Properties

Label 3381.2.a.bl.1.7
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
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] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, 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("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 4x^{9} - 6x^{8} + 36x^{7} + x^{6} - 100x^{5} + 26x^{4} + 108x^{3} - 33x^{2} - 36x + 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(1.46930\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.46930 q^{2} +1.00000 q^{3} +0.158829 q^{4} -2.14842 q^{5} +1.46930 q^{6} -2.70522 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.46930 q^{2} +1.00000 q^{3} +0.158829 q^{4} -2.14842 q^{5} +1.46930 q^{6} -2.70522 q^{8} +1.00000 q^{9} -3.15666 q^{10} +2.16247 q^{11} +0.158829 q^{12} +3.08320 q^{13} -2.14842 q^{15} -4.29243 q^{16} +0.354774 q^{17} +1.46930 q^{18} +4.96083 q^{19} -0.341232 q^{20} +3.17731 q^{22} -1.00000 q^{23} -2.70522 q^{24} -0.384310 q^{25} +4.53014 q^{26} +1.00000 q^{27} -2.50055 q^{29} -3.15666 q^{30} -1.83663 q^{31} -0.896404 q^{32} +2.16247 q^{33} +0.521268 q^{34} +0.158829 q^{36} -4.50032 q^{37} +7.28892 q^{38} +3.08320 q^{39} +5.81195 q^{40} +10.5155 q^{41} +7.96942 q^{43} +0.343465 q^{44} -2.14842 q^{45} -1.46930 q^{46} +7.83688 q^{47} -4.29243 q^{48} -0.564665 q^{50} +0.354774 q^{51} +0.489703 q^{52} +8.86015 q^{53} +1.46930 q^{54} -4.64589 q^{55} +4.96083 q^{57} -3.67404 q^{58} +6.79270 q^{59} -0.341232 q^{60} -2.35137 q^{61} -2.69856 q^{62} +7.26778 q^{64} -6.62400 q^{65} +3.17731 q^{66} -2.20673 q^{67} +0.0563485 q^{68} -1.00000 q^{69} +3.08130 q^{71} -2.70522 q^{72} +3.33525 q^{73} -6.61229 q^{74} -0.384310 q^{75} +0.787926 q^{76} +4.53014 q^{78} -5.15275 q^{79} +9.22193 q^{80} +1.00000 q^{81} +15.4504 q^{82} +4.70908 q^{83} -0.762202 q^{85} +11.7094 q^{86} -2.50055 q^{87} -5.84998 q^{88} +1.87944 q^{89} -3.15666 q^{90} -0.158829 q^{92} -1.83663 q^{93} +11.5147 q^{94} -10.6579 q^{95} -0.896404 q^{96} +8.12795 q^{97} +2.16247 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 4 q^{2} + 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 4 q^{2} + 10 q^{3} + 8 q^{4} + 4 q^{5} + 4 q^{6} + 12 q^{8} + 10 q^{9} + 8 q^{10} + 2 q^{11} + 8 q^{12} + 4 q^{15} + 4 q^{16} + 12 q^{17} + 4 q^{18} + 26 q^{19} + 24 q^{20} - 8 q^{22} - 10 q^{23} + 12 q^{24} - 2 q^{25} + 4 q^{26} + 10 q^{27} + 16 q^{29} + 8 q^{30} + 12 q^{31} + 8 q^{32} + 2 q^{33} + 28 q^{34} + 8 q^{36} - 8 q^{37} + 32 q^{38} + 4 q^{40} + 10 q^{41} - 4 q^{43} - 16 q^{44} + 4 q^{45} - 4 q^{46} + 2 q^{47} + 4 q^{48} - 8 q^{50} + 12 q^{51} + 24 q^{52} + 14 q^{53} + 4 q^{54} + 16 q^{55} + 26 q^{57} - 8 q^{58} + 38 q^{59} + 24 q^{60} + 14 q^{61} - 8 q^{62} + 8 q^{64} + 12 q^{65} - 8 q^{66} + 8 q^{68} - 10 q^{69} + 24 q^{71} + 12 q^{72} + 8 q^{73} - 8 q^{74} - 2 q^{75} + 64 q^{76} + 4 q^{78} - 16 q^{79} + 28 q^{80} + 10 q^{81} - 40 q^{82} + 28 q^{83} - 4 q^{85} + 20 q^{86} + 16 q^{87} - 68 q^{88} + 32 q^{89} + 8 q^{90} - 8 q^{92} + 12 q^{93} + 56 q^{94} + 8 q^{95} + 8 q^{96} + 4 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.46930 1.03895 0.519474 0.854486i \(-0.326128\pi\)
0.519474 + 0.854486i \(0.326128\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.158829 0.0794147
\(5\) −2.14842 −0.960801 −0.480400 0.877049i \(-0.659509\pi\)
−0.480400 + 0.877049i \(0.659509\pi\)
\(6\) 1.46930 0.599837
\(7\) 0 0
\(8\) −2.70522 −0.956441
\(9\) 1.00000 0.333333
\(10\) −3.15666 −0.998223
\(11\) 2.16247 0.652011 0.326005 0.945368i \(-0.394297\pi\)
0.326005 + 0.945368i \(0.394297\pi\)
\(12\) 0.158829 0.0458501
\(13\) 3.08320 0.855127 0.427563 0.903985i \(-0.359372\pi\)
0.427563 + 0.903985i \(0.359372\pi\)
\(14\) 0 0
\(15\) −2.14842 −0.554719
\(16\) −4.29243 −1.07311
\(17\) 0.354774 0.0860453 0.0430227 0.999074i \(-0.486301\pi\)
0.0430227 + 0.999074i \(0.486301\pi\)
\(18\) 1.46930 0.346316
\(19\) 4.96083 1.13809 0.569046 0.822306i \(-0.307313\pi\)
0.569046 + 0.822306i \(0.307313\pi\)
\(20\) −0.341232 −0.0763017
\(21\) 0 0
\(22\) 3.17731 0.677406
\(23\) −1.00000 −0.208514
\(24\) −2.70522 −0.552201
\(25\) −0.384310 −0.0768621
\(26\) 4.53014 0.888433
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −2.50055 −0.464340 −0.232170 0.972675i \(-0.574583\pi\)
−0.232170 + 0.972675i \(0.574583\pi\)
\(30\) −3.15666 −0.576324
\(31\) −1.83663 −0.329869 −0.164934 0.986305i \(-0.552741\pi\)
−0.164934 + 0.986305i \(0.552741\pi\)
\(32\) −0.896404 −0.158463
\(33\) 2.16247 0.376438
\(34\) 0.521268 0.0893967
\(35\) 0 0
\(36\) 0.158829 0.0264716
\(37\) −4.50032 −0.739847 −0.369924 0.929062i \(-0.620616\pi\)
−0.369924 + 0.929062i \(0.620616\pi\)
\(38\) 7.28892 1.18242
\(39\) 3.08320 0.493708
\(40\) 5.81195 0.918949
\(41\) 10.5155 1.64225 0.821124 0.570749i \(-0.193347\pi\)
0.821124 + 0.570749i \(0.193347\pi\)
\(42\) 0 0
\(43\) 7.96942 1.21533 0.607663 0.794195i \(-0.292107\pi\)
0.607663 + 0.794195i \(0.292107\pi\)
\(44\) 0.343465 0.0517792
\(45\) −2.14842 −0.320267
\(46\) −1.46930 −0.216636
