\) −38.5035 −1.75927 −0.879634 0.475651i \(-0.842213\pi\)
−0.879634 + 0.475651i \(0.842213\pi\)
\(480\) −3.29066 −0.150197
\(481\) 3.22924 0.147241
\(482\) −10.7491 −0.489609
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 12.8111 0.581722
\(486\) 1.00000 0.0453609
\(487\) −18.7137 −0.847997 −0.423998 0.905663i \(-0.639374\pi\)
−0.423998 + 0.905663i \(0.639374\pi\)
\(488\) −2.06791 −0.0936097
\(489\) −2.34315 −0.105961
\(490\) 0 0
\(491\) −31.3137 −1.41317 −0.706584 0.707629i \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(492\) −0.462230 −0.0208390
\(493\) −1.30739 −0.0588817
\(494\) 8.99684 0.404787
\(495\) 3.29066 0.147904
\(496\) 0.704871 0.0316496
\(497\) 0 0
\(498\) −7.53330 −0.337575
\(499\) −31.4169 −1.40641 −0.703207 0.710985i \(-0.748249\pi\)
−0.703207 + 0.710985i \(0.748249\pi\)
\(500\) −2.72607 −0.121914
\(501\) −17.4432 −0.779305
\(502\) −1.75921 −0.0785174
\(503\) 43.2921 1.93030 0.965150 0.261697i \(-0.0842821\pi\)
0.965150 + 0.261697i \(0.0842821\pi\)
\(504\) 0 0
\(505\) 36.0943 1.60617
\(506\) −2.90080 −0.128956
\(507\) 2.50578 0.111285
\(508\) −22.3016 −0.989474
\(509\) 38.1562 1.69125 0.845623 0.533781i \(-0.179229\pi\)
0.845623 + 0.533781i \(0.179229\pi\)
\(510\) 1.52104 0.0673529
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.77725 0.122619
\(514\) −26.5442 −1.17082
\(515\) 42.1442 1.85710
\(516\) −6.75605 −0.297419
\(517\) 3.94882 0.173669
\(518\) 0 0
\(519\) −15.7484 −0.691279
\(520\) 10.6600 0.467473
\(521\) 3.79366 0.166203 0.0831017 0.996541i \(-0.473517\pi\)
0.0831017 + 0.996541i \(0.473517\pi\)
\(522\) −2.82843 −0.123797
\(523\) 8.27670 0.361915 0.180957 0.983491i \(-0.442080\pi\)
0.180957 + 0.983491i \(0.442080\pi\)
\(524\) 3.53330 0.154353
\(525\) 0 0
\(526\) −3.68683 −0.160754
\(527\) −0.325813 −0.0141926
\(528\) 1.00000 0.0435194
\(529\) −14.5853 −0.634145
\(530\) 40.5098 1.75963
\(531\) −10.8284 −0.469914
\(532\) 0 0
\(533\) 1.49739 0.0648590
\(534\) 11.3753 0.492257
\(535\) 39.3598 1.70167
\(536\) −15.3374 −0.662473
\(537\) 17.3310 0.747890
\(538\) 0.471173 0.0203137
\(539\) 0 0
\(540\) 3.29066 0.141607
\(541\) 16.4916 0.709029 0.354515 0.935050i \(-0.384646\pi\)
0.354515 + 0.935050i \(0.384646\pi\)
\(542\) 29.7927 1.27970
\(543\) 23.9207 1.02654
\(544\) 0.462230 0.0198180
\(545\) −42.6609 −1.82739
\(546\) 0 0
\(547\) 4.00894 0.171410 0.0857050 0.996321i \(-0.472686\pi\)
0.0857050 + 0.996321i \(0.472686\pi\)
\(548\) −10.1358 −0.432980
\(549\) 2.06791 0.0882561
\(550\) 5.82843 0.248525
\(551\) −7.85525 −0.334645
\(552\) −2.90080 −0.123466
\(553\) 0 0
\(554\) 32.3650 1.37506
\(555\) 3.28025 0.139239
\(556\) −6.63249 −0.281280
\(557\) 19.7800 0.838106 0.419053 0.907962i \(-0.362362\pi\)
