Properties

Label 3344.2.a.y.1.1
Level $3344$
Weight $2$
Character 3344.1
Self dual yes
Analytic conductor $26.702$
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] = [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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.57500224.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 7x^{4} + 12x^{3} + 11x^{2} - 18x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.1
Root \(2.72058\) 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)
 
\(f(q)\) \(=\) \(q-1.72058 q^{3} -2.22719 q^{5} +1.96772 q^{7} -0.0396114 q^{9} +O(q^{10})\) \(q-1.72058 q^{3} -2.22719 q^{5} +1.96772 q^{7} -0.0396114 q^{9} -1.00000 q^{11} +7.04754 q^{13} +3.83206 q^{15} -7.60817 q^{17} +1.00000 q^{19} -3.38562 q^{21} +6.60084 q^{23} -0.0396114 q^{25} +5.22989 q^{27} -6.25147 q^{29} -5.36962 q^{31} +1.72058 q^{33} -4.38250 q^{35} +7.61473 q^{37} -12.1258 q^{39} -9.40485 q^{41} -6.26919 q^{43} +0.0882223 q^{45} +10.4279 q^{47} -3.12807 q^{49} +13.0905 q^{51} +4.34870 q^{53} +2.22719 q^{55} -1.72058 q^{57} -2.10518 q^{59} -8.77442 q^{61} -0.0779443 q^{63} -15.6962 q^{65} +13.7675 q^{67} -11.3573 q^{69} -1.14707 q^{71} -6.04471 q^{73} +0.0681546 q^{75} -1.96772 q^{77} +5.68740 q^{79} -8.87960 q^{81} -9.02439 q^{83} +16.9449 q^{85} +10.7561 q^{87} -4.83170 q^{89} +13.8676 q^{91} +9.23885 q^{93} -2.22719 q^{95} +12.0691 q^{97} +0.0396114 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{3} - 4 q^{5} + 4 q^{7} + 2 q^{9} - 6 q^{11} + 2 q^{13} - 10 q^{17} + 6 q^{19} + 10 q^{21} + 14 q^{23} + 2 q^{25} + 16 q^{27} + 6 q^{29} - 4 q^{31} - 4 q^{33} - 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41} + 26 q^{43} - 2 q^{45} + 24 q^{47} + 20 q^{49} + 14 q^{51} - 8 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} - 6 q^{61} + 40 q^{63} - 34 q^{65} + 44 q^{67} + 8 q^{69} + 16 q^{71} + 12 q^{73} + 28 q^{75} - 4 q^{77} - 6 q^{79} + 10 q^{81} + 28 q^{83} - 10 q^{85} + 24 q^{87} + 64 q^{91} - 14 q^{93} - 4 q^{95} + 4 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.72058 −0.993376 −0.496688 0.867929i \(-0.665451\pi\)
−0.496688 + 0.867929i \(0.665451\pi\)
\(4\) 0 0
\(5\) −2.22719 −0.996031 −0.498015 0.867168i \(-0.665938\pi\)
−0.498015 + 0.867168i \(0.665938\pi\)
\(6\) 0 0
\(7\) 1.96772 0.743729 0.371864 0.928287i \(-0.378719\pi\)
0.371864 + 0.928287i \(0.378719\pi\)
\(8\) 0 0
\(9\) −0.0396114 −0.0132038
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 7.04754 1.95464 0.977318 0.211779i \(-0.0679257\pi\)