\(47\) 7.83688 1.14313 0.571564 0.820558i \(-0.306337\pi\)
0.571564 + 0.820558i \(0.306337\pi\)
\(48\) −4.29243 −0.619559
\(49\) 0 0
\(50\) −0.564665 −0.0798557
\(51\) 0.354774 0.0496783
\(52\) 0.489703 0.0679096
\(53\) 8.86015 1.21703 0.608517 0.793541i \(-0.291765\pi\)
0.608517 + 0.793541i \(0.291765\pi\)
\(54\) 1.46930 0.199946
\(55\) −4.64589 −0.626452
\(56\) 0 0
\(57\) 4.96083 0.657078
\(58\) −3.67404 −0.482425
\(59\) 6.79270 0.884334 0.442167 0.896933i \(-0.354210\pi\)
0.442167 + 0.896933i \(0.354210\pi\)
\(60\) −0.341232 −0.0440528
\(61\) −2.35137 −0.301062 −0.150531 0.988605i \(-0.548098\pi\)
−0.150531 + 0.988605i \(0.548098\pi\)
\(62\) −2.69856 −0.342717
\(63\) 0 0
\(64\) 7.26778 0.908473
\(65\) −6.62400 −0.821606
\(66\) 3.17731 0.391100
\(67\) −2.20673 −0.269595 −0.134798 0.990873i \(-0.543038\pi\)
−0.134798 + 0.990873i \(0.543038\pi\)
\(68\) 0.0563485 0.00683326
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 3.08130 0.365683 0.182841 0.983142i \(-0.441471\pi\)
0.182841 + 0.983142i \(0.441471\pi\)
\(72\) −2.70522 −0.318814
\(73\) 3.33525 0.390362 0.195181 0.980767i \(-0.437471\pi\)
0.195181 + 0.980767i \(0.437471\pi\)
\(74\) −6.61229 −0.768664
\(75\) −0.384310 −0.0443763
\(76\) 0.787926 0.0903813
\(77\) 0 0
\(78\) 4.53014 0.512937
\(79\) −5.15275 −0.579729 −0.289865 0.957068i \(-0.593610\pi\)
−0.289865 + 0.957068i \(0.593610\pi\)
\(80\) 9.22193 1.03104
\(81\) 1.00000 0.111111
\(82\) 15.4504 1.70621
\(83\) 4.70908 0.516889 0.258445 0.966026i \(-0.416790\pi\)
0.258445 + 0.966026i \(0.416790\pi\)
\(84\) 0 0
\(85\) −0.762202 −0.0826724
\(86\) 11.7094 1.26266
\(87\) −2.50055 −0.268087
\(88\) −5.84998 −0.623610
\(89\) 1.87944 0.199220 0.0996101 0.995027i \(-0.468240\pi\)
0.0996101 + 0.995027i \(0.468240\pi\)
\(90\) −3.15666 −0.332741
\(91\) 0 0
\(92\) −0.158829 −0.0165591
\(93\) −1.83663 −0.190450
\(94\) 11.5147 1.18765
\(95\) −10.6579 −1.09348
\(96\) −0.896404 −0.0914888
\(97\) 8.12795 0.825268 0.412634 0.910897i \(-0.364609\pi\)
0.412634 + 0.910897i \(0.364609\pi\)
\(98\) 0 0
\(99\) 2.16247 0.217337
\(100\) −0.0610398 −0.00610398
\(101\) 18.1335 1.80435 0.902174 0.431371i \(-0.141970\pi\)
0.902174 + 0.431371i \(0.141970\pi\)
\(102\) 0.521268 0.0516132
\(103\) 3.90946 0.385210 0.192605 0.981276i \(-0.438306\pi\)
0.192605 + 0.981276i \(0.438306\pi\)
\(104\) −8.34076 −0.817878
\(105\) 0 0
\(106\) 13.0182 1.26444
\(107\) −5.42225 −0.524188 −0.262094 0.965042i \(-0.584413\pi\)
−0.262094 + 0.965042i \(0.584413\pi\)
\(108\) 0.158829 0.0152834
\(109\) 15.2859 1.46412 0.732060 0.681240i \(-0.238559\pi\)
0.732060 + 0.681240i \(0.238559\pi\)
\(110\) −6.82619 −0.650852
\(111\) −4.50032 −0.427151
\(112\) 0 0
\(113\) 5.97105 0.561709 0.280854 0.959750i \(-0.409382\pi\)
0.280854 + 0.959750i \(0.409382\pi\)
\(114\) 7.28892 0.682670
\(115\) 2.14842 0.200341
\(116\) −0.397160 −0.0368754
\(117\) 3.08320 0.285042
\(118\) 9.98048 0.918778
\(119\) 0 0
\(120\) 5.81195 0.530556
\(121\) −6.32370 −0.574882
\(122\) −3.45486 −0.312788
\(123\) 10.5155 0.948153
\(124\) −0.291711 −0.0261964
\(125\) 11.5677 1.03465
\(126\) 0 0
\(127\) −7.84667 −0.696279 −0.348140 0.937443i \(-0.613186\pi\)
−0.348140 + 0.937443i \(0.613186\pi\)
\(128\) 12.4713 1.10232
\(129\) 7.96942 0.701669
\(130\) −9.73262 −0.853607
\(131\) −5.14450 −0.449477 −0.224738 0.974419i \(-0.572153\pi\)
−0.224738 + 0.974419i \(0.572153\pi\)
\(132\) 0.343465 0.0298948
\(133\) 0 0
\(134\) −3.24234 −0.280096
\(135\) −2.14842 −0.184906
\(136\) −0.959743 −0.0822973
\(137\) 11.7090 1.00037 0.500183 0.865920i \(-0.333266\pi\)
0.500183 + 0.865920i \(0.333266\pi\)
\(138\) −1.46930 −0.125075
\(139\) −3.57676 −0.303377 −0.151688 0.988428i \(-0.548471\pi\)
−0.151688 + 0.988428i \(0.548471\pi\)
\(140\) 0 0
\(141\) 7.83688 0.659985
\(142\) 4.52734 0.379926
\(143\) 6.66735 0.557552
\(144\) −4.29243 −0.357703
\(145\) 5.37221 0.446138
\(146\) 4.90047 0.405566
\(147\) 0 0
\(148\) −0.714783 −0.0587548
\(149\) −9.49709 −0.778032 −0.389016 0.921231i \(-0.627185\pi\)
−0.389016 + 0.921231i \(0.627185\pi\)
\(150\) −0.564665 −0.0461047
\(151\) −12.8737 −1.04765 −0.523825 0.851826i \(-0.675496\pi\)
−0.523825 + 0.851826i \(0.675496\pi\)
\(152\) −13.4202 −1.08852
\(153\) 0.354774 0.0286818
\(154\) 0 0
\(155\) 3.94585 0.316938
\(156\) 0.489703 0.0392077
\(157\) −7.53352 −0.601241 −0.300620 0.953744i \(-0.597194\pi\)
−0.300620 + 0.953744i \(0.597194\pi\)
\(158\) −7.57091 −0.602309
\(159\) 8.86015 0.702655
\(160\) 1.92585 0.152252
\(161\) 0 0
\(162\) 1.46930 0.115439
\(163\) −18.0136 −1.41094 −0.705468 0.708742i \(-0.749263\pi\)
−0.705468 + 0.708742i \(0.749263\pi\)
\(164\) 1.67017 0.130419
\(165\) −4.64589 −0.361682
\(166\) 6.91904 0.537021
\(167\) 16.8365 1.30285 0.651425 0.758713i \(-0.274171\pi\)
0.651425 + 0.758713i \(0.274171\pi\)
\(168\) 0 0
\(169\) −3.49385 −0.268758
\(170\) −1.11990 −0.0858924
\(171\) 4.96083 0.379364
\(172\) 1.26578 0.0965147
\(173\) −16.8181 −1.27865 −0.639327 0.768935i \(-0.720787\pi\)
−0.639327 + 0.768935i \(0.720787\pi\)