0.419053 + 0.907962i \(0.362362\pi\)
\(558\) −0.704871 −0.0298396
\(559\) 21.8861 0.925683
\(560\) 0 0
\(561\) −0.462230 −0.0195154
\(562\) 30.6211 1.29167
\(563\) −39.9199 −1.68242 −0.841212 0.540705i \(-0.818157\pi\)
−0.841212 + 0.540705i \(0.818157\pi\)
\(564\) 3.94882 0.166275
\(565\) 44.1269 1.85643
\(566\) −18.3381 −0.770807
\(567\) 0 0
\(568\) −12.1358 −0.509207
\(569\) 35.7238 1.49762 0.748809 0.662786i \(-0.230626\pi\)
0.748809 + 0.662786i \(0.230626\pi\)
\(570\) 9.13897 0.382789
\(571\) 1.44320 0.0603959 0.0301980 0.999544i \(-0.490386\pi\)
0.0301980 + 0.999544i \(0.490386\pi\)
\(572\) −3.23948 −0.135449
\(573\) 7.10552 0.296837
\(574\) 0 0
\(575\) −16.9071 −0.705076
\(576\) 1.00000 0.0416667
\(577\) −7.64606 −0.318310 −0.159155 0.987254i \(-0.550877\pi\)
−0.159155 + 0.987254i \(0.550877\pi\)
\(578\) 16.7863 0.698220
\(579\) 5.50578 0.228812
\(580\) −9.30739 −0.386468
\(581\) 0 0
\(582\) −3.89317 −0.161377
\(583\) −12.3105 −0.509851
\(584\) 0.359873 0.0148917
\(585\) −10.6600 −0.440737
\(586\) −6.09156 −0.251640
\(587\) −19.3382 −0.798174 −0.399087 0.916913i \(-0.630673\pi\)
−0.399087 + 0.916913i \(0.630673\pi\)
\(588\) 0 0
\(589\) −1.95760 −0.0806616
\(590\) −35.6326 −1.46697
\(591\) 1.86419 0.0766824
\(592\) 0.996838 0.0409698
\(593\) 30.1191 1.23684 0.618421 0.785847i \(-0.287773\pi\)
0.618421 + 0.785847i \(0.287773\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) −20.8132 −0.852540
\(597\) 8.95198 0.366380
\(598\) 9.39710 0.384276
\(599\) −30.9080 −1.26287 −0.631433 0.775430i \(-0.717533\pi\)
−0.631433 + 0.775430i \(0.717533\pi\)
\(600\) 5.82843 0.237945
\(601\) −24.3970 −0.995176 −0.497588 0.867414i \(-0.665781\pi\)
−0.497588 + 0.867414i \(0.665781\pi\)
\(602\) 0 0
\(603\) 15.3374 0.624586
\(604\) 0.144755 0.00588999
\(605\) −3.29066 −0.133784
\(606\) −10.9687 −0.445573
\(607\) −39.9348 −1.62090 −0.810452 0.585805i \(-0.800778\pi\)
−0.810452 + 0.585805i \(0.800778\pi\)
\(608\) 2.77725 0.112632
\(609\) 0 0
\(610\) 6.80477 0.275517
\(611\) −12.7921 −0.517514
\(612\) −0.462230 −0.0186846
\(613\) 23.4034 0.945255 0.472628 0.881262i \(-0.343306\pi\)
0.472628 + 0.881262i \(0.343306\pi\)
\(614\) −9.32511 −0.376331
\(615\) 1.52104 0.0613343
\(616\) 0 0
\(617\) −35.7439 −1.43900 −0.719499 0.694494i \(-0.755628\pi\)
−0.719499 + 0.694494i \(0.755628\pi\)
\(618\) −12.8072 −0.515182
\(619\) 12.2382 0.491894 0.245947 0.969283i \(-0.420901\pi\)
0.245947 + 0.969283i \(0.420901\pi\)
\(620\) −2.31949 −0.0931529
\(621\) 2.90080 0.116405
\(622\) 25.0641 1.00498
\(623\) 0 0
\(624\) −3.23948 −0.129683
\(625\) −20.1716 −0.806863
\(626\) 19.1734 0.766324
\(627\) −2.77725 −0.110913
\(628\) −6.94119 −0.276984