0.977318 + 0.211779i \(0.0679257\pi\)
\(14\) 0 0
\(15\) 3.83206 0.989433
\(16\) 0 0
\(17\) −7.60817 −1.84525 −0.922627 0.385694i \(-0.873962\pi\)
−0.922627 + 0.385694i \(0.873962\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −3.38562 −0.738803
\(22\) 0 0
\(23\) 6.60084 1.37637 0.688185 0.725535i \(-0.258408\pi\)
0.688185 + 0.725535i \(0.258408\pi\)
\(24\) 0 0
\(25\) −0.0396114 −0.00792229
\(26\) 0 0
\(27\) 5.22989 1.00649
\(28\) 0 0
\(29\) −6.25147 −1.16087 −0.580435 0.814307i \(-0.697117\pi\)
−0.580435 + 0.814307i \(0.697117\pi\)
\(30\) 0 0
\(31\) −5.36962 −0.964413 −0.482206 0.876058i \(-0.660164\pi\)
−0.482206 + 0.876058i \(0.660164\pi\)
\(32\) 0 0
\(33\) 1.72058 0.299514
\(34\) 0 0
\(35\) −4.38250 −0.740777
\(36\) 0 0
\(37\) 7.61473 1.25185 0.625927 0.779881i \(-0.284721\pi\)
0.625927 + 0.779881i \(0.284721\pi\)
\(38\) 0 0
\(39\) −12.1258 −1.94169
\(40\) 0 0
\(41\) −9.40485 −1.46879 −0.734396 0.678722i \(-0.762534\pi\)
−0.734396 + 0.678722i \(0.762534\pi\)
\(42\) 0 0
\(43\) −6.26919 −0.956042 −0.478021 0.878348i \(-0.658646\pi\)
−0.478021 + 0.878348i \(0.658646\pi\)
\(44\) 0 0
\(45\) 0.0882223 0.0131514
\(46\) 0 0
\(47\) 10.4279 1.52107 0.760535 0.649297i \(-0.224937\pi\)
0.760535 + 0.649297i \(0.224937\pi\)
\(48\) 0 0
\(49\) −3.12807 −0.446867
\(50\) 0 0
\(51\) 13.0905 1.83303
\(52\) 0 0
\(53\) 4.34870 0.597340 0.298670 0.954356i \(-0.403457\pi\)
0.298670 + 0.954356i \(0.403457\pi\)
\(54\) 0 0
\(55\) 2.22719 0.300315
\(56\) 0 0
\(57\) −1.72058 −0.227896
\(58\) 0 0
\(59\) −2.10518 −0.274071 −0.137035 0.990566i \(-0.543757\pi\)
−0.137035 + 0.990566i \(0.543757\pi\)
\(60\) 0 0
\(61\) −8.77442 −1.12345 −0.561725 0.827324i \(-0.689862\pi\)
−0.561725 + 0.827324i \(0.689862\pi\)
\(62\) 0 0
\(63\) −0.0779443 −0.00982006
\(64\) 0 0
\(65\) −15.6962 −1.94688
\(66\) 0 0
\(67\) 13.7675 1.68197 0.840984 0.541060i \(-0.181977\pi\)
0.840984 + 0.541060i \(0.181977\pi\)
\(68\) 0 0
\(69\) −11.3573 −1.36725
\(70\) 0 0
\(71\) −1.14707 −0.136132 −0.0680659 0.997681i \(-0.521683\pi\)
−0.0680659 + 0.997681i \(0.521683\pi\)
\(72\) 0 0
\(73\) −6.04471 −0.707480 −0.353740 0.935344i \(-0.615090\pi\)
−0.353740 + 0.935344i \(0.615090\pi\)
\(74\) 0 0
\(75\) 0.0681546 0.00786981
\(76\) 0 0
\(77\) −1.96772 −0.224243
\(78\) 0 0
\(79\) 5.68740 0.639882 0.319941 0.947437i \(-0.396337\pi\)
0.319941 + 0.947437i \(0.396337\pi\)
\(80\) 0 0
\(81\) −8.87960 −0.986622
\(82\) 0 0
\(83\) −9.02439 −0.990555 −0.495277 0.868735i \(-0.664934\pi\)
