Properties

Label 1672.2.a.j.1.6
Level $1672$
Weight $2$
Character 1672.1
Self dual yes
Analytic conductor $13.351$
Analytic rank $0$
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] = [1672,2,Mod(1,1672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1672, 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("1672.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1672 = 2^{3} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1672.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: \(13.3509872180\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-2.21174\) of defining polynomial
Character \(\chi\) \(=\) 1672.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.21174 q^{3} +2.30747 q^{5} -1.34499 q^{7} +7.31527 q^{9} +O(q^{10})\) \(q+3.21174 q^{3} +2.30747 q^{5} -1.34499 q^{7} +7.31527 q^{9} +1.00000 q^{11} -1.34499 q^{13} +7.41101 q^{15} +0.904266 q^{17} +1.00000 q^{19} -4.31976 q^{21} +0.548933 q^{23} +0.324437 q^{25} +13.8595 q^{27} -4.32803 q^{29} +1.89392 q^{31} +3.21174 q^{33} -3.10353 q^{35} -0.546801 q^{37} -4.31976 q^{39} -4.86420 q^{41} +5.24127 q^{43} +16.8798 q^{45} -5.11346 q^{47} -5.19100 q^{49} +2.90427 q^{51} -3.63971 q^{53} +2.30747 q^{55} +3.21174 q^{57} +13.7786 q^{59} +3.90408 q^{61} -9.83897 q^{63} -3.10353 q^{65} -5.72084 q^{67} +1.76303 q^{69} -4.49402 q^{71} +1.28228 q^{73} +1.04201 q^{75} -1.34499 q^{77} -6.80150 q^{79} +22.5674 q^{81} -7.06560 q^{83} +2.08657 q^{85} -13.9005 q^{87} +15.8923 q^{89} +1.80900 q^{91} +6.08279 q^{93} +2.30747 q^{95} +12.2842 q^{97} +7.31527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} + q^{5} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} + q^{5} + 4 q^{9} + 6 q^{11} + 7 q^{15} + 3 q^{17} + 6 q^{19} + 7 q^{21} + 7 q^{23} + 3 q^{25} + 13 q^{27} - 4 q^{29} + 7 q^{31} + 4 q^{33} + 6 q^{35} - 2 q^{37} + 7 q^{39} + 7 q^{41} + 21 q^{43} + 11 q^{45} + 16 q^{47} + 2 q^{49} + 15 q^{51} + 17 q^{53} + q^{55} + 4 q^{57} + 19 q^{59} - 6 q^{61} + 2 q^{63} + 6 q^{65} + 14 q^{67} + q^{69} + q^{71} - 5 q^{73} + 18 q^{75} + 2 q^{81} + 11 q^{83} - 26 q^{85} + 14 q^{87} + 12 q^{89} + 44 q^{91} - 6 q^{93} + q^{95} + 14 q^{97} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.21174 1.85430 0.927149 0.374692i \(-0.122252\pi\)
0.927149 + 0.374692i \(0.122252\pi\)
\(4\) 0 0
\(5\) 2.30747 1.03193 0.515967 0.856609i \(-0.327433\pi\)
0.515967 + 0.856609i \(0.327433\pi\)
\(6\) 0 0
\(7\) −1.34499 −0.508359 −0.254179 0.967157i \(-0.581805\pi\)
−0.254179 + 0.967157i \(0.581805\pi\)
\(8\) 0 0
\(9\) 7.31527 2.43842
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −1.34499 −0.373033 −0.186517 0.982452i \(-0.559720\pi\)
−0.186517 + 0.982452i \(0.559720\pi\)
\(14\) 0 0
\(15\) 7.41101 1.91351
\(16\) 0 0
\(17\) 0.904266 0.219317 0.109658 0.993969i \(-0.465024\pi\)
0.109658 + 0.993969i \(0.465024\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −4.31976 −0.942649
\(22\) 0 0
\(23\) 0.548933 0.114460 0.0572302 0.998361i \(-0.481773\pi\)
0.0572302 + 0.998361i \(0.481773\pi\)
\(24\) 0 0
\(25\) 0.324437 0.0648873
\(26\) 0 0
\(27\) 13.8595 2.66727
\(28\) 0 0
\(29\) −4.32803 −0.803695 −0.401847 0.915707i \(-0.631632\pi\)
−0.401847 + 0.915707i \(0.631632\pi\)
\(30\) 0 0
\(31\) 1.89392 0.340159 0.170079 0.985430i \(-0.445598\pi\)
0.170079 + 0.985430i \(0.445598\pi\)
\(32\) 0 0
\(33\) 3.21174 0.559092
\(34\) 0 0
\(35\) −3.10353 −0.524593
\(36\) 0 0
\(37\) −0.546801 −0.0898936 −0.0449468 0.998989i \(-0.514312\pi\)
−0.0449468 + 0.998989i \(0.514312\pi\)
\(38\) 0 0
\(39\) −4.31976 −0.691715
\(40\) 0 0
\(41\) −4.86420 −0.759661 −0.379831 0.925056i \(-0.624018\pi\)
−0.379831 + 0.925056i \(0.624018\pi\)
\(42\) 0 0
\(43\) 5.24127 0.799287 0.399643 0.916671i \(-0.369134\pi\)
0.399643 + 0.916671i \(0.369134\pi\)
\(44\) 0 0
\(45\) 16.8798 2.51629
\(46\) 0 0
\(47\) −5.11346 −0.745875 −0.372938 0.927856i \(-0.621649\pi\)
−0.372938 + 0.927856i \(0.621649\pi\)
\(48\) 0 0
\(49\) −5.19100 −0.741571
\(50\) 0 0
\(51\) 2.90427 0.406679
\(52\) 0 0
\(53\) −3.63971 −0.499952 −0.249976 0.968252i \(-0.580423\pi\)
−0.249976 + 0.968252i \(0.580423\pi\)
\(54\) 0 0
\(55\) 2.30747 0.311140
\(56\) 0 0
\(57\) 3.21174 0.425405
\(58\) 0 0
\(59\) 13.7786 1.79382 0.896912 0.442209i \(-0.145805\pi\)