\(629\) −0.460769 −0.0183721
\(630\) 0 0
\(631\) −13.8100 −0.549767 −0.274884 0.961477i \(-0.588639\pi\)
−0.274884 + 0.961477i \(0.588639\pi\)
\(632\) −0.168411 −0.00669904
\(633\) −16.0634 −0.638464
\(634\) −28.1658 −1.11861
\(635\) 73.3869 2.91227
\(636\) −12.3105 −0.488145
\(637\) 0 0
\(638\) 2.82843 0.111979
\(639\) 12.1358 0.480085
\(640\) 3.29066 0.130075
\(641\) −47.6188 −1.88083 −0.940415 0.340030i \(-0.889563\pi\)
−0.940415 + 0.340030i \(0.889563\pi\)
\(642\) −11.9611 −0.472066
\(643\) −23.2921 −0.918552 −0.459276 0.888294i \(-0.651891\pi\)
−0.459276 + 0.888294i \(0.651891\pi\)
\(644\) 0 0
\(645\) 22.2318 0.875378
\(646\) −1.28373 −0.0505076
\(647\) −41.0463 −1.61369 −0.806847 0.590760i \(-0.798828\pi\)
−0.806847 + 0.590760i \(0.798828\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 10.8284 0.425053
\(650\) −18.8811 −0.740576
\(651\) 0 0
\(652\) 2.34315 0.0917647
\(653\) −33.9348 −1.32797 −0.663986 0.747745i \(-0.731136\pi\)
−0.663986 + 0.747745i \(0.731136\pi\)
\(654\) 12.9642 0.506942
\(655\) −11.6269 −0.454300
\(656\) 0.462230 0.0180471
\(657\) −0.359873 −0.0140400
\(658\) 0 0
\(659\) 30.1721 1.17534 0.587669 0.809101i \(-0.300046\pi\)
0.587669 + 0.809101i \(0.300046\pi\)
\(660\) −3.29066 −0.128089
\(661\) 7.82728 0.304446 0.152223 0.988346i \(-0.451357\pi\)
0.152223 + 0.988346i \(0.451357\pi\)
\(662\) −5.13897 −0.199732
\(663\) 1.49739 0.0581536
\(664\) 7.53330 0.292349
\(665\) 0 0
\(666\) −0.996838 −0.0386267
\(667\) −8.20471 −0.317688
\(668\) 17.4432 0.674898
\(669\) −15.3196 −0.592292
\(670\) 50.4700 1.94983
\(671\) −2.06791 −0.0798306
\(672\) 0 0
\(673\) 22.1721 0.854672 0.427336 0.904093i \(-0.359452\pi\)
0.427336 + 0.904093i \(0.359452\pi\)
\(674\) −19.8163 −0.763296
\(675\) −5.82843 −0.224336
\(676\) −2.50578 −0.0963760
\(677\) −0.494668 −0.0190116 −0.00950581 0.999955i \(-0.503026\pi\)
−0.00950581 + 0.999955i \(0.503026\pi\)
\(678\) −13.4097 −0.514998
\(679\) 0 0
\(680\) −1.52104 −0.0583293
\(681\) 11.3885 0.436410
\(682\) 0.704871 0.0269909
\(683\) −2.51156 −0.0961021 −0.0480510 0.998845i \(-0.515301\pi\)
−0.0480510 + 0.998845i \(0.515301\pi\)
\(684\) −2.77725 −0.106191
\(685\) 33.3535 1.27437
\(686\) 0 0
\(687\) −0.462230 −0.0176352
\(688\) 6.75605 0.257572
\(689\) 39.8798 1.51930
\(690\) 9.54555 0.363393
\(691\) 23.2587 0.884801 0.442401 0.896818i \(-0.354127\pi\)
0.442401 + 0.896818i \(0.354127\pi\)
\(692\) 15.7484 0.598665
\(693\) 0 0
\(694\) −4.10583 −0.155855
\(695\) 21.8253 0.827880
\(696\) 2.82843 0.107211
\(697\) −0.213657 −0.00809283
\(698\) 20.9897 0.794474
\(699\) 11.6569 0.440903
\(700\) 0 0
\(701\) 20.3253 0.767674 0.383837 0.923401i \(-0.374602\pi\)
0.383837 + 0.923401i \(0.374602\pi\)