−0.495277 + 0.868735i \(0.664934\pi\)
\(84\) 0 0
\(85\) 16.9449 1.83793
\(86\) 0 0
\(87\) 10.7561 1.15318
\(88\) 0 0
\(89\) −4.83170 −0.512160 −0.256080 0.966656i \(-0.582431\pi\)
−0.256080 + 0.966656i \(0.582431\pi\)
\(90\) 0 0
\(91\) 13.8676 1.45372
\(92\) 0 0
\(93\) 9.23885 0.958025
\(94\) 0 0
\(95\) −2.22719 −0.228505
\(96\) 0 0
\(97\) 12.0691 1.22543 0.612717 0.790303i \(-0.290077\pi\)
0.612717 + 0.790303i \(0.290077\pi\)
\(98\) 0 0
\(99\) 0.0396114 0.00398110
\(100\) 0 0
\(101\) 17.1105 1.70256 0.851280 0.524712i \(-0.175827\pi\)
0.851280 + 0.524712i \(0.175827\pi\)
\(102\) 0 0
\(103\) 11.4053 1.12379 0.561897 0.827207i \(-0.310072\pi\)
0.561897 + 0.827207i \(0.310072\pi\)
\(104\) 0 0
\(105\) 7.54043 0.735870
\(106\) 0 0
\(107\) 6.56007 0.634186 0.317093 0.948394i \(-0.397293\pi\)
0.317093 + 0.948394i \(0.397293\pi\)
\(108\) 0 0
\(109\) −1.49644 −0.143333 −0.0716663 0.997429i \(-0.522832\pi\)
−0.0716663 + 0.997429i \(0.522832\pi\)
\(110\) 0 0
\(111\) −13.1017 −1.24356
\(112\) 0 0
\(113\) 3.54633 0.333611 0.166805 0.985990i \(-0.446655\pi\)
0.166805 + 0.985990i \(0.446655\pi\)
\(114\) 0 0
\(115\) −14.7013 −1.37091
\(116\) 0 0
\(117\) −0.279163 −0.0258086
\(118\) 0 0
\(119\) −14.9708 −1.37237
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 16.1818 1.45906
\(124\) 0 0
\(125\) 11.2242 1.00392
\(126\) 0 0
\(127\) 4.70447 0.417454 0.208727 0.977974i \(-0.433068\pi\)
0.208727 + 0.977974i \(0.433068\pi\)
\(128\) 0 0
\(129\) 10.7866 0.949710
\(130\) 0 0
\(131\) 1.53122 0.133784 0.0668918 0.997760i \(-0.478692\pi\)
0.0668918 + 0.997760i \(0.478692\pi\)
\(132\) 0 0
\(133\) 1.96772 0.170623
\(134\) 0 0
\(135\) −11.6480 −1.00250
\(136\) 0 0
\(137\) −16.4781 −1.40782 −0.703910 0.710289i \(-0.748564\pi\)
−0.703910 + 0.710289i \(0.748564\pi\)
\(138\) 0 0
\(139\) −2.61989 −0.222216 −0.111108 0.993808i \(-0.535440\pi\)
−0.111108 + 0.993808i \(0.535440\pi\)
\(140\) 0 0
\(141\) −17.9421 −1.51099
\(142\) 0 0
\(143\) −7.04754 −0.589345
\(144\) 0 0
\(145\) 13.9232 1.15626
\(146\) 0 0
\(147\) 5.38209 0.443907
\(148\) 0 0
\(149\) 11.0818 0.907860 0.453930 0.891037i \(-0.350022\pi\)
0.453930 + 0.891037i \(0.350022\pi\)
\(150\) 0 0
\(151\) 1.83904 0.149659 0.0748293 0.997196i \(-0.476159\pi\)
0.0748293 + 0.997196i \(0.476159\pi\)
\(152\) 0 0
\(153\) 0.301371 0.0243644
\(154\) 0 0
\(155\) 11.9592 0.960585
\(156\) 0 0
\(157\) −2.36554 −0.188791 −0.0943954 0.995535i \(-0.530092\pi\)
−0.0943954 + 0.995535i \(0.530092\pi\)
\(158\) 0 0
\(159\) −7.48228 −0.593384
