Properties

Label 3344.2.a.v.1.6
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
Analytic rank $1$
Dimension $6$
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] = [3344,2,Mod(1,3344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3344.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.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: \(26.7019744359\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.576096652.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 8x^{4} + 11x^{3} + 16x^{2} - 6x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(2.89799\) of defining polynomial
Character \(\chi\) \(=\) 3344.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.89799 q^{3} -1.20786 q^{5} -1.24216 q^{7} +0.602364 q^{9} -1.00000 q^{11} +1.24216 q^{13} -2.29250 q^{15} -0.690134 q^{17} -1.00000 q^{19} -2.35761 q^{21} +3.95037 q^{23} -3.54108 q^{25} -4.55069 q^{27} -2.09107 q^{29} +5.19253 q^{31} -1.89799 q^{33} +1.50035 q^{35} -6.64051 q^{37} +2.35761 q^{39} +6.34801 q^{41} -10.9114 q^{43} -0.727569 q^{45} -10.2803 q^{47} -5.45703 q^{49} -1.30987 q^{51} +6.93872 q^{53} +1.20786 q^{55} -1.89799 q^{57} -13.1182 q^{59} +8.96381 q^{61} -0.748234 q^{63} -1.50035 q^{65} +12.4308 q^{67} +7.49776 q^{69} -8.61984 q^{71} -6.72185 q^{73} -6.72094 q^{75} +1.24216 q^{77} -9.82770 q^{79} -10.4442 q^{81} -14.7484 q^{83} +0.833582 q^{85} -3.96882 q^{87} -6.81604 q^{89} -1.54297 q^{91} +9.85537 q^{93} +1.20786 q^{95} +5.45498 q^{97} -0.602364 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} + q^{5} + 4 q^{9} - 6 q^{11} - 7 q^{15} + 3 q^{17} - 6 q^{19} + 7 q^{21} - 7 q^{23} + 3 q^{25} - 13 q^{27} - 4 q^{29} - 7 q^{31} + 4 q^{33} - 6 q^{35} - 2 q^{37} - 7 q^{39} + 7 q^{41} - 21 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.89799 1.09580 0.547902 0.836542i \(-0.315427\pi\)
0.547902 + 0.836542i \(0.315427\pi\)
\(4\) 0 0
\(5\) −1.20786 −0.540170 −0.270085 0.962837i \(-0.587052\pi\)
−0.270085 + 0.962837i \(0.587052\pi\)
\(6\) 0 0
\(7\) −1.24216 −0.469493 −0.234747 0.972057i \(-0.575426\pi\)
−0.234747 + 0.972057i \(0.575426\pi\)
\(8\) 0 0
\(9\) 0.602364 0.200788
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 1.24216 0.344514 0.172257 0.985052i \(-0.444894\pi\)
0.172257 + 0.985052i \(0.444894\pi\)
\(14\) 0 0
\(15\) −2.29250 −0.591920
\(16\) 0 0
\(17\) −0.690134 −0.167382 −0.0836910 0.996492i \(-0.526671\pi\)
−0.0836910 + 0.996492i \(0.526671\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −2.35761 −0.514473
\(22\) 0 0
\(23\) 3.95037 0.823709 0.411855 0.911250i \(-0.364881\pi\)
0.411855 + 0.911250i \(0.364881\pi\)
\(24\) 0 0
\(25\) −3.54108 −0.708217
\(26\) 0 0
\(27\) −4.55069 −0.875780
\(28\) 0 0
\(29\) −2.09107 −0.388301 −0.194151 0.980972i \(-0.562195\pi\)
−0.194151 + 0.980972i \(0.562195\pi\)
\(30\) 0 0
\(31\) 5.19253 0.932607 0.466303 0.884625i \(-0.345586\pi\)
0.466303 + 0.884625i \(0.345586\pi\)
\(32\) 0 0
\(33\) −1.89799 −0.330398
\(34\) 0 0
\(35\) 1.50035 0.253606
\(36\) 0 0
\(37\) −6.64051 −1.09169 −0.545846 0.837885i \(-0.683792\pi\)
−0.545846 + 0.837885i \(0.683792\pi\)
\(38\) 0 0
\(39\) 2.35761 0.377520
\(40\) 0 0
\(41\) 6.34801 0.991392 0.495696 0.868496i \(-0.334913\pi\)
0.495696 + 0.868496i \(0.334913\pi\)
\(42\) 0 0
\(43\) −10.9114 −1.66398 −0.831989 0.554793i \(-0.812798\pi\)
−0.831989 + 0.554793i \(0.812798\pi\)
\(44\) 0 0
\(45\) −0.727569 −0.108460
\(46\) 0 0
\(47\) −10.2803 −1.49954 −0.749768 0.661701i \(-0.769835\pi\)
−0.749768 + 0.661701i \(0.769835\pi\)
\(48\) 0 0
\(49\) −5.45703 −0.779576
\(50\) 0 0
\(51\) −1.30987 −0.183418
\(52\) 0 0
\(53\) 6.93872 0.953107 0.476553 0.879146i \(-0.341886\pi\)
0.476553 + 0.879146i \(0.341886\pi\)
\(54\) 0 0
\(55\) 1.20786 0.162867
\(56\) 0 0
\(57\) −1.89799 −0.251395
\(58\) 0 0
\(59\) −13.1182 −1.70784 −0.853922 0.520401i \(-0.825783\pi\)
−0.853922 + 0.520401i \(0.825783\pi\)
\(60\) 0 0
\(61\) 8.96381 1.14770 0.573849 0.818961i \(-0.305450\pi\)
0.573849 + 0.818961i \(0.305450\pi\)
\(62\) 0 0
\(63\) −0.748234 −0.0942686
\(64\) 0 0
\(65\) −1.50035 −0.186096
\(66\) 0 0
\(67\) 12.4308 1.51866 0.759330 0.650706i \(-0.225527\pi\)
0.759330 + 0.650706i \(0.225527\pi\)