\(702\) 3.23948 0.122266
\(703\) −2.76847 −0.104415
\(704\) −1.00000 −0.0376889
\(705\) −12.9942 −0.489391
\(706\) −10.7453 −0.404403
\(707\) 0 0
\(708\) 10.8284 0.406957
\(709\) 24.7869 0.930891 0.465446 0.885077i \(-0.345894\pi\)
0.465446 + 0.885077i \(0.345894\pi\)
\(710\) 39.9348 1.49873
\(711\) 0.168411 0.00631591
\(712\) −11.3753 −0.426307
\(713\) −2.04469 −0.0765744
\(714\) 0 0
\(715\) 10.6600 0.398662
\(716\) −17.3310 −0.647691
\(717\) 2.69261 0.100558
\(718\) 26.5911 0.992372
\(719\) 38.5212 1.43660 0.718299 0.695734i \(-0.244921\pi\)
0.718299 + 0.695734i \(0.244921\pi\)
\(720\) −3.29066 −0.122636
\(721\) 0 0
\(722\) 11.2869 0.420055
\(723\) −10.7491 −0.399764
\(724\) −23.9207 −0.889006
\(725\) 16.4853 0.612248
\(726\) 1.00000 0.0371135
\(727\) −28.2657 −1.04832 −0.524158 0.851621i \(-0.675620\pi\)
−0.524158 + 0.851621i \(0.675620\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.18422 −0.0438299
\(731\) −3.12285 −0.115503
\(732\) −2.06791 −0.0764320
\(733\) 4.28418 0.158240 0.0791199 0.996865i \(-0.474789\pi\)
0.0791199 + 0.996865i \(0.474789\pi\)
\(734\) 20.5601 0.758888
\(735\) 0 0
\(736\) 2.90080 0.106925
\(737\) −15.3374 −0.564959
\(738\) −0.462230 −0.0170149
\(739\) 20.1958 0.742913 0.371457 0.928450i \(-0.378858\pi\)
0.371457 + 0.928450i \(0.378858\pi\)
\(740\) −3.28025 −0.120585
\(741\) 8.99684 0.330507
\(742\) 0 0
\(743\) −32.8258 −1.20426 −0.602131 0.798397i \(-0.705682\pi\)
−0.602131 + 0.798397i \(0.705682\pi\)
\(744\) 0.704871 0.0258418
\(745\) 68.4890 2.50924
\(746\) 29.4495 1.07822
\(747\) −7.53330 −0.275629
\(748\) 0.462230 0.0169008
\(749\) 0 0
\(750\) −2.72607 −0.0995420
\(751\) −18.1468 −0.662187 −0.331093 0.943598i \(-0.607418\pi\)
−0.331093 + 0.943598i \(0.607418\pi\)
\(752\) −3.94882 −0.143999
\(753\) −1.75921 −0.0641092
\(754\) −9.16263 −0.333683
\(755\) −0.476339 −0.0173357
\(756\) 0 0
\(757\) −10.9905 −0.399457 −0.199729 0.979851i \(-0.564006\pi\)
−0.199729 + 0.979851i \(0.564006\pi\)
\(758\) −1.86157 −0.0676152
\(759\) −2.90080 −0.105293
\(760\) −9.13897 −0.331505
\(761\) 18.4596 0.669160 0.334580 0.942367i \(-0.391405\pi\)
0.334580 + 0.942367i \(0.391405\pi\)
\(762\) −22.3016 −0.807902
\(763\) 0 0
\(764\) −7.10552 −0.257069
\(765\) 1.52104 0.0549934
\(766\) 9.55063 0.345078
\(767\) −35.0785 −1.26661
\(768\) −1.00000 −0.0360844
\(769\) 47.8567 1.72576 0.862878 0.505411i \(-0.168659\pi\)
0.862878 + 0.505411i \(0.168659\pi\)
\(770\) 0 0
\(771\) −26.5442 −0.955968
\(772\) −5.50578 −0.198157
\(773\) 45.4201 1.63365 0.816825 0.576886i \(-0.195732\pi\)
0.816825 + 0.576886i \(0.195732\pi\)
\(774\) −6.75605 −0.242841
\(775\) 4.10829 0.147574
\(776\) 3.89317 0.139757
\(777\) 0 0