\(160\) 0 0
\(161\) 12.9886 1.02365
\(162\) 0 0
\(163\) 19.2217 1.50556 0.752781 0.658271i \(-0.228712\pi\)
0.752781 + 0.658271i \(0.228712\pi\)
\(164\) 0 0
\(165\) −3.83206 −0.298325
\(166\) 0 0
\(167\) −13.0347 −1.00865 −0.504326 0.863513i \(-0.668259\pi\)
−0.504326 + 0.863513i \(0.668259\pi\)
\(168\) 0 0
\(169\) 36.6678 2.82060
\(170\) 0 0
\(171\) −0.0396114 −0.00302916
\(172\) 0 0
\(173\) 12.9243 0.982614 0.491307 0.870987i \(-0.336519\pi\)
0.491307 + 0.870987i \(0.336519\pi\)
\(174\) 0 0
\(175\) −0.0779443 −0.00589204
\(176\) 0 0
\(177\) 3.62212 0.272255
\(178\) 0 0
\(179\) 7.25316 0.542126 0.271063 0.962562i \(-0.412625\pi\)
0.271063 + 0.962562i \(0.412625\pi\)
\(180\) 0 0
\(181\) 22.4890 1.67159 0.835797 0.549039i \(-0.185006\pi\)
0.835797 + 0.549039i \(0.185006\pi\)
\(182\) 0 0
\(183\) 15.0971 1.11601
\(184\) 0 0
\(185\) −16.9595 −1.24689
\(186\) 0 0
\(187\) 7.60817 0.556365
\(188\) 0 0
\(189\) 10.2910 0.748558
\(190\) 0 0
\(191\) −1.63679 −0.118434 −0.0592169 0.998245i \(-0.518860\pi\)
−0.0592169 + 0.998245i \(0.518860\pi\)
\(192\) 0 0
\(193\) 8.67334 0.624321 0.312160 0.950029i \(-0.398947\pi\)
0.312160 + 0.950029i \(0.398947\pi\)
\(194\) 0 0
\(195\) 27.0066 1.93398
\(196\) 0 0
\(197\) 14.4678 1.03079 0.515393 0.856954i \(-0.327646\pi\)
0.515393 + 0.856954i \(0.327646\pi\)
\(198\) 0 0
\(199\) 8.92883 0.632948 0.316474 0.948601i \(-0.397501\pi\)
0.316474 + 0.948601i \(0.397501\pi\)
\(200\) 0 0
\(201\) −23.6881 −1.67083
\(202\) 0 0
\(203\) −12.3012 −0.863372
\(204\) 0 0
\(205\) 20.9464 1.46296
\(206\) 0 0
\(207\) −0.261469 −0.0181733
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 13.9296 0.958953 0.479477 0.877555i \(-0.340827\pi\)
0.479477 + 0.877555i \(0.340827\pi\)
\(212\) 0 0
\(213\) 1.97362 0.135230
\(214\) 0 0
\(215\) 13.9627 0.952248
\(216\) 0 0
\(217\) −10.5659 −0.717262
\(218\) 0 0
\(219\) 10.4004 0.702794
\(220\) 0 0
\(221\) −53.6189 −3.60680
\(222\) 0 0
\(223\) 1.23753 0.0828713 0.0414356 0.999141i \(-0.486807\pi\)
0.0414356 + 0.999141i \(0.486807\pi\)
\(224\) 0 0
\(225\) 0.00156907 0.000104604 0
\(226\) 0 0
\(227\) 19.4085 1.28819 0.644093 0.764947i \(-0.277235\pi\)
0.644093 + 0.764947i \(0.277235\pi\)
\(228\) 0 0
\(229\) −13.6297 −0.900675 −0.450338 0.892858i \(-0.648696\pi\)
−0.450338 + 0.892858i \(0.648696\pi\)
\(230\) 0 0
\(231\) 3.38562 0.222757
\(232\) 0 0
\(233\) 6.87370 0.450311 0.225156 0.974323i \(-0.427711\pi\)
0.225156 + 0.974323i \(0.427711\pi\)
\(234\) 0 0
\(235\) −23.2250 −1.51503