\(174\) −3.67404 −0.278528
\(175\) 0 0
\(176\) −9.28227 −0.699678
\(177\) 6.79270 0.510571
\(178\) 2.76145 0.206980
\(179\) 3.16675 0.236694 0.118347 0.992972i \(-0.462241\pi\)
0.118347 + 0.992972i \(0.462241\pi\)
\(180\) −0.341232 −0.0254339
\(181\) −23.1662 −1.72193 −0.860965 0.508664i \(-0.830139\pi\)
−0.860965 + 0.508664i \(0.830139\pi\)
\(182\) 0 0
\(183\) −2.35137 −0.173818
\(184\) 2.70522 0.199432
\(185\) 9.66855 0.710846
\(186\) −2.69856 −0.197868
\(187\) 0.767190 0.0561025
\(188\) 1.24473 0.0907811
\(189\) 0 0
\(190\) −15.6596 −1.13607
\(191\) −4.73960 −0.342945 −0.171473 0.985189i \(-0.554853\pi\)
−0.171473 + 0.985189i \(0.554853\pi\)
\(192\) 7.26778 0.524507
\(193\) 20.9142 1.50544 0.752720 0.658341i \(-0.228742\pi\)
0.752720 + 0.658341i \(0.228742\pi\)
\(194\) 11.9424 0.857411
\(195\) −6.62400 −0.474355
\(196\) 0 0
\(197\) −19.8120 −1.41155 −0.705773 0.708438i \(-0.749400\pi\)
−0.705773 + 0.708438i \(0.749400\pi\)
\(198\) 3.17731 0.225802
\(199\) 12.7673 0.905051 0.452525 0.891751i \(-0.350523\pi\)
0.452525 + 0.891751i \(0.350523\pi\)
\(200\) 1.03965 0.0735140
\(201\) −2.20673 −0.155651
\(202\) 26.6434 1.87463
\(203\) 0 0
\(204\) 0.0563485 0.00394519
\(205\) −22.5917 −1.57787
\(206\) 5.74415 0.400214
\(207\) −1.00000 −0.0695048
\(208\) −13.2344 −0.917644
\(209\) 10.7277 0.742048
\(210\) 0 0
\(211\) 3.77646 0.259983 0.129991 0.991515i \(-0.458505\pi\)
0.129991 + 0.991515i \(0.458505\pi\)
\(212\) 1.40725 0.0966504
\(213\) 3.08130 0.211127
\(214\) −7.96688 −0.544605
\(215\) −17.1216 −1.16769
\(216\) −2.70522 −0.184067
\(217\) 0 0
\(218\) 22.4594 1.52115
\(219\) 3.33525 0.225376
\(220\) −0.737905 −0.0497495
\(221\) 1.09384 0.0735797
\(222\) −6.61229 −0.443788
\(223\) 3.14153 0.210373 0.105186 0.994453i \(-0.466456\pi\)
0.105186 + 0.994453i \(0.466456\pi\)
\(224\) 0 0
\(225\) −0.384310 −0.0256207
\(226\) 8.77323 0.583587
\(227\) 19.6642 1.30516 0.652579 0.757721i \(-0.273687\pi\)
0.652579 + 0.757721i \(0.273687\pi\)
\(228\) 0.787926 0.0521816
\(229\) −3.40015 −0.224688 −0.112344 0.993669i \(-0.535836\pi\)
−0.112344 + 0.993669i \(0.535836\pi\)
\(230\) 3.15666 0.208144
\(231\) 0 0
\(232\) 6.76454 0.444114
\(233\) −19.1227 −1.25277 −0.626385 0.779514i \(-0.715466\pi\)
−0.626385 + 0.779514i \(0.715466\pi\)
\(234\) 4.53014 0.296144
\(235\) −16.8369 −1.09832
\(236\) 1.07888 0.0702291
\(237\) −5.15275 −0.334707
\(238\) 0 0
\(239\) 3.10915 0.201114 0.100557 0.994931i \(-0.467937\pi\)
0.100557 + 0.994931i \(0.467937\pi\)
\(240\) 9.22193 0.595273
\(241\) −20.0711 −1.29289 −0.646445 0.762961i \(-0.723745\pi\)
−0.646445 + 0.762961i \(0.723745\pi\)
\(242\) −9.29139 −0.597273
\(243\) 1.00000 0.0641500
\(244\) −0.373467 −0.0239088
\(245\) 0 0
\(246\) 15.4504 0.985082
\(247\) 15.2952 0.973213
\(248\) 4.96850 0.315500
\(249\) 4.70908 0.298426
\(250\) 16.9964 1.07495
\(251\) −4.73370 −0.298788 −0.149394 0.988778i \(-0.547732\pi\)
−0.149394 + 0.988778i \(0.547732\pi\)
\(252\) 0 0
\(253\) −2.16247 −0.135954
\(254\) −11.5291 −0.723398
\(255\) −0.762202 −0.0477309
\(256\) 3.78850 0.236781
\(257\) −3.68563 −0.229903 −0.114952 0.993371i \(-0.536671\pi\)
−0.114952 + 0.993371i \(0.536671\pi\)
\(258\) 11.7094 0.728998
\(259\) 0 0
\(260\) −1.05209 −0.0652476
\(261\) −2.50055 −0.154780
\(262\) −7.55879 −0.466983
\(263\) −4.58029 −0.282433 −0.141216 0.989979i \(-0.545101\pi\)
−0.141216 + 0.989979i \(0.545101\pi\)
\(264\) −5.84998 −0.360041
\(265\) −19.0353 −1.16933
\(266\) 0 0
\(267\) 1.87944 0.115020
\(268\) −0.350494 −0.0214098
\(269\) −18.9657 −1.15636 −0.578181 0.815909i \(-0.696237\pi\)
−0.578181 + 0.815909i \(0.696237\pi\)
\(270\) −3.15666 −0.192108
\(271\) 5.77817 0.350999 0.175499 0.984480i \(-0.443846\pi\)
0.175499 + 0.984480i \(0.443846\pi\)
\(272\) −1.52284 −0.0923359
\(273\) 0 0
\(274\) 17.2040 1.03933
\(275\) −0.831061 −0.0501149
\(276\) −0.158829 −0.00956041
\(277\) −15.5279 −0.932981 −0.466490 0.884526i \(-0.654482\pi\)
−0.466490 + 0.884526i \(0.654482\pi\)
\(278\) −5.25532 −0.315193
\(279\) −1.83663 −0.109956
\(280\) 0 0
\(281\) 22.5853 1.34733 0.673664 0.739038i \(-0.264720\pi\)
0.673664 + 0.739038i \(0.264720\pi\)
\(282\) 11.5147 0.685690
\(283\) −22.4715 −1.33579 −0.667897 0.744254i \(-0.732805\pi\)
−0.667897 + 0.744254i \(0.732805\pi\)
\(284\) 0.489401 0.0290406
\(285\) −10.6579 −0.631321
\(286\) 9.79631 0.579268
\(287\) 0 0
\(288\) −0.896404 −0.0528211
\(289\) −16.8741 −0.992596
\(290\) 7.89337 0.463514
\(291\) 8.12795 0.476469
\(292\) 0.529736 0.0310005
\(293\) 5.71918 0.334118 0.167059 0.985947i \(-0.446573\pi\)
0.167059 + 0.985947i \(0.446573\pi\)
\(294\) 0 0
\(295\) −14.5935 −0.849669
\(296\) 12.1744 0.707620
\(297\) 2.16247 0.125479
\(298\) −13.9540 −0.808335
\(299\) −3.08320 −0.178306
\(300\) −0.0610398 −0.00352413
\(301\) 0 0
\(302\) −18.9153 −1.08846
\(303\) 18.1335 1.04174
\(304\) −21.2940 −1.22130
\(305\) 5.05173 0.289261
\(306\) 0.521268 0.0297989
\(307\) 9.95544 0.568187 0.284093 0.958797i \(-0.408307\pi\)
0.284093 + 0.958797i \(0.408307\pi\)