\(68\) 0 0
\(69\) 7.49776 0.902625
\(70\) 0 0
\(71\) −8.61984 −1.02299 −0.511493 0.859287i \(-0.670908\pi\)
−0.511493 + 0.859287i \(0.670908\pi\)
\(72\) 0 0
\(73\) −6.72185 −0.786733 −0.393367 0.919382i \(-0.628690\pi\)
−0.393367 + 0.919382i \(0.628690\pi\)
\(74\) 0 0
\(75\) −6.72094 −0.776067
\(76\) 0 0
\(77\) 1.24216 0.141558
\(78\) 0 0
\(79\) −9.82770 −1.10570 −0.552851 0.833280i \(-0.686460\pi\)
−0.552851 + 0.833280i \(0.686460\pi\)
\(80\) 0 0
\(81\) −10.4442 −1.16047
\(82\) 0 0
\(83\) −14.7484 −1.61885 −0.809423 0.587225i \(-0.800220\pi\)
−0.809423 + 0.587225i \(0.800220\pi\)
\(84\) 0 0
\(85\) 0.833582 0.0904147
\(86\) 0 0
\(87\) −3.96882 −0.425502
\(88\) 0 0
\(89\) −6.81604 −0.722499 −0.361250 0.932469i \(-0.617650\pi\)
−0.361250 + 0.932469i \(0.617650\pi\)
\(90\) 0 0
\(91\) −1.54297 −0.161747
\(92\) 0 0
\(93\) 9.85537 1.02195
\(94\) 0 0
\(95\) 1.20786 0.123923
\(96\) 0 0
\(97\) 5.45498 0.553870 0.276935 0.960889i \(-0.410681\pi\)
0.276935 + 0.960889i \(0.410681\pi\)
\(98\) 0 0
\(99\) −0.602364 −0.0605398
\(100\) 0 0
\(101\) −5.16783 −0.514218 −0.257109 0.966382i \(-0.582770\pi\)
−0.257109 + 0.966382i \(0.582770\pi\)
\(102\) 0 0
\(103\) 4.39505 0.433057 0.216528 0.976276i \(-0.430527\pi\)
0.216528 + 0.976276i \(0.430527\pi\)
\(104\) 0 0
\(105\) 2.84765 0.277903
\(106\) 0 0
\(107\) −9.35443 −0.904327 −0.452164 0.891935i \(-0.649348\pi\)
−0.452164 + 0.891935i \(0.649348\pi\)
\(108\) 0 0
\(109\) 6.51783 0.624295 0.312147 0.950034i \(-0.398952\pi\)
0.312147 + 0.950034i \(0.398952\pi\)
\(110\) 0 0
\(111\) −12.6036 −1.19628
\(112\) 0 0
\(113\) 10.1370 0.953606 0.476803 0.879010i \(-0.341795\pi\)
0.476803 + 0.879010i \(0.341795\pi\)
\(114\) 0 0
\(115\) −4.77148 −0.444943
\(116\) 0 0
\(117\) 0.748234 0.0691742
\(118\) 0 0
\(119\) 0.857258 0.0785847
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 12.0485 1.08637
\(124\) 0 0
\(125\) 10.3164 0.922727
\(126\) 0 0
\(127\) −6.54810 −0.581050 −0.290525 0.956867i \(-0.593830\pi\)
−0.290525 + 0.956867i \(0.593830\pi\)
\(128\) 0 0
\(129\) −20.7098 −1.82339
\(130\) 0 0
\(131\) 4.23181 0.369735 0.184867 0.982763i \(-0.440814\pi\)
0.184867 + 0.982763i \(0.440814\pi\)
\(132\) 0 0
\(133\) 1.24216 0.107709
\(134\) 0 0
\(135\) 5.49657 0.473070
\(136\) 0 0
\(137\) −4.08669 −0.349149 −0.174575 0.984644i \(-0.555855\pi\)
−0.174575 + 0.984644i \(0.555855\pi\)
\(138\) 0 0
\(139\) −10.4334 −0.884946 −0.442473 0.896782i \(-0.645899\pi\)
−0.442473 + 0.896782i \(0.645899\pi\)
\(140\) 0 0
\(141\) −19.5119 −1.64320
\(142\) 0 0
\(143\) −1.24216 −0.103875
\(144\) 0 0
\(145\) 2.52571 0.209748
\(146\) 0 0
\(147\) −10.3574 −0.854263
\(148\) 0 0
\(149\) 14.5241 1.18986 0.594928 0.803779i \(-0.297180\pi\)
0.594928 + 0.803779i \(0.297180\pi\)
\(150\) 0 0
\(151\) −18.2065 −1.48162 −0.740812 0.671712i \(-0.765559\pi\)
−0.740812 + 0.671712i \(0.765559\pi\)
\(152\) 0 0
\(153\) −0.415712 −0.0336083
\(154\) 0 0
\(155\) −6.27183 −0.503766
\(156\) 0 0
\(157\) −10.7184 −0.855418 −0.427709 0.903917i \(-0.640679\pi\)
−0.427709 + 0.903917i \(0.640679\pi\)
\(158\) 0 0
\(159\) 13.1696 1.04442
\(160\) 0 0
\(161\) −4.90700 −0.386726
\(162\) 0 0
\(163\) −19.7222 −1.54476 −0.772380 0.635160i \(-0.780934\pi\)
−0.772380 + 0.635160i \(0.780934\pi\)
\(164\) 0 0
\(165\) 2.29250 0.178471
\(166\) 0 0
\(167\) 15.3603 1.18862 0.594308 0.804237i \(-0.297426\pi\)
0.594308 + 0.804237i \(0.297426\pi\)
\(168\) 0 0
\(169\) −11.4570 −0.881310
\(170\) 0 0
\(171\) −0.602364 −0.0460639
\(172\) 0 0
\(173\) 22.3724 1.70094 0.850471 0.526021i \(-0.176317\pi\)
0.850471 + 0.526021i \(0.176317\pi\)
\(174\) 0 0
\(175\) 4.39860 0.332503
\(176\) 0 0
\(177\) −24.8982 −1.87146
\(178\) 0 0
\(179\) −23.9764 −1.79208 −0.896039 0.443975i \(-0.853568\pi\)
−0.896039 + 0.443975i \(0.853568\pi\)
\(180\) 0 0
\(181\) −13.7766 −1.02401 −0.512004 0.858983i \(-0.671097\pi\)
−0.512004 + 0.858983i \(0.671097\pi\)
\(182\) 0 0
\(183\) 17.0132 1.25765
\(184\) 0 0
\(185\) 8.02077 0.589699
\(186\) 0 0
\(187\) 0.690134 0.0504676
\(188\) 0 0
\(189\) 5.65269 0.411173
\(190\) 0 0
\(191\) 17.4390 1.26184 0.630920 0.775848i \(-0.282678\pi\)
0.630920 + 0.775848i \(0.282678\pi\)
\(192\) 0 0