\(236\) 0 0
\(237\) −9.78561 −0.635644
\(238\) 0 0
\(239\) −26.9021 −1.74015 −0.870075 0.492919i \(-0.835930\pi\)
−0.870075 + 0.492919i \(0.835930\pi\)
\(240\) 0 0
\(241\) 10.2375 0.659453 0.329727 0.944076i \(-0.393043\pi\)
0.329727 + 0.944076i \(0.393043\pi\)
\(242\) 0 0
\(243\) −0.411627 −0.0264059
\(244\) 0 0
\(245\) 6.96682 0.445094
\(246\) 0 0
\(247\) 7.04754 0.448424
\(248\) 0 0
\(249\) 15.5272 0.983994
\(250\) 0 0
\(251\) 14.8369 0.936498 0.468249 0.883596i \(-0.344885\pi\)
0.468249 + 0.883596i \(0.344885\pi\)
\(252\) 0 0
\(253\) −6.60084 −0.414991
\(254\) 0 0
\(255\) −29.1550 −1.82576
\(256\) 0 0
\(257\) 5.88893 0.367341 0.183671 0.982988i \(-0.441202\pi\)
0.183671 + 0.982988i \(0.441202\pi\)
\(258\) 0 0
\(259\) 14.9837 0.931041
\(260\) 0 0
\(261\) 0.247630 0.0153279
\(262\) 0 0
\(263\) −16.7421 −1.03236 −0.516181 0.856480i \(-0.672647\pi\)
−0.516181 + 0.856480i \(0.672647\pi\)
\(264\) 0 0
\(265\) −9.68540 −0.594969
\(266\) 0 0
\(267\) 8.31332 0.508767
\(268\) 0 0
\(269\) 9.63411 0.587402 0.293701 0.955897i \(-0.405113\pi\)
0.293701 + 0.955897i \(0.405113\pi\)
\(270\) 0 0
\(271\) −26.2625 −1.59533 −0.797667 0.603098i \(-0.793933\pi\)
−0.797667 + 0.603098i \(0.793933\pi\)
\(272\) 0 0
\(273\) −23.8603 −1.44409
\(274\) 0 0
\(275\) 0.0396114 0.00238866
\(276\) 0 0
\(277\) 27.1065 1.62867 0.814336 0.580394i \(-0.197101\pi\)
0.814336 + 0.580394i \(0.197101\pi\)
\(278\) 0 0
\(279\) 0.212698 0.0127339
\(280\) 0 0
\(281\) −25.2426 −1.50585 −0.752925 0.658106i \(-0.771358\pi\)
−0.752925 + 0.658106i \(0.771358\pi\)
\(282\) 0 0
\(283\) −3.88126 −0.230717 −0.115359 0.993324i \(-0.536802\pi\)
−0.115359 + 0.993324i \(0.536802\pi\)
\(284\) 0 0
\(285\) 3.83206 0.226992
\(286\) 0 0
\(287\) −18.5061 −1.09238
\(288\) 0 0
\(289\) 40.8843 2.40496
\(290\) 0 0
\(291\) −20.7659 −1.21732
\(292\) 0 0
\(293\) 9.16065 0.535171 0.267585 0.963534i \(-0.413774\pi\)
0.267585 + 0.963534i \(0.413774\pi\)
\(294\) 0 0
\(295\) 4.68864 0.272983
\(296\) 0 0
\(297\) −5.22989 −0.303469
\(298\) 0 0
\(299\) 46.5197 2.69030
\(300\) 0 0
\(301\) −12.3360 −0.711036
\(302\) 0 0
\(303\) −29.4400 −1.69128
\(304\) 0 0
\(305\) 19.5423 1.11899
\(306\) 0 0
\(307\) 7.45110 0.425256 0.212628 0.977133i \(-0.431798\pi\)
0.212628 + 0.977133i \(0.431798\pi\)
\(308\) 0 0
\(309\) −19.6236 −1.11635
\(310\) 0 0
\(311\) 28.1280 1.59499 0.797496 0.603324i \(-0.206157\pi\)
0.797496 + 0.603324i \(0.206157\pi\)
\(312\) 0 0
\(313\) −3.71407 −0.209932 −0.104966 0.994476i \(-0.533473\pi\)