0.896912 + 0.442209i \(0.145805\pi\)
\(60\) 0 0
\(61\) 3.90408 0.499866 0.249933 0.968263i \(-0.419591\pi\)
0.249933 + 0.968263i \(0.419591\pi\)
\(62\) 0 0
\(63\) −9.83897 −1.23959
\(64\) 0 0
\(65\) −3.10353 −0.384946
\(66\) 0 0
\(67\) −5.72084 −0.698912 −0.349456 0.936953i \(-0.613634\pi\)
−0.349456 + 0.936953i \(0.613634\pi\)
\(68\) 0 0
\(69\) 1.76303 0.212244
\(70\) 0 0
\(71\) −4.49402 −0.533342 −0.266671 0.963788i \(-0.585924\pi\)
−0.266671 + 0.963788i \(0.585924\pi\)
\(72\) 0 0
\(73\) 1.28228 0.150080 0.0750400 0.997181i \(-0.476092\pi\)
0.0750400 + 0.997181i \(0.476092\pi\)
\(74\) 0 0
\(75\) 1.04201 0.120321
\(76\) 0 0
\(77\) −1.34499 −0.153276
\(78\) 0 0
\(79\) −6.80150 −0.765228 −0.382614 0.923908i \(-0.624976\pi\)
−0.382614 + 0.923908i \(0.624976\pi\)
\(80\) 0 0
\(81\) 22.5674 2.50749
\(82\) 0 0
\(83\) −7.06560 −0.775551 −0.387775 0.921754i \(-0.626756\pi\)
−0.387775 + 0.921754i \(0.626756\pi\)
\(84\) 0 0
\(85\) 2.08657 0.226320
\(86\) 0 0
\(87\) −13.9005 −1.49029
\(88\) 0 0
\(89\) 15.8923 1.68458 0.842289 0.539027i \(-0.181208\pi\)
0.842289 + 0.539027i \(0.181208\pi\)
\(90\) 0 0
\(91\) 1.80900 0.189635
\(92\) 0 0
\(93\) 6.08279 0.630756
\(94\) 0 0
\(95\) 2.30747 0.236742
\(96\) 0 0
\(97\) 12.2842 1.24727 0.623635 0.781715i \(-0.285655\pi\)
0.623635 + 0.781715i \(0.285655\pi\)
\(98\) 0 0
\(99\) 7.31527 0.735212
\(100\) 0 0
\(101\) −10.3276 −1.02763 −0.513815 0.857901i \(-0.671768\pi\)
−0.513815 + 0.857901i \(0.671768\pi\)
\(102\) 0 0
\(103\) 9.65577 0.951412 0.475706 0.879604i \(-0.342193\pi\)
0.475706 + 0.879604i \(0.342193\pi\)
\(104\) 0 0
\(105\) −9.96774 −0.972752
\(106\) 0 0
\(107\) −8.25466 −0.798008 −0.399004 0.916949i \(-0.630644\pi\)
−0.399004 + 0.916949i \(0.630644\pi\)
\(108\) 0 0
\(109\) −11.7058 −1.12121 −0.560604 0.828084i \(-0.689431\pi\)
−0.560604 + 0.828084i \(0.689431\pi\)
\(110\) 0 0
\(111\) −1.75618 −0.166690
\(112\) 0 0
\(113\) −17.1311 −1.61156 −0.805781 0.592213i \(-0.798254\pi\)
−0.805781 + 0.592213i \(0.798254\pi\)
\(114\) 0 0
\(115\) 1.26665 0.118116
\(116\) 0 0
\(117\) −9.83897 −0.909614
\(118\) 0 0
\(119\) −1.21623 −0.111492
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −15.6226 −1.40864
\(124\) 0 0
\(125\) −10.7887 −0.964974
\(126\) 0 0
\(127\) 8.51903 0.755941 0.377971 0.925818i \(-0.376622\pi\)
0.377971 + 0.925818i \(0.376622\pi\)
\(128\) 0 0
\(129\) 16.8336 1.48212
\(130\) 0 0
\(131\) −11.0052 −0.961530 −0.480765 0.876849i \(-0.659641\pi\)
−0.480765 + 0.876849i \(0.659641\pi\)
\(132\) 0 0
\(133\) −1.34499 −0.116626
\(134\) 0 0
\(135\) 31.9805 2.75244
\(136\) 0 0
\(137\) −5.62529 −0.480601 −0.240300 0.970699i \(-0.577246\pi\)
−0.240300 + 0.970699i \(0.577246\pi\)
\(138\) 0 0
\(139\) 11.0613 0.938211 0.469105 0.883142i \(-0.344576\pi\)
0.469105 + 0.883142i \(0.344576\pi\)
\(140\) 0 0
\(141\) −16.4231 −1.38308
\(142\) 0 0
\(143\) −1.34499 −0.112474
\(144\) 0 0
\(145\) −9.98681 −0.829360
\(146\) 0 0
\(147\) −16.6721 −1.37509
\(148\) 0 0
\(149\) −4.56486 −0.373968 −0.186984 0.982363i \(-0.559871\pi\)
−0.186984 + 0.982363i \(0.559871\pi\)
\(150\) 0 0
\(151\) −13.7194 −1.11647 −0.558236 0.829682i \(-0.688522\pi\)
−0.558236 + 0.829682i \(0.688522\pi\)
\(152\) 0 0
\(153\) 6.61495 0.534787
\(154\) 0 0
\(155\) 4.37018 0.351021
\(156\) 0 0
\(157\) 2.29850 0.183440 0.0917199 0.995785i \(-0.470764\pi\)
0.0917199 + 0.995785i \(0.470764\pi\)
\(158\) 0 0
\(159\) −11.6898 −0.927061
\(160\) 0 0
\(161\) −0.738310 −0.0581870
\(162\) 0 0
\(163\) −7.02945 −0.550589 −0.275294 0.961360i \(-0.588775\pi\)
−0.275294 + 0.961360i \(0.588775\pi\)
\(164\) 0 0
\(165\) 7.41101 0.576946
\(166\) 0 0
\(167\) 23.0788 1.78589 0.892946 0.450165i \(-0.148635\pi\)
0.892946 + 0.450165i \(0.148635\pi\)
\(168\) 0 0
\(169\) −11.1910 −0.860846
\(170\) 0 0
\(171\) 7.31527 0.559413
\(172\) 0 0
\(173\) −6.32553 −0.480921 −0.240461 0.970659i \(-0.577299\pi\)
−0.240461 + 0.970659i \(0.577299\pi\)
\(174\) 0 0
\(175\) −0.436365 −0.0329861
\(176\) 0 0
\(177\) 44.2534 3.32629
\(178\) 0 0
\(179\) 22.6706 1.69448 0.847239 0.531212i \(-0.178263\pi\)
0.847239 + 0.531212i \(0.178263\pi\)
\(180\) 0 0
\(181\) −19.2524 −1.43102 −0.715509 0.698604i \(-0.753805\pi\)
−0.715509 + 0.698604i \(0.753805\pi\)