\(308\) 0 0
\(309\) 3.90946 0.222401
\(310\) 5.79762 0.329283
\(311\) −17.6774 −1.00239 −0.501197 0.865333i \(-0.667107\pi\)
−0.501197 + 0.865333i \(0.667107\pi\)
\(312\) −8.34076 −0.472202
\(313\) −17.1317 −0.968339 −0.484169 0.874974i \(-0.660878\pi\)
−0.484169 + 0.874974i \(0.660878\pi\)
\(314\) −11.0690 −0.624658
\(315\) 0 0
\(316\) −0.818408 −0.0460390
\(317\) 6.44673 0.362084 0.181042 0.983475i \(-0.442053\pi\)
0.181042 + 0.983475i \(0.442053\pi\)
\(318\) 13.0182 0.730023
\(319\) −5.40737 −0.302754
\(320\) −15.6142 −0.872861
\(321\) −5.42225 −0.302640
\(322\) 0 0
\(323\) 1.75997 0.0979275
\(324\) 0.158829 0.00882386
\(325\) −1.18491 −0.0657268
\(326\) −26.4673 −1.46589
\(327\) 15.2859 0.845310
\(328\) −28.4468 −1.57071
\(329\) 0 0
\(330\) −6.82619 −0.375769
\(331\) −6.58020 −0.361681 −0.180840 0.983512i \(-0.557882\pi\)
−0.180840 + 0.983512i \(0.557882\pi\)
\(332\) 0.747941 0.0410486
\(333\) −4.50032 −0.246616
\(334\) 24.7378 1.35359
\(335\) 4.74098 0.259028
\(336\) 0 0
\(337\) 34.4661 1.87749 0.938744 0.344615i \(-0.111990\pi\)
0.938744 + 0.344615i \(0.111990\pi\)
\(338\) −5.13350 −0.279226
\(339\) 5.97105 0.324303
\(340\) −0.121060 −0.00656540
\(341\) −3.97167 −0.215078
\(342\) 7.28892 0.394140
\(343\) 0 0
\(344\) −21.5591 −1.16239
\(345\) 2.14842 0.115667
\(346\) −24.7107 −1.32846
\(347\) −23.6184 −1.26790 −0.633950 0.773374i \(-0.718567\pi\)
−0.633950 + 0.773374i \(0.718567\pi\)
\(348\) −0.397160 −0.0212900
\(349\) 14.7497 0.789536 0.394768 0.918781i \(-0.370825\pi\)
0.394768 + 0.918781i \(0.370825\pi\)
\(350\) 0 0
\(351\) 3.08320 0.164569
\(352\) −1.93845 −0.103320
\(353\) −20.6050 −1.09669 −0.548346 0.836252i \(-0.684742\pi\)
−0.548346 + 0.836252i \(0.684742\pi\)
\(354\) 9.98048 0.530457
\(355\) −6.61991 −0.351348
\(356\) 0.298510 0.0158210
\(357\) 0 0
\(358\) 4.65289 0.245913
\(359\) −26.7302 −1.41077 −0.705383 0.708826i \(-0.749225\pi\)
−0.705383 + 0.708826i \(0.749225\pi\)
\(360\) 5.81195 0.306316
\(361\) 5.60983 0.295254
\(362\) −34.0380 −1.78900
\(363\) −6.32370 −0.331908
\(364\) 0 0
\(365\) −7.16551 −0.375060
\(366\) −3.45486 −0.180589
\(367\) −33.4515 −1.74616 −0.873078 0.487580i \(-0.837880\pi\)
−0.873078 + 0.487580i \(0.837880\pi\)
\(368\) 4.29243 0.223758
\(369\) 10.5155 0.547416
\(370\) 14.2060 0.738533
\(371\) 0 0
\(372\) −0.291711 −0.0151245
\(373\) 26.3427 1.36397 0.681985 0.731366i \(-0.261117\pi\)
0.681985 + 0.731366i \(0.261117\pi\)
\(374\) 1.12723 0.0582876
\(375\) 11.5677 0.597355
\(376\) −21.2005 −1.09333
\(377\) −7.70969 −0.397069
\(378\) 0 0
\(379\) 19.2340 0.987984 0.493992 0.869466i \(-0.335537\pi\)
0.493992 + 0.869466i \(0.335537\pi\)
\(380\) −1.69279 −0.0868384
\(381\) −7.84667 −0.401997
\(382\) −6.96387 −0.356303
\(383\) −16.3969 −0.837842 −0.418921 0.908023i \(-0.637592\pi\)
−0.418921 + 0.908023i \(0.637592\pi\)
\(384\) 12.4713 0.636425
\(385\) 0 0
\(386\) 30.7292 1.56407
\(387\) 7.96942 0.405109
\(388\) 1.29096 0.0655384
\(389\) 22.6519 1.14849 0.574247 0.818682i \(-0.305295\pi\)
0.574247 + 0.818682i \(0.305295\pi\)
\(390\) −9.73262 −0.492830
\(391\) −0.354774 −0.0179417
\(392\) 0 0
\(393\) −5.14450 −0.259506
\(394\) −29.1097 −1.46652
\(395\) 11.0702 0.557004
\(396\) 0.343465 0.0172597
\(397\) 15.2035 0.763040 0.381520 0.924361i \(-0.375401\pi\)
0.381520 + 0.924361i \(0.375401\pi\)
\(398\) 18.7590 0.940302
\(399\) 0 0
\(400\) 1.64963 0.0824813
\(401\) 7.05495 0.352307 0.176154 0.984363i \(-0.443634\pi\)
0.176154 + 0.984363i \(0.443634\pi\)
\(402\) −3.24234 −0.161713
\(403\) −5.66271 −0.282080
\(404\) 2.88013 0.143292
\(405\) −2.14842 −0.106756
\(406\) 0 0
\(407\) −9.73182 −0.482388
\(408\) −0.959743 −0.0475144
\(409\) 27.4335 1.35650 0.678249 0.734832i \(-0.262739\pi\)
0.678249 + 0.734832i \(0.262739\pi\)
\(410\) −33.1939 −1.63933
\(411\) 11.7090 0.577561
\(412\) 0.620937 0.0305913
\(413\) 0 0
\(414\) −1.46930 −0.0722119
\(415\) −10.1171 −0.496627
\(416\) −2.76380 −0.135506
\(417\) −3.57676 −0.175155
\(418\) 15.7621 0.770950
\(419\) 8.06199 0.393854 0.196927 0.980418i \(-0.436904\pi\)
0.196927 + 0.980418i \(0.436904\pi\)
\(420\) 0 0
\(421\) −3.26503 −0.159128 −0.0795640 0.996830i \(-0.525353\pi\)
−0.0795640 + 0.996830i \(0.525353\pi\)
\(422\) 5.54874 0.270109
\(423\) 7.83688 0.381042
\(424\) −23.9687 −1.16402
\(425\) −0.136343 −0.00661362
\(426\) 4.52734 0.219350
\(427\) 0 0
\(428\) −0.861212 −0.0416283
\(429\) 6.66735 0.321903
\(430\) −25.1567 −1.21317
\(431\) 18.7612 0.903697 0.451848 0.892095i \(-0.350765\pi\)
0.451848 + 0.892095i \(0.350765\pi\)
\(432\) −4.29243 −0.206520
\(433\) 4.07674 0.195915 0.0979577 0.995191i \(-0.468769\pi\)
0.0979577 + 0.995191i \(0.468769\pi\)
\(434\) 0 0
\(435\) 5.37221 0.257578
\(436\) 2.42784 0.116273
\(437\) −4.96083 −0.237309
\(438\) 4.90047 0.234154
\(439\) 20.0760 0.958173 0.479087 0.877768i \(-0.340968\pi\)
0.479087 + 0.877768i \(0.340968\pi\)
\(440\) 12.5682 0.599165
\(441\) 0 0
\(442\) 1.60717 0.0764455
\(443\) 18.1100 0.860433 0.430216 0.902726i \(-0.358437\pi\)