\(193\) 15.6528 1.12671 0.563355 0.826215i \(-0.309510\pi\)
0.563355 + 0.826215i \(0.309510\pi\)
\(194\) 0 0
\(195\) −2.84765 −0.203925
\(196\) 0 0
\(197\) 6.77758 0.482883 0.241441 0.970415i \(-0.422380\pi\)
0.241441 + 0.970415i \(0.422380\pi\)
\(198\) 0 0
\(199\) −7.62848 −0.540769 −0.270384 0.962752i \(-0.587151\pi\)
−0.270384 + 0.962752i \(0.587151\pi\)
\(200\) 0 0
\(201\) 23.5935 1.66415
\(202\) 0 0
\(203\) 2.59744 0.182305
\(204\) 0 0
\(205\) −7.66748 −0.535520
\(206\) 0 0
\(207\) 2.37956 0.165391
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 14.2769 0.982863 0.491432 0.870916i \(-0.336474\pi\)
0.491432 + 0.870916i \(0.336474\pi\)
\(212\) 0 0
\(213\) −16.3604 −1.12099
\(214\) 0 0
\(215\) 13.1794 0.898830
\(216\) 0 0
\(217\) −6.44997 −0.437853
\(218\) 0 0
\(219\) −12.7580 −0.862106
\(220\) 0 0
\(221\) −0.857258 −0.0576654
\(222\) 0 0
\(223\) 12.6353 0.846124 0.423062 0.906101i \(-0.360955\pi\)
0.423062 + 0.906101i \(0.360955\pi\)
\(224\) 0 0
\(225\) −2.13302 −0.142201
\(226\) 0 0
\(227\) 19.9660 1.32519 0.662594 0.748979i \(-0.269456\pi\)
0.662594 + 0.748979i \(0.269456\pi\)
\(228\) 0 0
\(229\) −10.7465 −0.710150 −0.355075 0.934838i \(-0.615545\pi\)
−0.355075 + 0.934838i \(0.615545\pi\)
\(230\) 0 0
\(231\) 2.35761 0.155119
\(232\) 0 0
\(233\) 9.41198 0.616600 0.308300 0.951289i \(-0.400240\pi\)
0.308300 + 0.951289i \(0.400240\pi\)
\(234\) 0 0
\(235\) 12.4171 0.810004
\(236\) 0 0
\(237\) −18.6529 −1.21163
\(238\) 0 0
\(239\) 15.1921 0.982695 0.491348 0.870963i \(-0.336504\pi\)
0.491348 + 0.870963i \(0.336504\pi\)
\(240\) 0 0
\(241\) −1.01694 −0.0655069 −0.0327535 0.999463i \(-0.510428\pi\)
−0.0327535 + 0.999463i \(0.510428\pi\)
\(242\) 0 0
\(243\) −6.17101 −0.395870
\(244\) 0 0
\(245\) 6.59131 0.421103
\(246\) 0 0
\(247\) −1.24216 −0.0790369
\(248\) 0 0
\(249\) −27.9923 −1.77394
\(250\) 0 0
\(251\) 10.6269 0.670765 0.335382 0.942082i \(-0.391135\pi\)
0.335382 + 0.942082i \(0.391135\pi\)
\(252\) 0 0
\(253\) −3.95037 −0.248358
\(254\) 0 0
\(255\) 1.58213 0.0990768
\(256\) 0 0
\(257\) −7.78939 −0.485889 −0.242944 0.970040i \(-0.578113\pi\)
−0.242944 + 0.970040i \(0.578113\pi\)
\(258\) 0 0
\(259\) 8.24859 0.512542
\(260\) 0 0
\(261\) −1.25958 −0.0779662
\(262\) 0 0
\(263\) −31.5402 −1.94485 −0.972427 0.233206i \(-0.925078\pi\)
−0.972427 + 0.233206i \(0.925078\pi\)
\(264\) 0 0
\(265\) −8.38097 −0.514839
\(266\) 0 0
\(267\) −12.9368 −0.791718
\(268\) 0 0
\(269\) −10.9339 −0.666652 −0.333326 0.942812i \(-0.608171\pi\)
−0.333326 + 0.942812i \(0.608171\pi\)
\(270\) 0 0
\(271\) 16.0665 0.975969 0.487984 0.872852i \(-0.337732\pi\)
0.487984 + 0.872852i \(0.337732\pi\)
\(272\) 0 0
\(273\) −2.92854 −0.177243
\(274\) 0 0
\(275\) 3.54108 0.213535
\(276\) 0 0
\(277\) 22.6174 1.35895 0.679473 0.733700i \(-0.262208\pi\)
0.679473 + 0.733700i \(0.262208\pi\)
\(278\) 0 0
\(279\) 3.12779 0.187256
\(280\) 0 0
\(281\) 25.6436 1.52977 0.764884 0.644168i \(-0.222796\pi\)
0.764884 + 0.644168i \(0.222796\pi\)
\(282\) 0 0
\(283\) 1.15062 0.0683972 0.0341986 0.999415i \(-0.489112\pi\)
0.0341986 + 0.999415i \(0.489112\pi\)
\(284\) 0 0
\(285\) 2.29250 0.135796
\(286\) 0 0
\(287\) −7.88526 −0.465452
\(288\) 0 0
\(289\) −16.5237 −0.971983
\(290\) 0 0
\(291\) 10.3535 0.606933
\(292\) 0 0
\(293\) −4.77069 −0.278706 −0.139353 0.990243i \(-0.544502\pi\)
−0.139353 + 0.990243i \(0.544502\pi\)
\(294\) 0 0
\(295\) 15.8449 0.922526
\(296\) 0 0
\(297\) 4.55069 0.264058
\(298\) 0 0
\(299\) 4.90700 0.283779
\(300\) 0 0
\(301\) 13.5538 0.781226
\(302\) 0 0
\(303\) −9.80849 −0.563483
\(304\) 0 0
\(305\) −10.8270 −0.619952
\(306\) 0 0
\(307\) −25.7117 −1.46745 −0.733723 0.679449i \(-0.762219\pi\)
−0.733723 + 0.679449i \(0.762219\pi\)
\(308\) 0 0
\(309\) 8.34175 0.474546
\(310\) 0 0
\(311\) 29.0241 1.64580 0.822902 0.568183i \(-0.192353\pi\)
0.822902 + 0.568183i \(0.192353\pi\)
\(312\) 0 0
\(313\) 1.26466 0.0714829 0.0357415 0.999361i \(-0.488621\pi\)
0.0357415 + 0.999361i \(0.488621\pi\)
\(314\) 0 0
\(315\) 0.903758 0.0509210
\(316\) 0 0
\(317\) 22.1536 1.24427 0.622136 0.782909i \(-0.286265\pi\)
0.622136 + 0.782909i \(0.286265\pi\)
\(318\) 0 0
\(319\) 2.09107 0.117077
\(320\) 0 0