−0.104966 + 0.994476i \(0.533473\pi\)
\(314\) 0 0
\(315\) 0.173597 0.00978108
\(316\) 0 0
\(317\) 4.79790 0.269477 0.134739 0.990881i \(-0.456981\pi\)
0.134739 + 0.990881i \(0.456981\pi\)
\(318\) 0 0
\(319\) 6.25147 0.350015
\(320\) 0 0
\(321\) −11.2871 −0.629985
\(322\) 0 0
\(323\) −7.60817 −0.423330
\(324\) 0 0
\(325\) −0.279163 −0.0154852
\(326\) 0 0
\(327\) 2.57474 0.142383
\(328\) 0 0
\(329\) 20.5193 1.13126
\(330\) 0 0
\(331\) −9.44840 −0.519331 −0.259665 0.965699i \(-0.583612\pi\)
−0.259665 + 0.965699i \(0.583612\pi\)
\(332\) 0 0
\(333\) −0.301631 −0.0165293
\(334\) 0 0
\(335\) −30.6629 −1.67529
\(336\) 0 0
\(337\) −27.1381 −1.47831 −0.739155 0.673536i \(-0.764775\pi\)
−0.739155 + 0.673536i \(0.764775\pi\)
\(338\) 0 0
\(339\) −6.10174 −0.331401
\(340\) 0 0
\(341\) 5.36962 0.290781
\(342\) 0 0
\(343\) −19.9292 −1.07608
\(344\) 0 0
\(345\) 25.2948 1.36183
\(346\) 0 0
\(347\) −5.46828 −0.293553 −0.146776 0.989170i \(-0.546890\pi\)
−0.146776 + 0.989170i \(0.546890\pi\)
\(348\) 0 0
\(349\) 3.27557 0.175337 0.0876687 0.996150i \(-0.472058\pi\)
0.0876687 + 0.996150i \(0.472058\pi\)
\(350\) 0 0
\(351\) 36.8578 1.96733
\(352\) 0 0
\(353\) 16.6340 0.885340 0.442670 0.896685i \(-0.354031\pi\)
0.442670 + 0.896685i \(0.354031\pi\)
\(354\) 0 0
\(355\) 2.55474 0.135592
\(356\) 0 0
\(357\) 25.7584 1.36328
\(358\) 0 0
\(359\) 29.1556 1.53877 0.769387 0.638783i \(-0.220562\pi\)
0.769387 + 0.638783i \(0.220562\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −1.72058 −0.0903069
\(364\) 0 0
\(365\) 13.4627 0.704672
\(366\) 0 0
\(367\) 27.9369 1.45830 0.729148 0.684356i \(-0.239917\pi\)
0.729148 + 0.684356i \(0.239917\pi\)
\(368\) 0 0
\(369\) 0.372540 0.0193936
\(370\) 0 0
\(371\) 8.55704 0.444259
\(372\) 0 0
\(373\) 14.7198 0.762163 0.381082 0.924541i \(-0.375552\pi\)
0.381082 + 0.924541i \(0.375552\pi\)
\(374\) 0 0
\(375\) −19.3121 −0.997272
\(376\) 0 0
\(377\) −44.0575 −2.26908
\(378\) 0 0
\(379\) −0.342859 −0.0176115 −0.00880574 0.999961i \(-0.502803\pi\)
−0.00880574 + 0.999961i \(0.502803\pi\)
\(380\) 0 0
\(381\) −8.09441 −0.414689
\(382\) 0 0
\(383\) 31.3669 1.60277 0.801386 0.598148i \(-0.204096\pi\)
0.801386 + 0.598148i \(0.204096\pi\)
\(384\) 0 0
\(385\) 4.38250 0.223353
\(386\) 0 0
\(387\) 0.248332 0.0126234
\(388\) 0 0
\(389\) −32.0391 −1.62445 −0.812225 0.583345i \(-0.801744\pi\)
−0.812225 + 0.583345i \(0.801744\pi\)
\(390\) 0 0
\(391\) −50.2203 −2.53975
\(392\) 0 0
\(393\) −2.63459 −0.132897
\(394\) 0 0
\(395\) −12.6669 −0.637342