0.430216 + 0.902726i \(0.358437\pi\)
\(444\) −0.714783 −0.0339221
\(445\) −4.03782 −0.191411
\(446\) 4.61584 0.218566
\(447\) −9.49709 −0.449197
\(448\) 0 0
\(449\) 21.9396 1.03539 0.517697 0.855564i \(-0.326790\pi\)
0.517697 + 0.855564i \(0.326790\pi\)
\(450\) −0.564665 −0.0266186
\(451\) 22.7396 1.07076
\(452\) 0.948378 0.0446079
\(453\) −12.8737 −0.604861
\(454\) 28.8925 1.35599
\(455\) 0 0
\(456\) −13.4202 −0.628456
\(457\) 25.7102 1.20267 0.601336 0.798997i \(-0.294636\pi\)
0.601336 + 0.798997i \(0.294636\pi\)
\(458\) −4.99583 −0.233440
\(459\) 0.354774 0.0165594
\(460\) 0.341232 0.0159100
\(461\) −29.6004 −1.37863 −0.689313 0.724464i \(-0.742088\pi\)
−0.689313 + 0.724464i \(0.742088\pi\)
\(462\) 0 0
\(463\) 7.00166 0.325395 0.162697 0.986676i \(-0.447981\pi\)
0.162697 + 0.986676i \(0.447981\pi\)
\(464\) 10.7334 0.498287
\(465\) 3.94585 0.182984
\(466\) −28.0969 −1.30156
\(467\) 21.5522 0.997317 0.498659 0.866798i \(-0.333826\pi\)
0.498659 + 0.866798i \(0.333826\pi\)
\(468\) 0.489703 0.0226365
\(469\) 0 0
\(470\) −24.7384 −1.14110
\(471\) −7.53352 −0.347126
\(472\) −18.3758 −0.845813
\(473\) 17.2337 0.792405
\(474\) −7.57091 −0.347743
\(475\) −1.90650 −0.0874761
\(476\) 0 0
\(477\) 8.86015 0.405678
\(478\) 4.56826 0.208947
\(479\) 1.45000 0.0662524 0.0331262 0.999451i \(-0.489454\pi\)
0.0331262 + 0.999451i \(0.489454\pi\)
\(480\) 1.92585 0.0879025
\(481\) −13.8754 −0.632663
\(482\) −29.4903 −1.34325
\(483\) 0 0
\(484\) −1.00439 −0.0456541
\(485\) −17.4622 −0.792918
\(486\) 1.46930 0.0666486
\(487\) −29.3715 −1.33095 −0.665474 0.746421i \(-0.731770\pi\)
−0.665474 + 0.746421i \(0.731770\pi\)
\(488\) 6.36099 0.287948
\(489\) −18.0136 −0.814605
\(490\) 0 0
\(491\) −22.0308 −0.994235 −0.497117 0.867683i \(-0.665608\pi\)
−0.497117 + 0.867683i \(0.665608\pi\)
\(492\) 1.67017 0.0752973
\(493\) −0.887129 −0.0399543
\(494\) 22.4732 1.01112
\(495\) −4.64589 −0.208817
\(496\) 7.88362 0.353985
\(497\) 0 0
\(498\) 6.91904 0.310049
\(499\) −13.0025 −0.582074 −0.291037 0.956712i \(-0.594000\pi\)
−0.291037 + 0.956712i \(0.594000\pi\)
\(500\) 1.83730 0.0821664
\(501\) 16.8365 0.752201
\(502\) −6.95520 −0.310426
\(503\) −27.4378 −1.22339 −0.611695 0.791094i \(-0.709512\pi\)
−0.611695 + 0.791094i \(0.709512\pi\)
\(504\) 0 0
\(505\) −38.9583 −1.73362
\(506\) −3.17731 −0.141249
\(507\) −3.49385 −0.155168
\(508\) −1.24628 −0.0552948
\(509\) 10.7217 0.475231 0.237615 0.971359i \(-0.423634\pi\)
0.237615 + 0.971359i \(0.423634\pi\)
\(510\) −1.11990 −0.0495900
\(511\) 0 0
\(512\) −19.3762 −0.856316
\(513\) 4.96083 0.219026
\(514\) −5.41528 −0.238858
\(515\) −8.39914 −0.370110
\(516\) 1.26578 0.0557228
\(517\) 16.9471 0.745331
\(518\) 0 0
\(519\) −16.8181 −0.738231
\(520\) 17.9194 0.785818
\(521\) 35.8465 1.57046 0.785232 0.619202i \(-0.212544\pi\)
0.785232 + 0.619202i \(0.212544\pi\)
\(522\) −3.67404 −0.160808
\(523\) 17.0049 0.743573 0.371787 0.928318i \(-0.378745\pi\)
0.371787 + 0.928318i \(0.378745\pi\)
\(524\) −0.817097 −0.0356951
\(525\) 0 0
\(526\) −6.72980 −0.293433
\(527\) −0.651589 −0.0283837
\(528\) −9.28227 −0.403959
\(529\) 1.00000 0.0434783
\(530\) −27.9684 −1.21487
\(531\) 6.79270 0.294778
\(532\) 0 0
\(533\) 32.4215 1.40433
\(534\) 2.76145 0.119500
\(535\) 11.6492 0.503641
\(536\) 5.96971 0.257852
\(537\) 3.16675 0.136655
\(538\) −27.8663 −1.20140
\(539\) 0 0
\(540\) −0.341232 −0.0146843
\(541\) −34.8174 −1.49692 −0.748459 0.663181i \(-0.769206\pi\)
−0.748459 + 0.663181i \(0.769206\pi\)
\(542\) 8.48983 0.364670
\(543\) −23.1662 −0.994157
\(544\) −0.318021 −0.0136350
\(545\) −32.8404 −1.40673
\(546\) 0 0
\(547\) −33.3988 −1.42803 −0.714015 0.700130i \(-0.753125\pi\)
−0.714015 + 0.700130i \(0.753125\pi\)
\(548\) 1.85973 0.0794437
\(549\) −2.35137 −0.100354
\(550\) −1.22107 −0.0520668
\(551\) −12.4048 −0.528461
\(552\) 2.70522 0.115142
\(553\) 0 0
\(554\) −22.8151 −0.969319
\(555\) 9.66855 0.410407
\(556\) −0.568095 −0.0240926
\(557\) 39.5811 1.67711 0.838553 0.544820i \(-0.183402\pi\)
0.838553 + 0.544820i \(0.183402\pi\)
\(558\) −2.69856 −0.114239
\(559\) 24.5714 1.03926
\(560\) 0 0
\(561\) 0.767190 0.0323908
\(562\) 33.1845 1.39980
\(563\) 15.5253 0.654314 0.327157 0.944970i \(-0.393909\pi\)
0.327157 + 0.944970i \(0.393909\pi\)
\(564\) 1.24473 0.0524125
\(565\) −12.8283 −0.539690
\(566\) −33.0173 −1.38782
\(567\) 0 0
\(568\) −8.33560 −0.349754
\(569\) 34.0553 1.42767 0.713836 0.700313i \(-0.246956\pi\)
0.713836 + 0.700313i \(0.246956\pi\)
\(570\) −15.6596 −0.655910
\(571\) 29.7572 1.24530 0.622650 0.782500i \(-0.286056\pi\)
0.622650 + 0.782500i \(0.286056\pi\)
\(572\) 1.05897 0.0442778
\(573\) −4.73960 −0.198000
\(574\) 0 0
\(575\) 0.384310 0.0160268
\(576\) 7.26778 0.302824
\(577\) −2.09294 −0.0871304 −0.0435652 0.999051i \(-0.513872\pi\)
−0.0435652 + 0.999051i \(0.513872\pi\)
\(578\) −24.7931 −1.03126
\(579\) 20.9142 0.869166
\(580\) 0.853265 0.0354299
\(581\) 0 0
\(582\) 11.9424 0.495026
\(583\) 19.1598 0.793519
\(584\) −9.02261 −0.373358
\(585\) −6.62400 −0.273869