\(321\) −17.7546 −0.990966
\(322\) 0 0
\(323\) 0.690134 0.0384001
\(324\) 0 0
\(325\) −4.39860 −0.243991
\(326\) 0 0
\(327\) 12.3708 0.684105
\(328\) 0 0
\(329\) 12.7698 0.704022
\(330\) 0 0
\(331\) 34.4869 1.89557 0.947785 0.318909i \(-0.103316\pi\)
0.947785 + 0.318909i \(0.103316\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −15.0146 −0.820334
\(336\) 0 0
\(337\) 14.1022 0.768198 0.384099 0.923292i \(-0.374512\pi\)
0.384099 + 0.923292i \(0.374512\pi\)
\(338\) 0 0
\(339\) 19.2399 1.04497
\(340\) 0 0
\(341\) −5.19253 −0.281191
\(342\) 0 0
\(343\) 15.4737 0.835499
\(344\) 0 0
\(345\) −9.05622 −0.487570
\(346\) 0 0
\(347\) 1.82662 0.0980581 0.0490290 0.998797i \(-0.484387\pi\)
0.0490290 + 0.998797i \(0.484387\pi\)
\(348\) 0 0
\(349\) −27.7964 −1.48791 −0.743954 0.668231i \(-0.767052\pi\)
−0.743954 + 0.668231i \(0.767052\pi\)
\(350\) 0 0
\(351\) −5.65269 −0.301719
\(352\) 0 0
\(353\) −18.6245 −0.991282 −0.495641 0.868527i \(-0.665067\pi\)
−0.495641 + 0.868527i \(0.665067\pi\)
\(354\) 0 0
\(355\) 10.4115 0.552586
\(356\) 0 0
\(357\) 1.62707 0.0861135
\(358\) 0 0
\(359\) 15.8766 0.837933 0.418967 0.908002i \(-0.362392\pi\)
0.418967 + 0.908002i \(0.362392\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 1.89799 0.0996186
\(364\) 0 0
\(365\) 8.11903 0.424969
\(366\) 0 0
\(367\) 0.497768 0.0259833 0.0129916 0.999916i \(-0.495865\pi\)
0.0129916 + 0.999916i \(0.495865\pi\)
\(368\) 0 0
\(369\) 3.82381 0.199060
\(370\) 0 0
\(371\) −8.61902 −0.447477
\(372\) 0 0
\(373\) 4.47904 0.231916 0.115958 0.993254i \(-0.463006\pi\)
0.115958 + 0.993254i \(0.463006\pi\)
\(374\) 0 0
\(375\) 19.5804 1.01113
\(376\) 0 0
\(377\) −2.59744 −0.133775
\(378\) 0 0
\(379\) −28.2849 −1.45290 −0.726449 0.687220i \(-0.758831\pi\)
−0.726449 + 0.687220i \(0.758831\pi\)
\(380\) 0 0
\(381\) −12.4282 −0.636717
\(382\) 0 0
\(383\) −36.8401 −1.88244 −0.941220 0.337795i \(-0.890319\pi\)
−0.941220 + 0.337795i \(0.890319\pi\)
\(384\) 0 0
\(385\) −1.50035 −0.0764651
\(386\) 0 0
\(387\) −6.57265 −0.334107
\(388\) 0 0
\(389\) −28.1779 −1.42868 −0.714339 0.699800i \(-0.753272\pi\)
−0.714339 + 0.699800i \(0.753272\pi\)
\(390\) 0 0
\(391\) −2.72628 −0.137874
\(392\) 0 0
\(393\) 8.03193 0.405157
\(394\) 0 0
\(395\) 11.8704 0.597267
\(396\) 0 0
\(397\) −12.4847 −0.626590 −0.313295 0.949656i \(-0.601433\pi\)
−0.313295 + 0.949656i \(0.601433\pi\)
\(398\) 0 0
\(399\) 2.35761 0.118028
\(400\) 0 0
\(401\) 20.5902 1.02822 0.514112 0.857723i \(-0.328122\pi\)
0.514112 + 0.857723i \(0.328122\pi\)
\(402\) 0 0
\(403\) 6.44997 0.321296
\(404\) 0 0
\(405\) 12.6151 0.626852
\(406\) 0 0
\(407\) 6.64051 0.329158
\(408\) 0 0
\(409\) 8.94974 0.442536 0.221268 0.975213i \(-0.428980\pi\)
0.221268 + 0.975213i \(0.428980\pi\)
\(410\) 0 0
\(411\) −7.75649 −0.382600
\(412\) 0 0
\(413\) 16.2949 0.801822
\(414\) 0 0
\(415\) 17.8139 0.874452
\(416\) 0 0
\(417\) −19.8024 −0.969728
\(418\) 0 0
\(419\) 13.8590 0.677057 0.338528 0.940956i \(-0.390071\pi\)
0.338528 + 0.940956i \(0.390071\pi\)
\(420\) 0 0
\(421\) 34.0707 1.66051 0.830253 0.557387i \(-0.188196\pi\)
0.830253 + 0.557387i \(0.188196\pi\)
\(422\) 0 0
\(423\) −6.19248 −0.301089
\(424\) 0 0
\(425\) 2.44382 0.118543
\(426\) 0 0
\(427\) −11.1345 −0.538837
\(428\) 0 0
\(429\) −2.35761 −0.113827
\(430\) 0 0
\(431\) 13.2036 0.635997 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(432\) 0 0
\(433\) −7.74315 −0.372112 −0.186056 0.982539i \(-0.559571\pi\)
−0.186056 + 0.982539i \(0.559571\pi\)
\(434\) 0 0
\(435\) 4.79376 0.229843
\(436\) 0 0
\(437\) −3.95037 −0.188972
\(438\) 0 0
\(439\) 5.56801 0.265747 0.132873 0.991133i \(-0.457580\pi\)
0.132873 + 0.991133i \(0.457580\pi\)
\(440\) 0 0
\(441\) −3.28712 −0.156529
\(442\) 0 0
\(443\) −20.4542 −0.971807 −0.485904 0.874012i \(-0.661509\pi\)
−0.485904 + 0.874012i \(0.661509\pi\)
\(444\) 0 0
\(445\) 8.23280 0.390272
\(446\) 0 0
\(447\) 27.5665 1.30385
\(448\) 0 0
\(449\) 21.8802 1.03259 0.516295 0.856411i \(-0.327311\pi\)
0.516295 + 0.856411i \(0.327311\pi\)
\(450\) 0 0
\(451\) −6.34801 −0.298916
\(452\) 0 0
\(453\) −34.5558 −1.62357
\(454\) 0 0
\(455\) 1.86368 0.0873708
\(456\) 0 0
\(457\) −21.8243 −1.02090 −0.510449 0.859908i \(-0.670521\pi\)