\(182\) 0 0
\(183\) 12.5389 0.926901
\(184\) 0 0
\(185\) −1.26173 −0.0927642
\(186\) 0 0
\(187\) 0.904266 0.0661264
\(188\) 0 0
\(189\) −18.6409 −1.35593
\(190\) 0 0
\(191\) 2.30738 0.166956 0.0834780 0.996510i \(-0.473397\pi\)
0.0834780 + 0.996510i \(0.473397\pi\)
\(192\) 0 0
\(193\) −22.7437 −1.63713 −0.818564 0.574415i \(-0.805230\pi\)
−0.818564 + 0.574415i \(0.805230\pi\)
\(194\) 0 0
\(195\) −9.96774 −0.713804
\(196\) 0 0
\(197\) 11.4344 0.814668 0.407334 0.913279i \(-0.366459\pi\)
0.407334 + 0.913279i \(0.366459\pi\)
\(198\) 0 0
\(199\) 8.50792 0.603110 0.301555 0.953449i \(-0.402494\pi\)
0.301555 + 0.953449i \(0.402494\pi\)
\(200\) 0 0
\(201\) −18.3738 −1.29599
\(202\) 0 0
\(203\) 5.82116 0.408565
\(204\) 0 0
\(205\) −11.2240 −0.783920
\(206\) 0 0
\(207\) 4.01560 0.279103
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 22.4322 1.54430 0.772149 0.635441i \(-0.219182\pi\)
0.772149 + 0.635441i \(0.219182\pi\)
\(212\) 0 0
\(213\) −14.4336 −0.988976
\(214\) 0 0
\(215\) 12.0941 0.824811
\(216\) 0 0
\(217\) −2.54731 −0.172923
\(218\) 0 0
\(219\) 4.11836 0.278293
\(220\) 0 0
\(221\) −1.21623 −0.0818124
\(222\) 0 0
\(223\) 8.13420 0.544706 0.272353 0.962197i \(-0.412198\pi\)
0.272353 + 0.962197i \(0.412198\pi\)
\(224\) 0 0
\(225\) 2.37334 0.158223
\(226\) 0 0
\(227\) 8.49167 0.563612 0.281806 0.959471i \(-0.409067\pi\)
0.281806 + 0.959471i \(0.409067\pi\)
\(228\) 0 0
\(229\) −23.5132 −1.55379 −0.776897 0.629628i \(-0.783207\pi\)
−0.776897 + 0.629628i \(0.783207\pi\)
\(230\) 0 0
\(231\) −4.31976 −0.284219
\(232\) 0 0
\(233\) −0.186550 −0.0122213 −0.00611064 0.999981i \(-0.501945\pi\)
−0.00611064 + 0.999981i \(0.501945\pi\)
\(234\) 0 0
\(235\) −11.7992 −0.769694
\(236\) 0 0
\(237\) −21.8446 −1.41896
\(238\) 0 0
\(239\) −16.0820 −1.04026 −0.520130 0.854087i \(-0.674117\pi\)
−0.520130 + 0.854087i \(0.674117\pi\)
\(240\) 0 0
\(241\) 29.5369 1.90264 0.951319 0.308207i \(-0.0997289\pi\)
0.951319 + 0.308207i \(0.0997289\pi\)
\(242\) 0 0
\(243\) 30.9020 1.98236
\(244\) 0 0
\(245\) −11.9781 −0.765252
\(246\) 0 0
\(247\) −1.34499 −0.0855797
\(248\) 0 0
\(249\) −22.6929 −1.43810
\(250\) 0 0
\(251\) −8.42537 −0.531804 −0.265902 0.964000i \(-0.585670\pi\)
−0.265902 + 0.964000i \(0.585670\pi\)
\(252\) 0 0
\(253\) 0.548933 0.0345111
\(254\) 0 0
\(255\) 6.70152 0.419665
\(256\) 0 0
\(257\) 23.5186 1.46705 0.733525 0.679662i \(-0.237874\pi\)
0.733525 + 0.679662i \(0.237874\pi\)
\(258\) 0 0
\(259\) 0.735443 0.0456982
\(260\) 0 0
\(261\) −31.6607 −1.95975
\(262\) 0 0
\(263\) 8.00203 0.493426 0.246713 0.969089i \(-0.420650\pi\)
0.246713 + 0.969089i \(0.420650\pi\)
\(264\) 0 0
\(265\) −8.39853 −0.515918
\(266\) 0 0
\(267\) 51.0418 3.12371
\(268\) 0 0
\(269\) −7.63352 −0.465424 −0.232712 0.972546i \(-0.574760\pi\)
−0.232712 + 0.972546i \(0.574760\pi\)
\(270\) 0 0
\(271\) −14.2470 −0.865441 −0.432720 0.901528i \(-0.642446\pi\)
−0.432720 + 0.901528i \(0.642446\pi\)
\(272\) 0 0
\(273\) 5.81004 0.351640
\(274\) 0 0
\(275\) 0.324437 0.0195643
\(276\) 0 0
\(277\) 23.9558 1.43936 0.719682 0.694304i \(-0.244288\pi\)
0.719682 + 0.694304i \(0.244288\pi\)
\(278\) 0 0
\(279\) 13.8546 0.829451
\(280\) 0 0
\(281\) −3.08649 −0.184125 −0.0920623 0.995753i \(-0.529346\pi\)
−0.0920623 + 0.995753i \(0.529346\pi\)
\(282\) 0 0
\(283\) −6.50272 −0.386547 −0.193273 0.981145i \(-0.561910\pi\)
−0.193273 + 0.981145i \(0.561910\pi\)
\(284\) 0 0
\(285\) 7.41101 0.438990
\(286\) 0 0
\(287\) 6.54231 0.386180
\(288\) 0 0
\(289\) −16.1823 −0.951900
\(290\) 0 0
\(291\) 39.4536 2.31281
\(292\) 0 0
\(293\) 5.43591 0.317569 0.158785 0.987313i \(-0.449242\pi\)
0.158785 + 0.987313i \(0.449242\pi\)
\(294\) 0 0
\(295\) 31.7938 1.85111
\(296\) 0 0
\(297\) 13.8595 0.804211
\(298\) 0 0
\(299\) −0.738310 −0.0426976
\(300\) 0 0
\(301\) −7.04946 −0.406324
\(302\) 0 0
\(303\) −33.1694 −1.90553
\(304\) 0 0
\(305\) 9.00856 0.515829
\(306\) 0 0
\(307\) 9.01422 0.514469 0.257234 0.966349i \(-0.417189\pi\)
0.257234 + 0.966349i \(0.417189\pi\)
\(308\) 0 0
\(309\) 31.0118 1.76420
\(310\) 0 0
\(311\) 28.2244 1.60046 0.800230 0.599693i \(-0.204711\pi\)
0.800230 + 0.599693i \(0.204711\pi\)
\(312\) 0 0
\(313\) −3.14170 −0.177580 −0.0887898 0.996050i \(-0.528300\pi\)