\(586\) 8.40317 0.347132
\(587\) −35.6427 −1.47113 −0.735565 0.677454i \(-0.763083\pi\)
−0.735565 + 0.677454i \(0.763083\pi\)
\(588\) 0 0
\(589\) −9.11122 −0.375421
\(590\) −21.4422 −0.882762
\(591\) −19.8120 −0.814956
\(592\) 19.3173 0.793936
\(593\) −24.5094 −1.00648 −0.503240 0.864147i \(-0.667859\pi\)
−0.503240 + 0.864147i \(0.667859\pi\)
\(594\) 3.17731 0.130367
\(595\) 0 0
\(596\) −1.50842 −0.0617871
\(597\) 12.7673 0.522531
\(598\) −4.53014 −0.185251
\(599\) 5.61551 0.229443 0.114722 0.993398i \(-0.463402\pi\)
0.114722 + 0.993398i \(0.463402\pi\)
\(600\) 1.03965 0.0424433
\(601\) 13.1302 0.535594 0.267797 0.963475i \(-0.413704\pi\)
0.267797 + 0.963475i \(0.413704\pi\)
\(602\) 0 0
\(603\) −2.20673 −0.0898652
\(604\) −2.04473 −0.0831989
\(605\) 13.5859 0.552347
\(606\) 26.6434 1.08232
\(607\) 0.0961849 0.00390403 0.00195201 0.999998i \(-0.499379\pi\)
0.00195201 + 0.999998i \(0.499379\pi\)
\(608\) −4.44691 −0.180346
\(609\) 0 0
\(610\) 7.42248 0.300527
\(611\) 24.1627 0.977519
\(612\) 0.0563485 0.00227775
\(613\) 2.52392 0.101940 0.0509701 0.998700i \(-0.483769\pi\)
0.0509701 + 0.998700i \(0.483769\pi\)
\(614\) 14.6275 0.590317
\(615\) −22.5917 −0.910986
\(616\) 0 0
\(617\) 22.7394 0.915453 0.457727 0.889093i \(-0.348664\pi\)
0.457727 + 0.889093i \(0.348664\pi\)
\(618\) 5.74415 0.231063
\(619\) −23.8721 −0.959500 −0.479750 0.877405i \(-0.659273\pi\)
−0.479750 + 0.877405i \(0.659273\pi\)
\(620\) 0.626717 0.0251696
\(621\) −1.00000 −0.0401286
\(622\) −25.9734 −1.04144
\(623\) 0 0
\(624\) −13.2344 −0.529802
\(625\) −22.9308 −0.917230
\(626\) −25.1715 −1.00605
\(627\) 10.7277 0.428422
\(628\) −1.19654 −0.0477473
\(629\) −1.59659 −0.0636604
\(630\) 0 0
\(631\) 13.5166 0.538086 0.269043 0.963128i \(-0.413293\pi\)
0.269043 + 0.963128i \(0.413293\pi\)
\(632\) 13.9393 0.554477
\(633\) 3.77646 0.150101
\(634\) 9.47215 0.376187
\(635\) 16.8579 0.668985
\(636\) 1.40725 0.0558012
\(637\) 0 0
\(638\) −7.94502 −0.314546
\(639\) 3.08130 0.121894
\(640\) −26.7936 −1.05911
\(641\) 17.3683 0.686005 0.343002 0.939335i \(-0.388556\pi\)
0.343002 + 0.939335i \(0.388556\pi\)
\(642\) −7.96688 −0.314428
\(643\) 10.2840 0.405561 0.202780 0.979224i \(-0.435002\pi\)
0.202780 + 0.979224i \(0.435002\pi\)
\(644\) 0 0
\(645\) −17.1216 −0.674164
\(646\) 2.58592 0.101742
\(647\) −43.6516 −1.71612 −0.858060 0.513550i \(-0.828330\pi\)
−0.858060 + 0.513550i \(0.828330\pi\)
\(648\) −2.70522 −0.106271
\(649\) 14.6890 0.576595
\(650\) −1.74098 −0.0682868
\(651\) 0 0
\(652\) −2.86109 −0.112049
\(653\) 5.79970 0.226960 0.113480 0.993540i \(-0.463800\pi\)
0.113480 + 0.993540i \(0.463800\pi\)
\(654\) 22.4594 0.878234
\(655\) 11.0525 0.431858
\(656\) −45.1372 −1.76231
\(657\) 3.33525 0.130121
\(658\) 0 0
\(659\) 4.50649 0.175548 0.0877740 0.996140i \(-0.472025\pi\)
0.0877740 + 0.996140i \(0.472025\pi\)
\(660\) −0.737905 −0.0287229
\(661\) −19.2687 −0.749467 −0.374734 0.927133i \(-0.622266\pi\)
−0.374734 + 0.927133i \(0.622266\pi\)
\(662\) −9.66826 −0.375768
\(663\) 1.09384 0.0424812
\(664\) −12.7391 −0.494374
\(665\) 0 0
\(666\) −6.61229 −0.256221
\(667\) 2.50055 0.0968215
\(668\) 2.67414 0.103465
\(669\) 3.14153 0.121459
\(670\) 6.96590 0.269116
\(671\) −5.08478 −0.196296
\(672\) 0 0
\(673\) −40.9683 −1.57921 −0.789606 0.613615i \(-0.789715\pi\)
−0.789606 + 0.613615i \(0.789715\pi\)
\(674\) 50.6409 1.95061
\(675\) −0.384310 −0.0147921
\(676\) −0.554927 −0.0213433
\(677\) 20.8977 0.803165 0.401582 0.915823i \(-0.368460\pi\)
0.401582 + 0.915823i \(0.368460\pi\)
\(678\) 8.77323 0.336934
\(679\) 0 0
\(680\) 2.06193 0.0790713
\(681\) 19.6642 0.753533
\(682\) −5.83556 −0.223455
\(683\) 50.4475 1.93032 0.965159 0.261663i \(-0.0842709\pi\)
0.965159 + 0.261663i \(0.0842709\pi\)
\(684\) 0.787926 0.0301271
\(685\) −25.1558 −0.961152
\(686\) 0 0
\(687\) −3.40015 −0.129724
\(688\) −34.2082 −1.30418
\(689\) 27.3176 1.04072
\(690\) 3.15666 0.120172
\(691\) 21.7766 0.828422 0.414211 0.910181i \(-0.364057\pi\)
0.414211 + 0.910181i \(0.364057\pi\)
\(692\) −2.67120 −0.101544
\(693\) 0 0
\(694\) −34.7023 −1.31728
\(695\) 7.68437 0.291485
\(696\) 6.76454 0.256409
\(697\) 3.73063 0.141308
\(698\) 21.6717 0.820287
\(699\) −19.1227 −0.723287
\(700\) 0 0
\(701\) 5.05307 0.190852 0.0954259 0.995437i \(-0.469579\pi\)
0.0954259 + 0.995437i \(0.469579\pi\)
\(702\) 4.53014 0.170979
\(703\) −22.3253 −0.842015
\(704\) 15.7164 0.592334
\(705\) −16.8369 −0.634114
\(706\) −30.2748 −1.13941
\(707\) 0 0
\(708\) 1.07888 0.0405468
\(709\) −19.4725 −0.731304 −0.365652 0.930752i \(-0.619154\pi\)
−0.365652 + 0.930752i \(0.619154\pi\)
\(710\) −9.72660 −0.365033
\(711\) −5.15275 −0.193243
\(712\) −5.08431 −0.190542
\(713\) 1.83663 0.0687824
\(714\) 0 0
\(715\) −14.3242 −0.535696
\(716\) 0.502973 0.0187970
\(717\) 3.10915 0.116113
\(718\) −39.2746 −1.46571
\(719\) 21.1483 0.788698 0.394349 0.918961i \(-0.370970\pi\)
0.394349 + 0.918961i \(0.370970\pi\)
\(720\) 9.22193 0.343681
\(721\) 0 0
\(722\) 8.24249 0.306754