−0.510449 + 0.859908i \(0.670521\pi\)
\(458\) 0 0
\(459\) 3.14058 0.146590
\(460\) 0 0
\(461\) −16.1526 −0.752300 −0.376150 0.926559i \(-0.622752\pi\)
−0.376150 + 0.926559i \(0.622752\pi\)
\(462\) 0 0
\(463\) −15.9853 −0.742899 −0.371450 0.928453i \(-0.621139\pi\)
−0.371450 + 0.928453i \(0.621139\pi\)
\(464\) 0 0
\(465\) −11.9039 −0.552029
\(466\) 0 0
\(467\) −0.504024 −0.0233234 −0.0116617 0.999932i \(-0.503712\pi\)
−0.0116617 + 0.999932i \(0.503712\pi\)
\(468\) 0 0
\(469\) −15.4410 −0.713001
\(470\) 0 0
\(471\) −20.3433 −0.937371
\(472\) 0 0
\(473\) 10.9114 0.501708
\(474\) 0 0
\(475\) 3.54108 0.162476
\(476\) 0 0
\(477\) 4.17963 0.191372
\(478\) 0 0
\(479\) 5.39563 0.246533 0.123266 0.992374i \(-0.460663\pi\)
0.123266 + 0.992374i \(0.460663\pi\)
\(480\) 0 0
\(481\) −8.24859 −0.376103
\(482\) 0 0
\(483\) −9.31344 −0.423776
\(484\) 0 0
\(485\) −6.58883 −0.299184
\(486\) 0 0
\(487\) 10.2897 0.466270 0.233135 0.972444i \(-0.425102\pi\)
0.233135 + 0.972444i \(0.425102\pi\)
\(488\) 0 0
\(489\) −37.4325 −1.69276
\(490\) 0 0
\(491\) −29.1413 −1.31513 −0.657563 0.753399i \(-0.728413\pi\)
−0.657563 + 0.753399i \(0.728413\pi\)
\(492\) 0 0
\(493\) 1.44312 0.0649946
\(494\) 0 0
\(495\) 0.727569 0.0327018
\(496\) 0 0
\(497\) 10.7072 0.480285
\(498\) 0 0
\(499\) −11.2496 −0.503603 −0.251801 0.967779i \(-0.581023\pi\)
−0.251801 + 0.967779i \(0.581023\pi\)
\(500\) 0 0
\(501\) 29.1537 1.30249
\(502\) 0 0
\(503\) −18.8290 −0.839542 −0.419771 0.907630i \(-0.637890\pi\)
−0.419771 + 0.907630i \(0.637890\pi\)
\(504\) 0 0
\(505\) 6.24199 0.277765
\(506\) 0 0
\(507\) −21.7453 −0.965744
\(508\) 0 0
\(509\) 18.4338 0.817064 0.408532 0.912744i \(-0.366041\pi\)
0.408532 + 0.912744i \(0.366041\pi\)
\(510\) 0 0
\(511\) 8.34963 0.369366
\(512\) 0 0
\(513\) 4.55069 0.200918
\(514\) 0 0
\(515\) −5.30858 −0.233924
\(516\) 0 0
\(517\) 10.2803 0.452127
\(518\) 0 0
\(519\) 42.4626 1.86390
\(520\) 0 0
\(521\) 2.55309 0.111853 0.0559265 0.998435i \(-0.482189\pi\)
0.0559265 + 0.998435i \(0.482189\pi\)
\(522\) 0 0
\(523\) 16.5592 0.724082 0.362041 0.932162i \(-0.382080\pi\)
0.362041 + 0.932162i \(0.382080\pi\)
\(524\) 0 0
\(525\) 8.34850 0.364358
\(526\) 0 0
\(527\) −3.58354 −0.156102
\(528\) 0 0
\(529\) −7.39456 −0.321503
\(530\) 0 0
\(531\) −7.90193 −0.342915
\(532\) 0 0
\(533\) 7.88526 0.341548
\(534\) 0 0
\(535\) 11.2988 0.488490
\(536\) 0 0
\(537\) −45.5069 −1.96377
\(538\) 0 0
\(539\) 5.45703 0.235051
\(540\) 0 0
\(541\) −15.2044 −0.653688 −0.326844 0.945078i \(-0.605985\pi\)
−0.326844 + 0.945078i \(0.605985\pi\)
\(542\) 0 0
\(543\) −26.1479 −1.12211
\(544\) 0 0
\(545\) −7.87260 −0.337225
\(546\) 0 0
\(547\) 12.4098 0.530604 0.265302 0.964165i \(-0.414528\pi\)
0.265302 + 0.964165i \(0.414528\pi\)
\(548\) 0 0
\(549\) 5.39947 0.230444
\(550\) 0 0
\(551\) 2.09107 0.0890824
\(552\) 0 0
\(553\) 12.2076 0.519120
\(554\) 0 0
\(555\) 15.2233 0.646195
\(556\) 0 0
\(557\) −24.5846 −1.04168 −0.520841 0.853654i \(-0.674382\pi\)
−0.520841 + 0.853654i \(0.674382\pi\)
\(558\) 0 0
\(559\) −13.5538 −0.573263
\(560\) 0 0
\(561\) 1.30987 0.0553026
\(562\) 0 0
\(563\) 9.33678 0.393498 0.196749 0.980454i \(-0.436962\pi\)
0.196749 + 0.980454i \(0.436962\pi\)
\(564\) 0 0
\(565\) −12.2440 −0.515109
\(566\) 0 0
\(567\) 12.9735 0.544834
\(568\) 0 0
\(569\) 42.4314 1.77882 0.889409 0.457112i \(-0.151116\pi\)
0.889409 + 0.457112i \(0.151116\pi\)
\(570\) 0 0
\(571\) −24.9246 −1.04306 −0.521531 0.853232i \(-0.674639\pi\)
−0.521531 + 0.853232i \(0.674639\pi\)
\(572\) 0 0
\(573\) 33.0990 1.38273
\(574\) 0 0
\(575\) −13.9886 −0.583365
\(576\) 0 0
\(577\) 31.0685 1.29340 0.646700 0.762745i \(-0.276149\pi\)
0.646700 + 0.762745i \(0.276149\pi\)
\(578\) 0 0
\(579\) 29.7088 1.23465
\(580\) 0 0
\(581\) 18.3199 0.760038
\(582\) 0 0
\(583\) −6.93872 −0.287372
\(584\) 0 0
\(585\) −0.903758 −0.0373658
\(586\) 0 0
\(587\) 7.06575 0.291635 0.145817 0.989312i \(-0.453419\pi\)
0.145817 + 0.989312i \(0.453419\pi\)
\(588\) 0 0
\(589\) −5.19253 −0.213955
\(590\) 0 0
\(591\) 12.8638 0.529145
\(592\) 0 0
\(593\) 19.6153 0.805502 0.402751 0.915310i \(-0.368054\pi\)
0.402751 + 0.915310i \(0.368054\pi\)