−0.0887898 + 0.996050i \(0.528300\pi\)
\(314\) 0 0
\(315\) −22.7032 −1.27918
\(316\) 0 0
\(317\) 18.9722 1.06559 0.532793 0.846246i \(-0.321143\pi\)
0.532793 + 0.846246i \(0.321143\pi\)
\(318\) 0 0
\(319\) −4.32803 −0.242323
\(320\) 0 0
\(321\) −26.5118 −1.47974
\(322\) 0 0
\(323\) 0.904266 0.0503147
\(324\) 0 0
\(325\) −0.436365 −0.0242051
\(326\) 0 0
\(327\) −37.5959 −2.07906
\(328\) 0 0
\(329\) 6.87756 0.379172
\(330\) 0 0
\(331\) −3.88149 −0.213346 −0.106673 0.994294i \(-0.534020\pi\)
−0.106673 + 0.994294i \(0.534020\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −13.2007 −0.721230
\(336\) 0 0
\(337\) 24.6995 1.34547 0.672734 0.739885i \(-0.265120\pi\)
0.672734 + 0.739885i \(0.265120\pi\)
\(338\) 0 0
\(339\) −55.0208 −2.98832
\(340\) 0 0
\(341\) 1.89392 0.102562
\(342\) 0 0
\(343\) 16.3968 0.885343
\(344\) 0 0
\(345\) 4.06815 0.219022
\(346\) 0 0
\(347\) −33.7555 −1.81209 −0.906046 0.423179i \(-0.860914\pi\)
−0.906046 + 0.423179i \(0.860914\pi\)
\(348\) 0 0
\(349\) −33.1420 −1.77405 −0.887026 0.461720i \(-0.847232\pi\)
−0.887026 + 0.461720i \(0.847232\pi\)
\(350\) 0 0
\(351\) −18.6409 −0.994980
\(352\) 0 0
\(353\) 14.0448 0.747527 0.373764 0.927524i \(-0.378067\pi\)
0.373764 + 0.927524i \(0.378067\pi\)
\(354\) 0 0
\(355\) −10.3698 −0.550374
\(356\) 0 0
\(357\) −3.90621 −0.206739
\(358\) 0 0
\(359\) 35.9721 1.89853 0.949266 0.314473i \(-0.101828\pi\)
0.949266 + 0.314473i \(0.101828\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 3.21174 0.168573
\(364\) 0 0
\(365\) 2.95884 0.154873
\(366\) 0 0
\(367\) −33.4752 −1.74739 −0.873697 0.486471i \(-0.838284\pi\)
−0.873697 + 0.486471i \(0.838284\pi\)
\(368\) 0 0
\(369\) −35.5830 −1.85238
\(370\) 0 0
\(371\) 4.89538 0.254155
\(372\) 0 0
\(373\) −18.2471 −0.944801 −0.472400 0.881384i \(-0.656612\pi\)
−0.472400 + 0.881384i \(0.656612\pi\)
\(374\) 0 0
\(375\) −34.6506 −1.78935
\(376\) 0 0
\(377\) 5.82116 0.299805
\(378\) 0 0
\(379\) −11.7401 −0.603050 −0.301525 0.953458i \(-0.597496\pi\)
−0.301525 + 0.953458i \(0.597496\pi\)
\(380\) 0 0
\(381\) 27.3609 1.40174
\(382\) 0 0
\(383\) 2.42259 0.123788 0.0618942 0.998083i \(-0.480286\pi\)
0.0618942 + 0.998083i \(0.480286\pi\)
\(384\) 0 0
\(385\) −3.10353 −0.158171
\(386\) 0 0
\(387\) 38.3413 1.94900
\(388\) 0 0
\(389\) −20.7267 −1.05088 −0.525442 0.850829i \(-0.676100\pi\)
−0.525442 + 0.850829i \(0.676100\pi\)
\(390\) 0 0
\(391\) 0.496381 0.0251031
\(392\) 0 0
\(393\) −35.3459 −1.78296
\(394\) 0 0
\(395\) −15.6943 −0.789665
\(396\) 0 0
\(397\) −5.89658 −0.295941 −0.147971 0.988992i \(-0.547274\pi\)
−0.147971 + 0.988992i \(0.547274\pi\)
\(398\) 0 0
\(399\) −4.31976 −0.216259
\(400\) 0 0
\(401\) −31.9214 −1.59408 −0.797040 0.603927i \(-0.793602\pi\)
−0.797040 + 0.603927i \(0.793602\pi\)
\(402\) 0 0
\(403\) −2.54731 −0.126891
\(404\) 0 0
\(405\) 52.0737 2.58756
\(406\) 0 0
\(407\) −0.546801 −0.0271039
\(408\) 0 0
\(409\) 17.9584 0.887985 0.443993 0.896030i \(-0.353562\pi\)
0.443993 + 0.896030i \(0.353562\pi\)
\(410\) 0 0
\(411\) −18.0670 −0.891178
\(412\) 0 0
\(413\) −18.5321 −0.911906
\(414\) 0 0
\(415\) −16.3037 −0.800317
\(416\) 0 0
\(417\) 35.5261 1.73972
\(418\) 0 0
\(419\) −1.88319 −0.0920000 −0.0460000 0.998941i \(-0.514647\pi\)
−0.0460000 + 0.998941i \(0.514647\pi\)
\(420\) 0 0
\(421\) −28.1722 −1.37303 −0.686514 0.727117i \(-0.740860\pi\)
−0.686514 + 0.727117i \(0.740860\pi\)
\(422\) 0 0
\(423\) −37.4064 −1.81876
\(424\) 0 0
\(425\) 0.293377 0.0142309
\(426\) 0 0
\(427\) −5.25095 −0.254111
\(428\) 0 0
\(429\) −4.31976 −0.208560
\(430\) 0 0
\(431\) 2.23684 0.107745 0.0538724 0.998548i \(-0.482844\pi\)
0.0538724 + 0.998548i \(0.482844\pi\)
\(432\) 0 0
\(433\) 6.74878 0.324326 0.162163 0.986764i \(-0.448153\pi\)
0.162163 + 0.986764i \(0.448153\pi\)
\(434\) 0 0
\(435\) −32.0750 −1.53788
\(436\) 0 0
\(437\) 0.548933 0.0262590
\(438\) 0 0
\(439\) −37.0940 −1.77040 −0.885201 0.465210i \(-0.845979\pi\)
−0.885201 + 0.465210i \(0.845979\pi\)
\(440\) 0 0
\(441\) −37.9736 −1.80827
\(442\) 0 0
\(443\) −13.9416 −0.662387 −0.331193 0.943563i \(-0.607451\pi\)
−0.331193 + 0.943563i \(0.607451\pi\)
\(444\) 0 0
\(445\) 36.6710 1.73837
\(446\) 0 0
\(447\) −14.6611 −0.693448
\(448\) 0 0