\(723\) −20.0711 −0.746450
\(724\) −3.67947 −0.136747
\(725\) 0.960985 0.0356901
\(726\) −9.29139 −0.344836
\(727\) 36.0941 1.33866 0.669328 0.742967i \(-0.266582\pi\)
0.669328 + 0.742967i \(0.266582\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −10.5283 −0.389668
\(731\) 2.82734 0.104573
\(732\) −0.373467 −0.0138037
\(733\) −12.5279 −0.462728 −0.231364 0.972867i \(-0.574319\pi\)
−0.231364 + 0.972867i \(0.574319\pi\)
\(734\) −49.1502 −1.81417
\(735\) 0 0
\(736\) 0.896404 0.0330419
\(737\) −4.77201 −0.175779
\(738\) 15.4504 0.568738
\(739\) 28.8783 1.06230 0.531152 0.847276i \(-0.321759\pi\)
0.531152 + 0.847276i \(0.321759\pi\)
\(740\) 1.53565 0.0564516
\(741\) 15.2952 0.561885
\(742\) 0 0
\(743\) 4.69965 0.172413 0.0862066 0.996277i \(-0.472525\pi\)
0.0862066 + 0.996277i \(0.472525\pi\)
\(744\) 4.96850 0.182154
\(745\) 20.4037 0.747533
\(746\) 38.7051 1.41710
\(747\) 4.70908 0.172296
\(748\) 0.121852 0.00445536
\(749\) 0 0
\(750\) 16.9964 0.620622
\(751\) −49.0318 −1.78920 −0.894598 0.446871i \(-0.852538\pi\)
−0.894598 + 0.446871i \(0.852538\pi\)
\(752\) −33.6393 −1.22670
\(753\) −4.73370 −0.172505
\(754\) −11.3278 −0.412535
\(755\) 27.6582 1.00658
\(756\) 0 0
\(757\) −9.25720 −0.336459 −0.168229 0.985748i \(-0.553805\pi\)
−0.168229 + 0.985748i \(0.553805\pi\)
\(758\) 28.2604 1.02646
\(759\) −2.16247 −0.0784929
\(760\) 28.8321 1.04585
\(761\) 9.75714 0.353696 0.176848 0.984238i \(-0.443410\pi\)
0.176848 + 0.984238i \(0.443410\pi\)
\(762\) −11.5291 −0.417654
\(763\) 0 0
\(764\) −0.752788 −0.0272349
\(765\) −0.762202 −0.0275575
\(766\) −24.0919 −0.870475
\(767\) 20.9433 0.756218
\(768\) 3.78850 0.136706
\(769\) −3.88254 −0.140008 −0.0700041 0.997547i \(-0.522301\pi\)
−0.0700041 + 0.997547i \(0.522301\pi\)
\(770\) 0 0
\(771\) −3.68563 −0.132735
\(772\) 3.32180 0.119554
\(773\) −32.6917 −1.17584 −0.587919 0.808920i \(-0.700053\pi\)
−0.587919 + 0.808920i \(0.700053\pi\)
\(774\) 11.7094 0.420887
\(775\) 0.705836 0.0253544
\(776\) −21.9879 −0.789320
\(777\) 0 0
\(778\) 33.2823 1.19323
\(779\) 52.1657 1.86903
\(780\) −1.05209 −0.0376707
\(781\) 6.66323 0.238429
\(782\) −0.521268 −0.0186405
\(783\) −2.50055 −0.0893622
\(784\) 0 0
\(785\) 16.1851 0.577672
\(786\) −7.55879 −0.269613
\(787\) 29.4878 1.05113 0.525563 0.850755i \(-0.323855\pi\)
0.525563 + 0.850755i \(0.323855\pi\)
\(788\) −3.14673 −0.112097
\(789\) −4.58029 −0.163063
\(790\) 16.2655 0.578699
\(791\) 0 0
\(792\) −5.84998 −0.207870
\(793\) −7.24976 −0.257447
\(794\) 22.3384 0.792759
\(795\) −19.0353 −0.675112
\(796\) 2.02783 0.0718744
\(797\) 31.2478 1.10685 0.553427 0.832897i \(-0.313320\pi\)
0.553427 + 0.832897i \(0.313320\pi\)
\(798\) 0 0
\(799\) 2.78032 0.0983607
\(800\) 0.344497 0.0121798
\(801\) 1.87944 0.0664068
\(802\) 10.3658 0.366029
\(803\) 7.21240 0.254520
\(804\) −0.350494 −0.0123610
\(805\) 0 0
\(806\) −8.32020 −0.293066
\(807\) −18.9657 −0.667626
\(808\) −49.0551 −1.72575
\(809\) −6.47854 −0.227773 −0.113887 0.993494i \(-0.536330\pi\)
−0.113887 + 0.993494i \(0.536330\pi\)
\(810\) −3.15666 −0.110914
\(811\) −16.2398 −0.570257 −0.285128 0.958489i \(-0.592036\pi\)
−0.285128 + 0.958489i \(0.592036\pi\)
\(812\) 0 0
\(813\) 5.77817 0.202649
\(814\) −14.2989 −0.501177
\(815\) 38.7008 1.35563
\(816\) −1.52284 −0.0533102
\(817\) 39.5349 1.38315
\(818\) 40.3079 1.40933
\(819\) 0 0
\(820\) −3.58823 −0.125306
\(821\) −39.9374 −1.39382 −0.696912 0.717157i \(-0.745443\pi\)
−0.696912 + 0.717157i \(0.745443\pi\)
\(822\) 17.2040 0.600057
\(823\) 51.5045 1.79533 0.897667 0.440675i \(-0.145261\pi\)
0.897667 + 0.440675i \(0.145261\pi\)
\(824\) −10.5760 −0.368431
\(825\) −0.831061 −0.0289338
\(826\) 0 0
\(827\) −11.0594 −0.384571 −0.192286 0.981339i \(-0.561590\pi\)
−0.192286 + 0.981339i \(0.561590\pi\)
\(828\) −0.158829 −0.00551970
\(829\) 9.76537 0.339165 0.169583 0.985516i \(-0.445758\pi\)
0.169583 + 0.985516i \(0.445758\pi\)
\(830\) −14.8650 −0.515971
\(831\) −15.5279 −0.538657
\(832\) 22.4081 0.776859
\(833\) 0 0
\(834\) −5.25532 −0.181977
\(835\) −36.1719 −1.25178
\(836\) 1.70387 0.0589295
\(837\) −1.83663 −0.0634833
\(838\) 11.8454 0.409194
\(839\) 27.3798 0.945254 0.472627 0.881262i \(-0.343306\pi\)
0.472627 + 0.881262i \(0.343306\pi\)
\(840\) 0 0
\(841\) −22.7473 −0.784389
\(842\) −4.79730 −0.165326
\(843\) 22.5853 0.777880
\(844\) 0.599814 0.0206464
\(845\) 7.50625 0.258223
\(846\) 11.5147 0.395884
\(847\) 0 0
\(848\) −38.0316 −1.30601
\(849\) −22.4715 −0.771221
\(850\) −0.200329 −0.00687121
\(851\) 4.50032 0.154269
\(852\) 0.489401 0.0167666
\(853\) −29.0973 −0.996274 −0.498137 0.867098i \(-0.665982\pi\)
−0.498137 + 0.867098i \(0.665982\pi\)
\(854\) 0 0
\(855\) −10.6579 −0.364493
\(856\) 14.6684 0.501355
\(857\) −44.9912 −1.53687 −0.768435 0.639928i \(-0.778964\pi\)
−0.768435 + 0.639928i \(0.778964\pi\)
\(858\) 9.79631 0.334440
\(859\) −9.64929 −0.329230 −0.164615 0.986358i \(-0.552638\pi\)
−0.164615 + 0.986358i \(0.552638\pi\)
\(860\) −2.71942 −0.0927314
\(861\) 0 0
\(862\) 27.5658 0.938895