\(778\) 3.01788 0.108196
\(779\) −1.28373 −0.0459944
\(780\) 10.6600 0.381690
\(781\) −12.1358 −0.434254
\(782\) −1.34084 −0.0479483
\(783\) −2.82843 −0.101080
\(784\) 0 0
\(785\) 22.8411 0.815233
\(786\) 3.53330 0.126029
\(787\) 13.5298 0.482286 0.241143 0.970490i \(-0.422478\pi\)
0.241143 + 0.970490i \(0.422478\pi\)
\(788\) −1.86419 −0.0664089
\(789\) −3.68683 −0.131255
\(790\) 0.554183 0.0197170
\(791\) 0 0
\(792\) 1.00000 0.0355335
\(793\) 6.69894 0.237886
\(794\) −34.8425 −1.23652
\(795\) 40.5098 1.43673
\(796\) −8.95198 −0.317295
\(797\) 13.8031 0.488930 0.244465 0.969658i \(-0.421388\pi\)
0.244465 + 0.969658i \(0.421388\pi\)
\(798\) 0 0
\(799\) 1.82527 0.0645732
\(800\) −5.82843 −0.206066
\(801\) 11.3753 0.401926
\(802\) −9.44952 −0.333674
\(803\) 0.359873 0.0126997
\(804\) −15.3374 −0.540907
\(805\) 0 0
\(806\) −2.28342 −0.0804299
\(807\) 0.471173 0.0165861
\(808\) 10.9687 0.385878
\(809\) 38.0816 1.33888 0.669439 0.742867i \(-0.266534\pi\)
0.669439 + 0.742867i \(0.266534\pi\)
\(810\) 3.29066 0.115622
\(811\) 42.0359 1.47608 0.738040 0.674757i \(-0.235751\pi\)
0.738040 + 0.674757i \(0.235751\pi\)
\(812\) 0 0
\(813\) 29.7927 1.04487
\(814\) 0.996838 0.0349392
\(815\) −7.71049 −0.270087
\(816\) 0.462230 0.0161813
\(817\) −18.7632 −0.656442
\(818\) 29.0372 1.01526
\(819\) 0 0
\(820\) −1.52104 −0.0531171
\(821\) 49.0974 1.71351 0.856756 0.515722i \(-0.172476\pi\)
0.856756 + 0.515722i \(0.172476\pi\)
\(822\) −10.1358 −0.353527
\(823\) −24.7073 −0.861243 −0.430622 0.902533i \(-0.641706\pi\)
−0.430622 + 0.902533i \(0.641706\pi\)
\(824\) 12.8072 0.446161
\(825\) 5.82843 0.202920
\(826\) 0 0
\(827\) 19.7774 0.687728 0.343864 0.939020i \(-0.388264\pi\)
0.343864 + 0.939020i \(0.388264\pi\)
\(828\) −2.90080 −0.100810
\(829\) −25.4789 −0.884919 −0.442459 0.896789i \(-0.645894\pi\)
−0.442459 + 0.896789i \(0.645894\pi\)
\(830\) −24.7895 −0.860456
\(831\) 32.3650 1.12273
\(832\) 3.23948 0.112309
\(833\) 0 0
\(834\) −6.63249 −0.229664
\(835\) −57.3996 −1.98639
\(836\) 2.77725 0.0960531
\(837\) −0.704871 −0.0243639
\(838\) 18.2318 0.629809
\(839\) −11.9064 −0.411055 −0.205528 0.978651i \(-0.565891\pi\)
−0.205528 + 0.978651i \(0.565891\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) −1.79529 −0.0618696
\(843\) 30.6211 1.05465
\(844\) 16.0634 0.552926
\(845\) 8.24565 0.283659
\(846\) 3.94882 0.135763
\(847\) 0 0
\(848\) 12.3105 0.422746
\(849\) −18.3381 −0.629361
\(850\) 2.69408 0.0924061
\(851\) −2.89163 −0.0991239
\(852\) −12.1358 −0.415766
\(853\) 41.2968 1.41398 0.706988 0.707225i \(-0.250053\pi\)
0.706988 + 0.707225i \(0.250053\pi\)
\(854\) 0 0
\(855\) 9.13897 0.312546
\(856\) 11.9611 0.408821