\(449\) −10.4412 −0.492753 −0.246376 0.969174i \(-0.579240\pi\)
−0.246376 + 0.969174i \(0.579240\pi\)
\(450\) 0 0
\(451\) −4.86420 −0.229046
\(452\) 0 0
\(453\) −44.0632 −2.07027
\(454\) 0 0
\(455\) 4.17422 0.195691
\(456\) 0 0
\(457\) −18.8859 −0.883443 −0.441722 0.897152i \(-0.645632\pi\)
−0.441722 + 0.897152i \(0.645632\pi\)
\(458\) 0 0
\(459\) 12.5327 0.584976
\(460\) 0 0
\(461\) 35.0739 1.63355 0.816777 0.576954i \(-0.195759\pi\)
0.816777 + 0.576954i \(0.195759\pi\)
\(462\) 0 0
\(463\) 4.84941 0.225371 0.112686 0.993631i \(-0.464055\pi\)
0.112686 + 0.993631i \(0.464055\pi\)
\(464\) 0 0
\(465\) 14.0359 0.650899
\(466\) 0 0
\(467\) −0.400793 −0.0185465 −0.00927325 0.999957i \(-0.502952\pi\)
−0.00927325 + 0.999957i \(0.502952\pi\)
\(468\) 0 0
\(469\) 7.69448 0.355298
\(470\) 0 0
\(471\) 7.38217 0.340152
\(472\) 0 0
\(473\) 5.24127 0.240994
\(474\) 0 0
\(475\) 0.324437 0.0148862
\(476\) 0 0
\(477\) −26.6255 −1.21910
\(478\) 0 0
\(479\) 19.9633 0.912147 0.456074 0.889942i \(-0.349255\pi\)
0.456074 + 0.889942i \(0.349255\pi\)
\(480\) 0 0
\(481\) 0.735443 0.0335333
\(482\) 0 0
\(483\) −2.37126 −0.107896
\(484\) 0 0
\(485\) 28.3454 1.28710
\(486\) 0 0
\(487\) 20.9214 0.948039 0.474019 0.880514i \(-0.342803\pi\)
0.474019 + 0.880514i \(0.342803\pi\)
\(488\) 0 0
\(489\) −22.5768 −1.02096
\(490\) 0 0
\(491\) 22.6176 1.02072 0.510358 0.859962i \(-0.329513\pi\)
0.510358 + 0.859962i \(0.329513\pi\)
\(492\) 0 0
\(493\) −3.91369 −0.176264
\(494\) 0 0
\(495\) 16.8798 0.758691
\(496\) 0 0
\(497\) 6.04442 0.271129
\(498\) 0 0
\(499\) 25.4367 1.13870 0.569351 0.822095i \(-0.307195\pi\)
0.569351 + 0.822095i \(0.307195\pi\)
\(500\) 0 0
\(501\) 74.1231 3.31158
\(502\) 0 0
\(503\) 0.0412059 0.00183728 0.000918639 1.00000i \(-0.499708\pi\)
0.000918639 1.00000i \(0.499708\pi\)
\(504\) 0 0
\(505\) −23.8306 −1.06045
\(506\) 0 0
\(507\) −35.9426 −1.59627
\(508\) 0 0
\(509\) 22.7238 1.00722 0.503608 0.863932i \(-0.332006\pi\)
0.503608 + 0.863932i \(0.332006\pi\)
\(510\) 0 0
\(511\) −1.72466 −0.0762945
\(512\) 0 0
\(513\) 13.8595 0.611913
\(514\) 0 0
\(515\) 22.2804 0.981794
\(516\) 0 0
\(517\) −5.11346 −0.224890
\(518\) 0 0
\(519\) −20.3160 −0.891772
\(520\) 0 0
\(521\) −42.8088 −1.87549 −0.937744 0.347327i \(-0.887089\pi\)
−0.937744 + 0.347327i \(0.887089\pi\)
\(522\) 0 0
\(523\) −12.3759 −0.541160 −0.270580 0.962698i \(-0.587215\pi\)
−0.270580 + 0.962698i \(0.587215\pi\)
\(524\) 0 0
\(525\) −1.40149 −0.0611660
\(526\) 0 0
\(527\) 1.71261 0.0746025
\(528\) 0 0
\(529\) −22.6987 −0.986899
\(530\) 0 0
\(531\) 100.794 4.37410
\(532\) 0 0
\(533\) 6.54231 0.283379
\(534\) 0 0
\(535\) −19.0474 −0.823491
\(536\) 0 0
\(537\) 72.8119 3.14207
\(538\) 0 0
\(539\) −5.19100 −0.223592
\(540\) 0 0
\(541\) −8.66550 −0.372559 −0.186279 0.982497i \(-0.559643\pi\)
−0.186279 + 0.982497i \(0.559643\pi\)
\(542\) 0 0
\(543\) −61.8336 −2.65353
\(544\) 0 0
\(545\) −27.0107 −1.15701
\(546\) 0 0
\(547\) 18.1983 0.778103 0.389051 0.921216i \(-0.372803\pi\)
0.389051 + 0.921216i \(0.372803\pi\)
\(548\) 0 0
\(549\) 28.5594 1.21889
\(550\) 0 0
\(551\) −4.32803 −0.184380
\(552\) 0 0
\(553\) 9.14795 0.389011
\(554\) 0 0
\(555\) −4.05235 −0.172013
\(556\) 0 0
\(557\) 31.5397 1.33638 0.668190 0.743991i \(-0.267069\pi\)
0.668190 + 0.743991i \(0.267069\pi\)
\(558\) 0 0
\(559\) −7.04946 −0.298161
\(560\) 0 0
\(561\) 2.90427 0.122618
\(562\) 0 0
\(563\) 23.2066 0.978044 0.489022 0.872272i \(-0.337354\pi\)
0.489022 + 0.872272i \(0.337354\pi\)
\(564\) 0 0
\(565\) −39.5297 −1.66303
\(566\) 0 0
\(567\) −30.3529 −1.27470
\(568\) 0 0
\(569\) 14.0539 0.589171 0.294586 0.955625i \(-0.404818\pi\)
0.294586 + 0.955625i \(0.404818\pi\)
\(570\) 0 0
\(571\) 42.5174 1.77930 0.889648 0.456646i \(-0.150949\pi\)
0.889648 + 0.456646i \(0.150949\pi\)
\(572\) 0 0
\(573\) 7.41069 0.309586
\(574\) 0 0
\(575\) 0.178094 0.00742704
\(576\) 0 0
\(577\) −12.9888 −0.540732 −0.270366 0.962758i \(-0.587145\pi\)
−0.270366 + 0.962758i \(0.587145\pi\)
\(578\) 0 0
\(579\) −73.0469 −3.03573
\(580\) 0 0
\(581\) 9.50317 0.394258
\(582\) 0 0
\(583\) −3.63971 −0.150741
\(584\) 0 0
\(585\) −22.7032 −0.938661
\(586\) 0 0
\(587\) −25.4076 −1.04868 −0.524341 0.851508i \(-0.675688\pi\)
−0.524341 + 0.851508i \(0.675688\pi\)