\(594\) 0 0
\(595\) −1.03544 −0.0424491
\(596\) 0 0
\(597\) −14.4788 −0.592577
\(598\) 0 0
\(599\) 22.0560 0.901182 0.450591 0.892731i \(-0.351213\pi\)
0.450591 + 0.892731i \(0.351213\pi\)
\(600\) 0 0
\(601\) −6.57976 −0.268394 −0.134197 0.990955i \(-0.542846\pi\)
−0.134197 + 0.990955i \(0.542846\pi\)
\(602\) 0 0
\(603\) 7.48784 0.304929
\(604\) 0 0
\(605\) −1.20786 −0.0491063
\(606\) 0 0
\(607\) −23.5127 −0.954350 −0.477175 0.878808i \(-0.658339\pi\)
−0.477175 + 0.878808i \(0.658339\pi\)
\(608\) 0 0
\(609\) 4.92992 0.199770
\(610\) 0 0
\(611\) −12.7698 −0.516611
\(612\) 0 0
\(613\) −1.17119 −0.0473040 −0.0236520 0.999720i \(-0.507529\pi\)
−0.0236520 + 0.999720i \(0.507529\pi\)
\(614\) 0 0
\(615\) −14.5528 −0.586825
\(616\) 0 0
\(617\) −20.2649 −0.815836 −0.407918 0.913019i \(-0.633745\pi\)
−0.407918 + 0.913019i \(0.633745\pi\)
\(618\) 0 0
\(619\) −48.3884 −1.94489 −0.972447 0.233125i \(-0.925105\pi\)
−0.972447 + 0.233125i \(0.925105\pi\)
\(620\) 0 0
\(621\) −17.9769 −0.721389
\(622\) 0 0
\(623\) 8.46664 0.339209
\(624\) 0 0
\(625\) 5.24470 0.209788
\(626\) 0 0
\(627\) 1.89799 0.0757984
\(628\) 0 0
\(629\) 4.58284 0.182730
\(630\) 0 0
\(631\) 30.5857 1.21760 0.608799 0.793325i \(-0.291652\pi\)
0.608799 + 0.793325i \(0.291652\pi\)
\(632\) 0 0
\(633\) 27.0974 1.07703
\(634\) 0 0
\(635\) 7.90916 0.313865
\(636\) 0 0
\(637\) −6.77852 −0.268575
\(638\) 0 0
\(639\) −5.19228 −0.205403
\(640\) 0 0
\(641\) 11.7378 0.463615 0.231807 0.972762i \(-0.425536\pi\)
0.231807 + 0.972762i \(0.425536\pi\)
\(642\) 0 0
\(643\) −31.4233 −1.23921 −0.619606 0.784913i \(-0.712708\pi\)
−0.619606 + 0.784913i \(0.712708\pi\)
\(644\) 0 0
\(645\) 25.0144 0.984942
\(646\) 0 0
\(647\) 21.0169 0.826260 0.413130 0.910672i \(-0.364436\pi\)
0.413130 + 0.910672i \(0.364436\pi\)
\(648\) 0 0
\(649\) 13.1182 0.514934
\(650\) 0 0
\(651\) −12.2420 −0.479801
\(652\) 0 0
\(653\) −24.5247 −0.959726 −0.479863 0.877343i \(-0.659314\pi\)
−0.479863 + 0.877343i \(0.659314\pi\)
\(654\) 0 0
\(655\) −5.11141 −0.199719
\(656\) 0 0
\(657\) −4.04900 −0.157967
\(658\) 0 0
\(659\) −34.6883 −1.35126 −0.675632 0.737239i \(-0.736129\pi\)
−0.675632 + 0.737239i \(0.736129\pi\)
\(660\) 0 0
\(661\) −5.13503 −0.199730 −0.0998648 0.995001i \(-0.531841\pi\)
−0.0998648 + 0.995001i \(0.531841\pi\)
\(662\) 0 0
\(663\) −1.62707 −0.0631900
\(664\) 0 0
\(665\) −1.50035 −0.0581812
\(666\) 0 0
\(667\) −8.26049 −0.319847
\(668\) 0 0
\(669\) 23.9817 0.927187
\(670\) 0 0
\(671\) −8.96381 −0.346044
\(672\) 0 0
\(673\) −17.9611 −0.692350 −0.346175 0.938170i \(-0.612520\pi\)
−0.346175 + 0.938170i \(0.612520\pi\)
\(674\) 0 0
\(675\) 16.1144 0.620242
\(676\) 0 0
\(677\) 9.95618 0.382647 0.191324 0.981527i \(-0.438722\pi\)
0.191324 + 0.981527i \(0.438722\pi\)
\(678\) 0 0
\(679\) −6.77598 −0.260038
\(680\) 0 0
\(681\) 37.8952 1.45215
\(682\) 0 0
\(683\) −37.3100 −1.42763 −0.713815 0.700335i \(-0.753034\pi\)
−0.713815 + 0.700335i \(0.753034\pi\)
\(684\) 0 0
\(685\) 4.93613 0.188600
\(686\) 0 0
\(687\) −20.3968 −0.778186
\(688\) 0 0
\(689\) 8.61902 0.328358
\(690\) 0 0
\(691\) 33.8278 1.28687 0.643434 0.765501i \(-0.277509\pi\)
0.643434 + 0.765501i \(0.277509\pi\)
\(692\) 0 0
\(693\) 0.748234 0.0284231
\(694\) 0 0
\(695\) 12.6020 0.478021
\(696\) 0 0
\(697\) −4.38097 −0.165941
\(698\) 0 0
\(699\) 17.8638 0.675673
\(700\) 0 0
\(701\) −48.3402 −1.82578 −0.912892 0.408201i \(-0.866156\pi\)
−0.912892 + 0.408201i \(0.866156\pi\)
\(702\) 0 0
\(703\) 6.64051 0.250451
\(704\) 0 0
\(705\) 23.5676 0.887606
\(706\) 0 0
\(707\) 6.41929 0.241422
\(708\) 0 0
\(709\) −21.6851 −0.814400 −0.407200 0.913339i \(-0.633495\pi\)
−0.407200 + 0.913339i \(0.633495\pi\)
\(710\) 0 0
\(711\) −5.91985 −0.222012
\(712\) 0 0
\(713\) 20.5124 0.768197
\(714\) 0 0
\(715\) 1.50035 0.0561100
\(716\) 0 0
\(717\) 28.8345 1.07684
\(718\) 0 0
\(719\) 5.43335 0.202630 0.101315 0.994854i \(-0.467695\pi\)
0.101315 + 0.994854i \(0.467695\pi\)
\(720\) 0 0
\(721\) −5.45936 −0.203317
\(722\) 0 0
\(723\) −1.93014 −0.0717828
\(724\) 0 0
\(725\) 7.40464 0.275001
\(726\) 0 0
\(727\) −35.0709 −1.30071 −0.650355 0.759631i \(-0.725380\pi\)
−0.650355 + 0.759631i \(0.725380\pi\)
\(728\) 0 0