\(857\) 1.80506 0.0616598 0.0308299 0.999525i \(-0.490185\pi\)
0.0308299 + 0.999525i \(0.490185\pi\)
\(858\) −3.23948 −0.110594
\(859\) −49.9974 −1.70589 −0.852944 0.522002i \(-0.825185\pi\)
−0.852944 + 0.522002i \(0.825185\pi\)
\(860\) −22.2318 −0.758100
\(861\) 0 0
\(862\) 12.9942 0.442585
\(863\) 22.5640 0.768087 0.384043 0.923315i \(-0.374531\pi\)
0.384043 + 0.923315i \(0.374531\pi\)
\(864\) 1.00000 0.0340207
\(865\) −51.8226 −1.76202
\(866\) −0.387396 −0.0131642
\(867\) 16.7863 0.570094
\(868\) 0 0
\(869\) −0.168411 −0.00571296
\(870\) −9.30739 −0.315550
\(871\) 49.6851 1.68351
\(872\) −12.9642 −0.439025
\(873\) −3.89317 −0.131764
\(874\) −8.05626 −0.272507
\(875\) 0 0
\(876\) 0.359873 0.0121590
\(877\) 9.98211 0.337072 0.168536 0.985695i \(-0.446096\pi\)
0.168536 + 0.985695i \(0.446096\pi\)
\(878\) −6.82843 −0.230448
\(879\) −6.09156 −0.205463
\(880\) 3.29066 0.110928
\(881\) 37.6108 1.26714 0.633571 0.773685i \(-0.281589\pi\)
0.633571 + 0.773685i \(0.281589\pi\)
\(882\) 0 0
\(883\) −24.2843 −0.817231 −0.408615 0.912707i \(-0.633988\pi\)
−0.408615 + 0.912707i \(0.633988\pi\)
\(884\) −1.49739 −0.0503625
\(885\) −35.6326 −1.19778
\(886\) −18.5911 −0.624581
\(887\) 7.66318 0.257304 0.128652 0.991690i \(-0.458935\pi\)
0.128652 + 0.991690i \(0.458935\pi\)
\(888\) 0.996838 0.0334517
\(889\) 0 0
\(890\) 37.4322 1.25473
\(891\) −1.00000 −0.0335013
\(892\) 15.3196 0.512940
\(893\) 10.9669 0.366992
\(894\) −20.8132 −0.696096
\(895\) 57.0305 1.90632
\(896\) 0 0
\(897\) 9.39710 0.313760
\(898\) −6.10868 −0.203849
\(899\) 1.99368 0.0664928
\(900\) 5.82843 0.194281
\(901\) −5.69031 −0.189572
\(902\) 0.462230 0.0153906
\(903\) 0 0
\(904\) 13.4097 0.446001
\(905\) 78.7148 2.61657
\(906\) 0.144755 0.00480916
\(907\) 23.9458 0.795108 0.397554 0.917579i \(-0.369859\pi\)
0.397554 + 0.917579i \(0.369859\pi\)
\(908\) −11.3885 −0.377942
\(909\) −10.9687 −0.363809
\(910\) 0 0
\(911\) −2.61129 −0.0865161 −0.0432580 0.999064i \(-0.513774\pi\)
−0.0432580 + 0.999064i \(0.513774\pi\)
\(912\) 2.77725 0.0919639
\(913\) 7.53330 0.249316
\(914\) −29.9348 −0.990155
\(915\) 6.80477 0.224959
\(916\) 0.462230 0.0152725
\(917\) 0 0
\(918\) −0.462230 −0.0152559
\(919\) 43.9222 1.44886 0.724429 0.689349i \(-0.242103\pi\)
0.724429 + 0.689349i \(0.242103\pi\)
\(920\) −9.54555 −0.314708
\(921\) −9.32511 −0.307273
\(922\) 5.65238 0.186151
\(923\) 39.3137 1.29403
\(924\) 0 0
\(925\) 5.81000 0.191032
\(926\) −7.79529 −0.256169
\(927\) −12.8072 −0.420645
\(928\) −2.82843 −0.0928477
\(929\) −40.3369 −1.32341 −0.661706 0.749764i \(-0.730167\pi\)
−0.661706 + 0.749764i \(0.730167\pi\)
\(930\) −2.31949 −0.0760591
\(931\) 0 0
\(932\) −11.6569 −0.381833
\(933\) 25.0641 0.820563
\(934\) −34.9464 −1.14348