\(588\) 0 0
\(589\) 1.89392 0.0780378
\(590\) 0 0
\(591\) 36.7243 1.51064
\(592\) 0 0
\(593\) 11.3346 0.465456 0.232728 0.972542i \(-0.425235\pi\)
0.232728 + 0.972542i \(0.425235\pi\)
\(594\) 0 0
\(595\) −2.80642 −0.115052
\(596\) 0 0
\(597\) 27.3252 1.11835
\(598\) 0 0
\(599\) −44.3934 −1.81387 −0.906933 0.421276i \(-0.861582\pi\)
−0.906933 + 0.421276i \(0.861582\pi\)
\(600\) 0 0
\(601\) −23.5195 −0.959380 −0.479690 0.877438i \(-0.659251\pi\)
−0.479690 + 0.877438i \(0.659251\pi\)
\(602\) 0 0
\(603\) −41.8495 −1.70424
\(604\) 0 0
\(605\) 2.30747 0.0938122
\(606\) 0 0
\(607\) −33.5476 −1.36165 −0.680827 0.732444i \(-0.738379\pi\)
−0.680827 + 0.732444i \(0.738379\pi\)
\(608\) 0 0
\(609\) 18.6960 0.757602
\(610\) 0 0
\(611\) 6.87756 0.278236
\(612\) 0 0
\(613\) −25.6571 −1.03628 −0.518141 0.855295i \(-0.673376\pi\)
−0.518141 + 0.855295i \(0.673376\pi\)
\(614\) 0 0
\(615\) −36.0487 −1.45362
\(616\) 0 0
\(617\) 7.96480 0.320651 0.160325 0.987064i \(-0.448746\pi\)
0.160325 + 0.987064i \(0.448746\pi\)
\(618\) 0 0
\(619\) 24.9617 1.00329 0.501647 0.865072i \(-0.332728\pi\)
0.501647 + 0.865072i \(0.332728\pi\)
\(620\) 0 0
\(621\) 7.60795 0.305297
\(622\) 0 0
\(623\) −21.3750 −0.856370
\(624\) 0 0
\(625\) −26.5169 −1.06068
\(626\) 0 0
\(627\) 3.21174 0.128265
\(628\) 0 0
\(629\) −0.494454 −0.0197152
\(630\) 0 0
\(631\) 13.8618 0.551831 0.275916 0.961182i \(-0.411019\pi\)
0.275916 + 0.961182i \(0.411019\pi\)
\(632\) 0 0
\(633\) 72.0465 2.86359
\(634\) 0 0
\(635\) 19.6574 0.780082
\(636\) 0 0
\(637\) 6.98185 0.276631
\(638\) 0 0
\(639\) −32.8750 −1.30052
\(640\) 0 0
\(641\) 38.4522 1.51877 0.759386 0.650640i \(-0.225499\pi\)
0.759386 + 0.650640i \(0.225499\pi\)
\(642\) 0 0
\(643\) −32.7622 −1.29202 −0.646008 0.763331i \(-0.723563\pi\)
−0.646008 + 0.763331i \(0.723563\pi\)
\(644\) 0 0
\(645\) 38.8431 1.52945
\(646\) 0 0
\(647\) 30.8413 1.21250 0.606248 0.795276i \(-0.292674\pi\)
0.606248 + 0.795276i \(0.292674\pi\)
\(648\) 0 0
\(649\) 13.7786 0.540858
\(650\) 0 0
\(651\) −8.18130 −0.320650
\(652\) 0 0
\(653\) −7.13589 −0.279249 −0.139624 0.990205i \(-0.544590\pi\)
−0.139624 + 0.990205i \(0.544590\pi\)
\(654\) 0 0
\(655\) −25.3942 −0.992235
\(656\) 0 0
\(657\) 9.38026 0.365959
\(658\) 0 0
\(659\) −4.67399 −0.182073 −0.0910364 0.995848i \(-0.529018\pi\)
−0.0910364 + 0.995848i \(0.529018\pi\)
\(660\) 0 0
\(661\) 31.3405 1.21900 0.609502 0.792784i \(-0.291369\pi\)
0.609502 + 0.792784i \(0.291369\pi\)
\(662\) 0 0
\(663\) −3.90621 −0.151705
\(664\) 0 0
\(665\) −3.10353 −0.120350
\(666\) 0 0
\(667\) −2.37580 −0.0919913
\(668\) 0 0
\(669\) 26.1249 1.01005
\(670\) 0 0
\(671\) 3.90408 0.150715
\(672\) 0 0
\(673\) −18.1600 −0.700018 −0.350009 0.936746i \(-0.613821\pi\)
−0.350009 + 0.936746i \(0.613821\pi\)
\(674\) 0 0
\(675\) 4.49654 0.173072
\(676\) 0 0
\(677\) −32.0875 −1.23322 −0.616611 0.787268i \(-0.711495\pi\)
−0.616611 + 0.787268i \(0.711495\pi\)
\(678\) 0 0
\(679\) −16.5221 −0.634061
\(680\) 0 0
\(681\) 27.2730 1.04510
\(682\) 0 0
\(683\) 4.30997 0.164917 0.0824583 0.996595i \(-0.473723\pi\)
0.0824583 + 0.996595i \(0.473723\pi\)
\(684\) 0 0
\(685\) −12.9802 −0.495948
\(686\) 0 0
\(687\) −75.5182 −2.88120
\(688\) 0 0
\(689\) 4.89538 0.186499
\(690\) 0 0
\(691\) 36.2358 1.37848 0.689238 0.724535i \(-0.257945\pi\)
0.689238 + 0.724535i \(0.257945\pi\)
\(692\) 0 0
\(693\) −9.83897 −0.373752
\(694\) 0 0
\(695\) 25.5238 0.968171
\(696\) 0 0
\(697\) −4.39853 −0.166606
\(698\) 0 0
\(699\) −0.599149 −0.0226619
\(700\) 0 0
\(701\) 6.19008 0.233796 0.116898 0.993144i \(-0.462705\pi\)
0.116898 + 0.993144i \(0.462705\pi\)
\(702\) 0 0
\(703\) −0.546801 −0.0206230
\(704\) 0 0
\(705\) −37.8959 −1.42724
\(706\) 0 0
\(707\) 13.8905 0.522405
\(708\) 0 0
\(709\) 30.6859 1.15243 0.576216 0.817297i \(-0.304529\pi\)
0.576216 + 0.817297i \(0.304529\pi\)
\(710\) 0 0
\(711\) −49.7548 −1.86595
\(712\) 0 0
\(713\) 1.03964 0.0389347
\(714\) 0 0
\(715\) −3.10353 −0.116066
\(716\) 0 0
\(717\) −51.6513 −1.92895
\(718\) 0 0
\(719\) −21.6704 −0.808171 −0.404085 0.914721i \(-0.632410\pi\)
−0.404085 + 0.914721i \(0.632410\pi\)
\(720\) 0 0
\(721\) −12.9869 −0.483658
\(722\) 0 0
\(723\) 94.8648 3.52806
\(724\) 0 0
\(725\) −1.40417 −0.0521496