\(729\) 19.6202 0.726675
\(730\) 0 0
\(731\) 7.53034 0.278520
\(732\) 0 0
\(733\) −2.41787 −0.0893061 −0.0446530 0.999003i \(-0.514218\pi\)
−0.0446530 + 0.999003i \(0.514218\pi\)
\(734\) 0 0
\(735\) 12.5102 0.461447
\(736\) 0 0
\(737\) −12.4308 −0.457893
\(738\) 0 0
\(739\) 33.2381 1.22268 0.611342 0.791367i \(-0.290630\pi\)
0.611342 + 0.791367i \(0.290630\pi\)
\(740\) 0 0
\(741\) −2.35761 −0.0866090
\(742\) 0 0
\(743\) 47.7003 1.74995 0.874977 0.484165i \(-0.160876\pi\)
0.874977 + 0.484165i \(0.160876\pi\)
\(744\) 0 0
\(745\) −17.5430 −0.642724
\(746\) 0 0
\(747\) −8.88390 −0.325045
\(748\) 0 0
\(749\) 11.6197 0.424576
\(750\) 0 0
\(751\) −11.3385 −0.413747 −0.206873 0.978368i \(-0.566329\pi\)
−0.206873 + 0.978368i \(0.566329\pi\)
\(752\) 0 0
\(753\) 20.1698 0.735027
\(754\) 0 0
\(755\) 21.9908 0.800329
\(756\) 0 0
\(757\) 3.67982 0.133745 0.0668727 0.997762i \(-0.478698\pi\)
0.0668727 + 0.997762i \(0.478698\pi\)
\(758\) 0 0
\(759\) −7.49776 −0.272152
\(760\) 0 0
\(761\) 46.9899 1.70338 0.851691 0.524045i \(-0.175578\pi\)
0.851691 + 0.524045i \(0.175578\pi\)
\(762\) 0 0
\(763\) −8.09620 −0.293102
\(764\) 0 0
\(765\) 0.502120 0.0181542
\(766\) 0 0
\(767\) −16.2949 −0.588376
\(768\) 0 0
\(769\) −44.1449 −1.59190 −0.795952 0.605360i \(-0.793029\pi\)
−0.795952 + 0.605360i \(0.793029\pi\)
\(770\) 0 0
\(771\) −14.7842 −0.532439
\(772\) 0 0
\(773\) −9.15592 −0.329315 −0.164658 0.986351i \(-0.552652\pi\)
−0.164658 + 0.986351i \(0.552652\pi\)
\(774\) 0 0
\(775\) −18.3872 −0.660488
\(776\) 0 0
\(777\) 15.6557 0.561646
\(778\) 0 0
\(779\) −6.34801 −0.227441
\(780\) 0 0
\(781\) 8.61984 0.308442
\(782\) 0 0
\(783\) 9.51579 0.340067
\(784\) 0 0
\(785\) 12.9462 0.462070
\(786\) 0 0
\(787\) 49.9487 1.78048 0.890240 0.455492i \(-0.150537\pi\)
0.890240 + 0.455492i \(0.150537\pi\)
\(788\) 0 0
\(789\) −59.8630 −2.13118
\(790\) 0 0
\(791\) −12.5918 −0.447712
\(792\) 0 0
\(793\) 11.1345 0.395398
\(794\) 0 0
\(795\) −15.9070 −0.564163
\(796\) 0 0
\(797\) −11.4356 −0.405071 −0.202536 0.979275i \(-0.564918\pi\)
−0.202536 + 0.979275i \(0.564918\pi\)
\(798\) 0 0
\(799\) 7.09478 0.250995
\(800\) 0 0
\(801\) −4.10574 −0.145069
\(802\) 0 0
\(803\) 6.72185 0.237209
\(804\) 0 0
\(805\) 5.92695 0.208898
\(806\) 0 0
\(807\) −20.7525 −0.730521
\(808\) 0 0
\(809\) 48.0943 1.69090 0.845452 0.534051i \(-0.179331\pi\)
0.845452 + 0.534051i \(0.179331\pi\)
\(810\) 0 0
\(811\) −3.41754 −0.120006 −0.0600030 0.998198i \(-0.519111\pi\)
−0.0600030 + 0.998198i \(0.519111\pi\)
\(812\) 0 0
\(813\) 30.4940 1.06947
\(814\) 0 0
\(815\) 23.8216 0.834433
\(816\) 0 0
\(817\) 10.9114 0.381743
\(818\) 0 0
\(819\) −0.929428 −0.0324768
\(820\) 0 0
\(821\) −1.63856 −0.0571863 −0.0285931 0.999591i \(-0.509103\pi\)
−0.0285931 + 0.999591i \(0.509103\pi\)
\(822\) 0 0
\(823\) −21.1955 −0.738827 −0.369414 0.929265i \(-0.620441\pi\)
−0.369414 + 0.929265i \(0.620441\pi\)
\(824\) 0 0
\(825\) 6.72094 0.233993
\(826\) 0 0
\(827\) 42.7585 1.48686 0.743429 0.668815i \(-0.233198\pi\)
0.743429 + 0.668815i \(0.233198\pi\)
\(828\) 0 0
\(829\) −19.3896 −0.673430 −0.336715 0.941607i \(-0.609316\pi\)
−0.336715 + 0.941607i \(0.609316\pi\)
\(830\) 0 0
\(831\) 42.9275 1.48914
\(832\) 0 0
\(833\) 3.76608 0.130487
\(834\) 0 0
\(835\) −18.5530 −0.642055
\(836\) 0 0
\(837\) −23.6296 −0.816759
\(838\) 0 0
\(839\) 5.49519 0.189715 0.0948574 0.995491i \(-0.469760\pi\)
0.0948574 + 0.995491i \(0.469760\pi\)
\(840\) 0 0
\(841\) −24.6274 −0.849222
\(842\) 0 0
\(843\) 48.6713 1.67633
\(844\) 0 0
\(845\) 13.8384 0.476057
\(846\) 0 0
\(847\) −1.24216 −0.0426812
\(848\) 0 0
\(849\) 2.18386 0.0749500
\(850\) 0 0
\(851\) −26.2325 −0.899237
\(852\) 0 0
\(853\) 34.1766 1.17018 0.585092 0.810967i \(-0.301058\pi\)
0.585092 + 0.810967i \(0.301058\pi\)
\(854\) 0 0
\(855\) 0.727569 0.0248823
\(856\) 0 0
\(857\) 28.9699 0.989593 0.494796 0.869009i \(-0.335243\pi\)
0.494796 + 0.869009i \(0.335243\pi\)
\(858\) 0 0
\(859\) −11.9817 −0.408811 −0.204405 0.978886i \(-0.565526\pi\)
−0.204405 + 0.978886i \(0.565526\pi\)
\(860\) 0 0
\(861\) −14.9661 −0.510045
\(862\) 0 0
\(863\) −30.5276 −1.03917 −0.519585 0.854419i \(-0.673914\pi\)