\(935\) −1.52104 −0.0497434
\(936\) −3.23948 −0.105886
\(937\) 10.6378 0.347522 0.173761 0.984788i \(-0.444408\pi\)
0.173761 + 0.984788i \(0.444408\pi\)
\(938\) 0 0
\(939\) 19.1734 0.625701
\(940\) 12.9942 0.423825
\(941\) −5.45258 −0.177749 −0.0888746 0.996043i \(-0.528327\pi\)
−0.0888746 + 0.996043i \(0.528327\pi\)
\(942\) −6.94119 −0.226156
\(943\) −1.34084 −0.0436638
\(944\) −10.8284 −0.352435
\(945\) 0 0
\(946\) 6.75605 0.219658
\(947\) −11.9152 −0.387192 −0.193596 0.981081i \(-0.562015\pi\)
−0.193596 + 0.981081i \(0.562015\pi\)
\(948\) −0.168411 −0.00546974
\(949\) −1.16580 −0.0378435
\(950\) 16.1870 0.525175
\(951\) −28.1658 −0.913338
\(952\) 0 0
\(953\) 19.5447 0.633115 0.316557 0.948573i \(-0.397473\pi\)
0.316557 + 0.948573i \(0.397473\pi\)
\(954\) −12.3105 −0.398569
\(955\) 23.3818 0.756618
\(956\) −2.69261 −0.0870854
\(957\) 2.82843 0.0914301
\(958\) 38.5035 1.24399
\(959\) 0 0
\(960\) 3.29066 0.106206
\(961\) −30.5032 −0.983973
\(962\) −3.22924 −0.104115
\(963\) −11.9611 −0.385440
\(964\) 10.7491 0.346206
\(965\) 18.1176 0.583227
\(966\) 0 0
\(967\) −55.0311 −1.76968 −0.884841 0.465893i \(-0.845733\pi\)
−0.884841 + 0.465893i \(0.845733\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −1.28373 −0.0412393
\(970\) −12.8111 −0.411339
\(971\) −7.36505 −0.236356 −0.118178 0.992992i \(-0.537705\pi\)
−0.118178 + 0.992992i \(0.537705\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 18.7137 0.599624
\(975\) −18.8811 −0.604678
\(976\) 2.06791 0.0661921
\(977\) −32.9192 −1.05318 −0.526590 0.850120i \(-0.676530\pi\)
−0.526590 + 0.850120i \(0.676530\pi\)
\(978\) 2.34315 0.0749255
\(979\) −11.3753 −0.363556
\(980\) 0 0
\(981\) 12.9642 0.413916
\(982\) 31.3137 0.999261
\(983\) 45.3957 1.44790 0.723949 0.689853i \(-0.242325\pi\)
0.723949 + 0.689853i \(0.242325\pi\)
\(984\) 0.462230 0.0147354
\(985\) 6.13440 0.195458
\(986\) 1.30739 0.0416356
\(987\) 0 0
\(988\) −8.99684 −0.286228
\(989\) −19.5980 −0.623180
\(990\) −3.29066 −0.104584
\(991\) −11.5657 −0.367398 −0.183699 0.982983i \(-0.558807\pi\)
−0.183699 + 0.982983i \(0.558807\pi\)
\(992\) −0.704871 −0.0223797
\(993\) −5.13897 −0.163080
\(994\) 0 0
\(995\) 29.4579 0.933879
\(996\) 7.53330 0.238702
\(997\) 2.58294 0.0818025 0.0409012 0.999163i \(-0.486977\pi\)
0.0409012 + 0.999163i \(0.486977\pi\)
\(998\) 31.4169 0.994485
\(999\) −0.996838 −0.0315386
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.1.1 4
3.2 odd 2 9702.2.a.eb.1.4 4
7.6 odd 2 3234.2.a.bk.1.4 yes 4
21.20 even 2 9702.2.a.ec.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bj.1.1 4 1.1 even 1 trivial
3234.2.a.bk.1.4 yes 4 7.6 odd 2
9702.2.a.eb.1.4 4 3.2 odd 2
9702.2.a.ec.1.1 4 21.20 even 2