\(726\) 0 0
\(727\) −1.88637 −0.0699614 −0.0349807 0.999388i \(-0.511137\pi\)
−0.0349807 + 0.999388i \(0.511137\pi\)
\(728\) 0 0
\(729\) 31.5469 1.16841
\(730\) 0 0
\(731\) 4.73950 0.175297
\(732\) 0 0
\(733\) −10.7015 −0.395270 −0.197635 0.980276i \(-0.563326\pi\)
−0.197635 + 0.980276i \(0.563326\pi\)
\(734\) 0 0
\(735\) −38.4705 −1.41901
\(736\) 0 0
\(737\) −5.72084 −0.210730
\(738\) 0 0
\(739\) 11.6797 0.429644 0.214822 0.976653i \(-0.431083\pi\)
0.214822 + 0.976653i \(0.431083\pi\)
\(740\) 0 0
\(741\) −4.31976 −0.158690
\(742\) 0 0
\(743\) 27.2152 0.998428 0.499214 0.866479i \(-0.333622\pi\)
0.499214 + 0.866479i \(0.333622\pi\)
\(744\) 0 0
\(745\) −10.5333 −0.385910
\(746\) 0 0
\(747\) −51.6868 −1.89112
\(748\) 0 0
\(749\) 11.1024 0.405674
\(750\) 0 0
\(751\) −4.26615 −0.155674 −0.0778371 0.996966i \(-0.524801\pi\)
−0.0778371 + 0.996966i \(0.524801\pi\)
\(752\) 0 0
\(753\) −27.0601 −0.986124
\(754\) 0 0
\(755\) −31.6572 −1.15212
\(756\) 0 0
\(757\) −14.2953 −0.519571 −0.259786 0.965666i \(-0.583652\pi\)
−0.259786 + 0.965666i \(0.583652\pi\)
\(758\) 0 0
\(759\) 1.76303 0.0639939
\(760\) 0 0
\(761\) −4.12180 −0.149415 −0.0747076 0.997205i \(-0.523802\pi\)
−0.0747076 + 0.997205i \(0.523802\pi\)
\(762\) 0 0
\(763\) 15.7441 0.569976
\(764\) 0 0
\(765\) 15.2638 0.551865
\(766\) 0 0
\(767\) −18.5321 −0.669156
\(768\) 0 0
\(769\) 24.0195 0.866164 0.433082 0.901355i \(-0.357426\pi\)
0.433082 + 0.901355i \(0.357426\pi\)
\(770\) 0 0
\(771\) 75.5357 2.72035
\(772\) 0 0
\(773\) 26.4504 0.951354 0.475677 0.879620i \(-0.342203\pi\)
0.475677 + 0.879620i \(0.342203\pi\)
\(774\) 0 0
\(775\) 0.614459 0.0220720
\(776\) 0 0
\(777\) 2.36205 0.0847381
\(778\) 0 0
\(779\) −4.86420 −0.174278
\(780\) 0 0
\(781\) −4.49402 −0.160809
\(782\) 0 0
\(783\) −59.9844 −2.14367
\(784\) 0 0
\(785\) 5.30372 0.189298
\(786\) 0 0
\(787\) 11.0855 0.395157 0.197578 0.980287i \(-0.436692\pi\)
0.197578 + 0.980287i \(0.436692\pi\)
\(788\) 0 0
\(789\) 25.7004 0.914959
\(790\) 0 0
\(791\) 23.0412 0.819252
\(792\) 0 0
\(793\) −5.25095 −0.186467
\(794\) 0 0
\(795\) −26.9739 −0.956666
\(796\) 0 0
\(797\) 48.0917 1.70349 0.851747 0.523953i \(-0.175543\pi\)
0.851747 + 0.523953i \(0.175543\pi\)
\(798\) 0 0
\(799\) −4.62393 −0.163583
\(800\) 0 0
\(801\) 116.256 4.10771
\(802\) 0 0
\(803\) 1.28228 0.0452508
\(804\) 0 0
\(805\) −1.70363 −0.0600451
\(806\) 0 0
\(807\) −24.5169 −0.863035
\(808\) 0 0
\(809\) 42.8287 1.50578 0.752888 0.658148i \(-0.228660\pi\)
0.752888 + 0.658148i \(0.228660\pi\)
\(810\) 0 0
\(811\) 30.4487 1.06920 0.534599 0.845106i \(-0.320463\pi\)
0.534599 + 0.845106i \(0.320463\pi\)
\(812\) 0 0
\(813\) −45.7575 −1.60479
\(814\) 0 0
\(815\) −16.2203 −0.568171
\(816\) 0 0
\(817\) 5.24127 0.183369
\(818\) 0 0
\(819\) 13.2333 0.462410
\(820\) 0 0
\(821\) 29.4900 1.02921 0.514604 0.857428i \(-0.327939\pi\)
0.514604 + 0.857428i \(0.327939\pi\)
\(822\) 0 0
\(823\) −12.9097 −0.450003 −0.225002 0.974358i \(-0.572239\pi\)
−0.225002 + 0.974358i \(0.572239\pi\)
\(824\) 0 0
\(825\) 1.04201 0.0362780
\(826\) 0 0
\(827\) 51.5466 1.79245 0.896225 0.443600i \(-0.146299\pi\)
0.896225 + 0.443600i \(0.146299\pi\)
\(828\) 0 0
\(829\) 6.89217 0.239375 0.119687 0.992812i \(-0.461811\pi\)
0.119687 + 0.992812i \(0.461811\pi\)
\(830\) 0 0
\(831\) 76.9397 2.66901
\(832\) 0 0
\(833\) −4.69404 −0.162639
\(834\) 0 0
\(835\) 53.2537 1.84292
\(836\) 0 0
\(837\) 26.2489 0.907295
\(838\) 0 0
\(839\) −23.2397 −0.802323 −0.401162 0.916007i \(-0.631393\pi\)
−0.401162 + 0.916007i \(0.631393\pi\)
\(840\) 0 0
\(841\) −10.2682 −0.354075
\(842\) 0 0
\(843\) −9.91300 −0.341422
\(844\) 0 0
\(845\) −25.8229 −0.888336
\(846\) 0 0
\(847\) −1.34499 −0.0462144
\(848\) 0 0
\(849\) −20.8850 −0.716773
\(850\) 0 0
\(851\) −0.300157 −0.0102893
\(852\) 0 0
\(853\) 39.0161 1.33589 0.667943 0.744212i \(-0.267175\pi\)
0.667943 + 0.744212i \(0.267175\pi\)
\(854\) 0 0
\(855\) 16.8798 0.577277
\(856\) 0 0
\(857\) 36.2608 1.23865 0.619323 0.785137i \(-0.287407\pi\)
0.619323 + 0.785137i \(0.287407\pi\)
\(858\) 0 0
\(859\) 14.1249 0.481937 0.240968 0.970533i \(-0.422535\pi\)
0.240968 + 0.970533i \(0.422535\pi\)
\(860\) 0 0
\(861\) 21.0122 0.716094
\(862\) 0 0
\(863\) 54.5442 1.85671 0.928353 0.371699i \(-0.121225\pi\)