−0.519585 + 0.854419i \(0.673914\pi\)
\(864\) 0 0
\(865\) −27.0226 −0.918798
\(866\) 0 0
\(867\) −31.3618 −1.06510
\(868\) 0 0
\(869\) 9.82770 0.333382
\(870\) 0 0
\(871\) 15.4410 0.523199
\(872\) 0 0
\(873\) 3.28589 0.111210
\(874\) 0 0
\(875\) −12.8146 −0.433214
\(876\) 0 0
\(877\) 23.8775 0.806287 0.403144 0.915137i \(-0.367917\pi\)
0.403144 + 0.915137i \(0.367917\pi\)
\(878\) 0 0
\(879\) −9.05471 −0.305408
\(880\) 0 0
\(881\) −6.69730 −0.225638 −0.112819 0.993616i \(-0.535988\pi\)
−0.112819 + 0.993616i \(0.535988\pi\)
\(882\) 0 0
\(883\) −49.0580 −1.65093 −0.825467 0.564451i \(-0.809088\pi\)
−0.825467 + 0.564451i \(0.809088\pi\)
\(884\) 0 0
\(885\) 30.0734 1.01091
\(886\) 0 0
\(887\) 4.05578 0.136180 0.0680898 0.997679i \(-0.478310\pi\)
0.0680898 + 0.997679i \(0.478310\pi\)
\(888\) 0 0
\(889\) 8.13380 0.272799
\(890\) 0 0
\(891\) 10.4442 0.349896
\(892\) 0 0
\(893\) 10.2803 0.344017
\(894\) 0 0
\(895\) 28.9600 0.968026
\(896\) 0 0
\(897\) 9.31344 0.310967
\(898\) 0 0
\(899\) −10.8579 −0.362132
\(900\) 0 0
\(901\) −4.78864 −0.159533
\(902\) 0 0
\(903\) 25.7249 0.856071
\(904\) 0 0
\(905\) 16.6402 0.553138
\(906\) 0 0
\(907\) −15.9954 −0.531119 −0.265560 0.964094i \(-0.585557\pi\)
−0.265560 + 0.964094i \(0.585557\pi\)
\(908\) 0 0
\(909\) −3.11291 −0.103249
\(910\) 0 0
\(911\) −2.84554 −0.0942770 −0.0471385 0.998888i \(-0.515010\pi\)
−0.0471385 + 0.998888i \(0.515010\pi\)
\(912\) 0 0
\(913\) 14.7484 0.488101
\(914\) 0 0
\(915\) −20.5495 −0.679346
\(916\) 0 0
\(917\) −5.25659 −0.173588
\(918\) 0 0
\(919\) 40.9831 1.35191 0.675955 0.736943i \(-0.263732\pi\)
0.675955 + 0.736943i \(0.263732\pi\)
\(920\) 0 0
\(921\) −48.8006 −1.60803
\(922\) 0 0
\(923\) −10.7072 −0.352433
\(924\) 0 0
\(925\) 23.5146 0.773155
\(926\) 0 0
\(927\) 2.64742 0.0869526
\(928\) 0 0
\(929\) −19.7454 −0.647824 −0.323912 0.946087i \(-0.604998\pi\)
−0.323912 + 0.946087i \(0.604998\pi\)
\(930\) 0 0
\(931\) 5.45703 0.178847
\(932\) 0 0
\(933\) 55.0874 1.80348
\(934\) 0 0
\(935\) −0.833582 −0.0272610
\(936\) 0 0
\(937\) 24.5669 0.802567 0.401283 0.915954i \(-0.368564\pi\)
0.401283 + 0.915954i \(0.368564\pi\)
\(938\) 0 0
\(939\) 2.40031 0.0783313
\(940\) 0 0
\(941\) −15.6233 −0.509305 −0.254653 0.967033i \(-0.581961\pi\)
−0.254653 + 0.967033i \(0.581961\pi\)
\(942\) 0 0
\(943\) 25.0770 0.816619
\(944\) 0 0
\(945\) −6.82764 −0.222103
\(946\) 0 0
\(947\) 8.37770 0.272239 0.136119 0.990692i \(-0.456537\pi\)
0.136119 + 0.990692i \(0.456537\pi\)
\(948\) 0 0
\(949\) −8.34963 −0.271040
\(950\) 0 0
\(951\) 42.0474 1.36348
\(952\) 0 0
\(953\) 50.7519 1.64402 0.822008 0.569476i \(-0.192854\pi\)
0.822008 + 0.569476i \(0.192854\pi\)
\(954\) 0 0
\(955\) −21.0638 −0.681608
\(956\) 0 0
\(957\) 3.96882 0.128294
\(958\) 0 0
\(959\) 5.07633 0.163923
\(960\) 0 0
\(961\) −4.03759 −0.130245
\(962\) 0 0
\(963\) −5.63477 −0.181578
\(964\) 0 0
\(965\) −18.9063 −0.608614
\(966\) 0 0
\(967\) 48.3207 1.55389 0.776944 0.629569i \(-0.216769\pi\)
0.776944 + 0.629569i \(0.216769\pi\)
\(968\) 0 0
\(969\) 1.30987 0.0420790
\(970\) 0 0
\(971\) 32.0788 1.02946 0.514729 0.857353i \(-0.327893\pi\)
0.514729 + 0.857353i \(0.327893\pi\)
\(972\) 0 0
\(973\) 12.9599 0.415476
\(974\) 0 0
\(975\) −8.34850 −0.267366
\(976\) 0 0
\(977\) −19.3545 −0.619207 −0.309603 0.950866i \(-0.600196\pi\)
−0.309603 + 0.950866i \(0.600196\pi\)
\(978\) 0 0
\(979\) 6.81604 0.217842
\(980\) 0 0
\(981\) 3.92610 0.125351
\(982\) 0 0
\(983\) 48.1841 1.53683 0.768417 0.639950i \(-0.221045\pi\)
0.768417 + 0.639950i \(0.221045\pi\)
\(984\) 0 0
\(985\) −8.18634 −0.260839
\(986\) 0 0
\(987\) 24.2370 0.771471
\(988\) 0 0
\(989\) −43.1042 −1.37063
\(990\) 0 0
\(991\) 45.3376 1.44020 0.720098 0.693872i \(-0.244097\pi\)
0.720098 + 0.693872i \(0.244097\pi\)
\(992\) 0 0
\(993\) 65.4557 2.07718
\(994\) 0 0
\(995\) 9.21410 0.292107
\(996\) 0 0
\(997\) −17.6016 −0.557447 −0.278724 0.960371i \(-0.589911\pi\)
−0.278724 + 0.960371i \(0.589911\pi\)
\(998\) 0 0
\(999\) 30.2189 0.956083
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 3344.2.a.v.1.6 6
4.3 odd 2 1672.2.a.j.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.j.1.1 6 4.3 odd 2
3344.2.a.v.1.6 6 1.1 even 1 trivial