\(863\) 12.9216 0.439856 0.219928 0.975516i \(-0.429418\pi\)
0.219928 + 0.975516i \(0.429418\pi\)
\(864\) −0.896404 −0.0304963
\(865\) 36.1322 1.22853
\(866\) 5.98993 0.203546
\(867\) −16.8741 −0.573076
\(868\) 0 0
\(869\) −11.1427 −0.377990
\(870\) 7.89337 0.267610
\(871\) −6.80381 −0.230538
\(872\) −41.3517 −1.40034
\(873\) 8.12795 0.275089
\(874\) −7.28892 −0.246552
\(875\) 0 0
\(876\) 0.529736 0.0178981
\(877\) 32.3173 1.09128 0.545639 0.838020i \(-0.316287\pi\)
0.545639 + 0.838020i \(0.316287\pi\)
\(878\) 29.4975 0.995493
\(879\) 5.71918 0.192903
\(880\) 19.9422 0.672251
\(881\) 46.4011 1.56329 0.781647 0.623721i \(-0.214380\pi\)
0.781647 + 0.623721i \(0.214380\pi\)
\(882\) 0 0
\(883\) −4.37111 −0.147100 −0.0735498 0.997292i \(-0.523433\pi\)
−0.0735498 + 0.997292i \(0.523433\pi\)
\(884\) 0.173734 0.00584331
\(885\) −14.5935 −0.490557
\(886\) 26.6090 0.893946
\(887\) 25.3672 0.851749 0.425874 0.904782i \(-0.359967\pi\)
0.425874 + 0.904782i \(0.359967\pi\)
\(888\) 12.1744 0.408545
\(889\) 0 0
\(890\) −5.93275 −0.198866
\(891\) 2.16247 0.0724456
\(892\) 0.498968 0.0167067
\(893\) 38.8774 1.30098
\(894\) −13.9540 −0.466692
\(895\) −6.80349 −0.227416
\(896\) 0 0
\(897\) −3.08320 −0.102945
\(898\) 32.2358 1.07572
\(899\) 4.59258 0.153171
\(900\) −0.0610398 −0.00203466
\(901\) 3.14335 0.104720
\(902\) 33.4111 1.11247
\(903\) 0 0
\(904\) −16.1530 −0.537241
\(905\) 49.7706 1.65443
\(906\) −18.9153 −0.628420
\(907\) −50.6441 −1.68161 −0.840804 0.541340i \(-0.817917\pi\)
−0.840804 + 0.541340i \(0.817917\pi\)
\(908\) 3.12325 0.103649
\(909\) 18.1335 0.601450
\(910\) 0 0
\(911\) −6.29108 −0.208433 −0.104216 0.994555i \(-0.533233\pi\)
−0.104216 + 0.994555i \(0.533233\pi\)
\(912\) −21.2940 −0.705116
\(913\) 10.1833 0.337017
\(914\) 37.7758 1.24951
\(915\) 5.05173 0.167005
\(916\) −0.540044 −0.0178436
\(917\) 0 0
\(918\) 0.521268 0.0172044
\(919\) 26.3435 0.868991 0.434496 0.900674i \(-0.356927\pi\)
0.434496 + 0.900674i \(0.356927\pi\)
\(920\) −5.81195 −0.191614
\(921\) 9.95544 0.328043
\(922\) −43.4917 −1.43232
\(923\) 9.50027 0.312705
\(924\) 0 0
\(925\) 1.72952 0.0568662
\(926\) 10.2875 0.338069
\(927\) 3.90946 0.128403
\(928\) 2.24150 0.0735808
\(929\) 51.3944 1.68619 0.843097 0.537762i \(-0.180730\pi\)
0.843097 + 0.537762i \(0.180730\pi\)
\(930\) 5.79762 0.190111
\(931\) 0 0
\(932\) −3.03725 −0.0994884
\(933\) −17.6774 −0.578733
\(934\) 31.6666 1.03616
\(935\) −1.64824 −0.0539033
\(936\) −8.34076 −0.272626
\(937\) −9.83635 −0.321339 −0.160670 0.987008i \(-0.551365\pi\)
−0.160670 + 0.987008i \(0.551365\pi\)
\(938\) 0 0
\(939\) −17.1317 −0.559071
\(940\) −2.67419 −0.0872225
\(941\) 43.7737 1.42698 0.713491 0.700664i \(-0.247113\pi\)
0.713491 + 0.700664i \(0.247113\pi\)
\(942\) −11.0690 −0.360647
\(943\) −10.5155 −0.342433
\(944\) −29.1572 −0.948986
\(945\) 0 0
\(946\) 25.3214 0.823268
\(947\) −29.6306 −0.962865 −0.481432 0.876483i \(-0.659883\pi\)
−0.481432 + 0.876483i \(0.659883\pi\)
\(948\) −0.818408 −0.0265806
\(949\) 10.2833 0.333809
\(950\) −2.80121 −0.0908832
\(951\) 6.44673 0.209049
\(952\) 0 0
\(953\) 38.6354 1.25152 0.625762 0.780014i \(-0.284788\pi\)
0.625762 + 0.780014i \(0.284788\pi\)
\(954\) 13.0182 0.421479
\(955\) 10.1826 0.329502
\(956\) 0.493825 0.0159714
\(957\) −5.40737 −0.174795
\(958\) 2.13048 0.0688328
\(959\) 0 0
\(960\) −15.6142 −0.503947
\(961\) −27.6268 −0.891187
\(962\) −20.3871 −0.657305
\(963\) −5.42225 −0.174729
\(964\) −3.18787 −0.102674
\(965\) −44.9325 −1.44643
\(966\) 0 0
\(967\) −46.5223 −1.49606 −0.748028 0.663668i \(-0.768999\pi\)
−0.748028 + 0.663668i \(0.768999\pi\)
\(968\) 17.1070 0.549841
\(969\) 1.75997 0.0565385
\(970\) −25.6571 −0.823801
\(971\) −5.98111 −0.191943 −0.0959715 0.995384i \(-0.530596\pi\)
−0.0959715 + 0.995384i \(0.530596\pi\)
\(972\) 0.158829 0.00509446
\(973\) 0 0
\(974\) −43.1553 −1.38279
\(975\) −1.18491 −0.0379474
\(976\) 10.0931 0.323073
\(977\) −10.9369 −0.349901 −0.174950 0.984577i \(-0.555977\pi\)
−0.174950 + 0.984577i \(0.555977\pi\)
\(978\) −26.4673 −0.846332
\(979\) 4.06424 0.129894
\(980\) 0 0
\(981\) 15.2859 0.488040
\(982\) −32.3697 −1.03296
\(983\) −56.6707 −1.80751 −0.903757 0.428045i \(-0.859202\pi\)
−0.903757 + 0.428045i \(0.859202\pi\)
\(984\) −28.4468 −0.906852
\(985\) 42.5644 1.35621
\(986\) −1.30345 −0.0415104
\(987\) 0 0
\(988\) 2.42934 0.0772874
\(989\) −7.96942 −0.253413
\(990\) −6.82619 −0.216951
\(991\) 9.57119 0.304039 0.152019 0.988378i \(-0.451422\pi\)
0.152019 + 0.988378i \(0.451422\pi\)
\(992\) 1.64636 0.0522721
\(993\) −6.58020 −0.208816
\(994\) 0 0
\(995\) −27.4295 −0.869574
\(996\) 0.747941 0.0236994
\(997\) −56.0202 −1.77418 −0.887089 0.461598i \(-0.847276\pi\)
−0.887089 + 0.461598i \(0.847276\pi\)
\(998\) −19.1046 −0.604745
\(999\) −4.50032 −0.142384
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 3381.2.a.bl.1.7 yes 10
7.6 odd 2 3381.2.a.bk.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3381.2.a.bk.1.7 10 7.6 odd 2
3381.2.a.bl.1.7 yes 10 1.1 even 1 trivial