0.928353 + 0.371699i \(0.121225\pi\)
\(864\) 0 0
\(865\) −14.5960 −0.496279
\(866\) 0 0
\(867\) −51.9733 −1.76511
\(868\) 0 0
\(869\) −6.80150 −0.230725
\(870\) 0 0
\(871\) 7.69448 0.260717
\(872\) 0 0
\(873\) 89.8622 3.04137
\(874\) 0 0
\(875\) 14.5108 0.490553
\(876\) 0 0
\(877\) 5.59673 0.188988 0.0944941 0.995525i \(-0.469877\pi\)
0.0944941 + 0.995525i \(0.469877\pi\)
\(878\) 0 0
\(879\) 17.4587 0.588869
\(880\) 0 0
\(881\) 11.9259 0.401793 0.200897 0.979612i \(-0.435614\pi\)
0.200897 + 0.979612i \(0.435614\pi\)
\(882\) 0 0
\(883\) −32.3029 −1.08708 −0.543540 0.839383i \(-0.682916\pi\)
−0.543540 + 0.839383i \(0.682916\pi\)
\(884\) 0 0
\(885\) 102.113 3.43251
\(886\) 0 0
\(887\) 44.1671 1.48299 0.741494 0.670960i \(-0.234118\pi\)
0.741494 + 0.670960i \(0.234118\pi\)
\(888\) 0 0
\(889\) −11.4580 −0.384290
\(890\) 0 0
\(891\) 22.5674 0.756036
\(892\) 0 0
\(893\) −5.11346 −0.171115
\(894\) 0 0
\(895\) 52.3117 1.74859
\(896\) 0 0
\(897\) −2.37126 −0.0791741
\(898\) 0 0
\(899\) −8.19696 −0.273384
\(900\) 0 0
\(901\) −3.29126 −0.109648
\(902\) 0 0
\(903\) −22.6410 −0.753447
\(904\) 0 0
\(905\) −44.4244 −1.47672
\(906\) 0 0
\(907\) 5.21952 0.173311 0.0866557 0.996238i \(-0.472382\pi\)
0.0866557 + 0.996238i \(0.472382\pi\)
\(908\) 0 0
\(909\) −75.5489 −2.50580
\(910\) 0 0
\(911\) −51.0254 −1.69055 −0.845274 0.534333i \(-0.820563\pi\)
−0.845274 + 0.534333i \(0.820563\pi\)
\(912\) 0 0
\(913\) −7.06560 −0.233837
\(914\) 0 0
\(915\) 28.9332 0.956501
\(916\) 0 0
\(917\) 14.8019 0.488802
\(918\) 0 0
\(919\) 36.0823 1.19024 0.595122 0.803635i \(-0.297104\pi\)
0.595122 + 0.803635i \(0.297104\pi\)
\(920\) 0 0
\(921\) 28.9513 0.953978
\(922\) 0 0
\(923\) 6.04442 0.198955
\(924\) 0 0
\(925\) −0.177402 −0.00583296
\(926\) 0 0
\(927\) 70.6346 2.31994
\(928\) 0 0
\(929\) −18.3679 −0.602632 −0.301316 0.953524i \(-0.597426\pi\)
−0.301316 + 0.953524i \(0.597426\pi\)
\(930\) 0 0
\(931\) −5.19100 −0.170128
\(932\) 0 0
\(933\) 90.6495 2.96773
\(934\) 0 0
\(935\) 2.08657 0.0682381
\(936\) 0 0
\(937\) 30.9327 1.01053 0.505264 0.862965i \(-0.331395\pi\)
0.505264 + 0.862965i \(0.331395\pi\)
\(938\) 0 0
\(939\) −10.0903 −0.329286
\(940\) 0 0
\(941\) 33.1792 1.08161 0.540806 0.841148i \(-0.318119\pi\)
0.540806 + 0.841148i \(0.318119\pi\)
\(942\) 0 0
\(943\) −2.67012 −0.0869512
\(944\) 0 0
\(945\) −43.0135 −1.39923
\(946\) 0 0
\(947\) 21.9471 0.713184 0.356592 0.934260i \(-0.383939\pi\)
0.356592 + 0.934260i \(0.383939\pi\)
\(948\) 0 0
\(949\) −1.72466 −0.0559849
\(950\) 0 0
\(951\) 60.9338 1.97591
\(952\) 0 0
\(953\) 41.4001 1.34108 0.670540 0.741873i \(-0.266062\pi\)
0.670540 + 0.741873i \(0.266062\pi\)
\(954\) 0 0
\(955\) 5.32421 0.172287
\(956\) 0 0
\(957\) −13.9005 −0.449339
\(958\) 0 0
\(959\) 7.56596 0.244318
\(960\) 0 0
\(961\) −27.4131 −0.884292
\(962\) 0 0
\(963\) −60.3851 −1.94588
\(964\) 0 0
\(965\) −52.4805 −1.68941
\(966\) 0 0
\(967\) −16.5235 −0.531358 −0.265679 0.964062i \(-0.585596\pi\)
−0.265679 + 0.964062i \(0.585596\pi\)
\(968\) 0 0
\(969\) 2.90427 0.0932985
\(970\) 0 0
\(971\) −14.0515 −0.450933 −0.225467 0.974251i \(-0.572391\pi\)
−0.225467 + 0.974251i \(0.572391\pi\)
\(972\) 0 0
\(973\) −14.8774 −0.476948
\(974\) 0 0
\(975\) −1.40149 −0.0448836
\(976\) 0 0
\(977\) 7.28151 0.232956 0.116478 0.993193i \(-0.462840\pi\)
0.116478 + 0.993193i \(0.462840\pi\)
\(978\) 0 0
\(979\) 15.8923 0.507919
\(980\) 0 0
\(981\) −85.6308 −2.73398
\(982\) 0 0
\(983\) 61.8115 1.97148 0.985740 0.168277i \(-0.0538203\pi\)
0.985740 + 0.168277i \(0.0538203\pi\)
\(984\) 0 0
\(985\) 26.3846 0.840683
\(986\) 0 0
\(987\) 22.0889 0.703098
\(988\) 0 0
\(989\) 2.87711 0.0914867
\(990\) 0 0
\(991\) −34.0846 −1.08273 −0.541367 0.840786i \(-0.682093\pi\)
−0.541367 + 0.840786i \(0.682093\pi\)
\(992\) 0 0
\(993\) −12.4663 −0.395607
\(994\) 0 0
\(995\) 19.6318 0.622370
\(996\) 0 0
\(997\) −17.6127 −0.557800 −0.278900 0.960320i \(-0.589970\pi\)
−0.278900 + 0.960320i \(0.589970\pi\)
\(998\) 0 0
\(999\) −7.57841 −0.239770
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 1672.2.a.j.1.6 6
4.3 odd 2 3344.2.a.v.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.j.1.6 6 1.1 even 1 trivial
3344.2.a.v.1.1 6 4.3 odd 2