\(396\) 0 0
\(397\) −33.4621 −1.67941 −0.839706 0.543041i \(-0.817273\pi\)
−0.839706 + 0.543041i \(0.817273\pi\)
\(398\) 0 0
\(399\) −3.38562 −0.169493
\(400\) 0 0
\(401\) 9.41337 0.470081 0.235041 0.971986i \(-0.424478\pi\)
0.235041 + 0.971986i \(0.424478\pi\)
\(402\) 0 0
\(403\) −37.8426 −1.88507
\(404\) 0 0
\(405\) 19.7766 0.982706
\(406\) 0 0
\(407\) −7.61473 −0.377448
\(408\) 0 0
\(409\) −10.0101 −0.494966 −0.247483 0.968892i \(-0.579603\pi\)
−0.247483 + 0.968892i \(0.579603\pi\)
\(410\) 0 0
\(411\) 28.3519 1.39849
\(412\) 0 0
\(413\) −4.14240 −0.203834
\(414\) 0 0
\(415\) 20.0990 0.986623
\(416\) 0 0
\(417\) 4.50772 0.220744
\(418\) 0 0
\(419\) −6.41377 −0.313333 −0.156667 0.987652i \(-0.550075\pi\)
−0.156667 + 0.987652i \(0.550075\pi\)
\(420\) 0 0
\(421\) 10.1476 0.494565 0.247282 0.968943i \(-0.420462\pi\)
0.247282 + 0.968943i \(0.420462\pi\)
\(422\) 0 0
\(423\) −0.413065 −0.0200839
\(424\) 0 0
\(425\) 0.301371 0.0146186
\(426\) 0 0
\(427\) −17.2656 −0.835542
\(428\) 0 0
\(429\) 12.1258 0.585441
\(430\) 0 0
\(431\) −5.59530 −0.269516 −0.134758 0.990879i \(-0.543026\pi\)
−0.134758 + 0.990879i \(0.543026\pi\)
\(432\) 0 0
\(433\) −34.1228 −1.63984 −0.819918 0.572481i \(-0.805981\pi\)
−0.819918 + 0.572481i \(0.805981\pi\)
\(434\) 0 0
\(435\) −23.9560 −1.14860
\(436\) 0 0
\(437\) 6.60084 0.315761
\(438\) 0 0
\(439\) −6.28891 −0.300153 −0.150077 0.988674i \(-0.547952\pi\)
−0.150077 + 0.988674i \(0.547952\pi\)
\(440\) 0 0
\(441\) 0.123907 0.00590035
\(442\) 0 0
\(443\) 22.1169 1.05081 0.525403 0.850853i \(-0.323914\pi\)
0.525403 + 0.850853i \(0.323914\pi\)
\(444\) 0 0
\(445\) 10.7611 0.510127
\(446\) 0 0
\(447\) −19.0672 −0.901847
\(448\) 0 0
\(449\) 3.54479 0.167289 0.0836444 0.996496i \(-0.473344\pi\)
0.0836444 + 0.996496i \(0.473344\pi\)
\(450\) 0 0
\(451\) 9.40485 0.442857
\(452\) 0 0
\(453\) −3.16421 −0.148667
\(454\) 0 0
\(455\) −30.8858 −1.44795
\(456\) 0 0
\(457\) 29.0898 1.36076 0.680382 0.732858i \(-0.261814\pi\)
0.680382 + 0.732858i \(0.261814\pi\)
\(458\) 0 0
\(459\) −39.7899 −1.85723
\(460\) 0 0
\(461\) 14.5291 0.676688 0.338344 0.941023i \(-0.390133\pi\)
0.338344 + 0.941023i \(0.390133\pi\)
\(462\) 0 0
\(463\) 20.6005 0.957386 0.478693 0.877982i \(-0.341111\pi\)
0.478693 + 0.877982i \(0.341111\pi\)
\(464\) 0 0
\(465\) −20.5767 −0.954222
\(466\) 0 0
\(467\) −11.8174 −0.546846 −0.273423 0.961894i \(-0.588156\pi\)
−0.273423 + 0.961894i \(0.588156\pi\)
\(468\) 0 0
\(469\) 27.0906 1.25093
\(470\) 0 0
\(471\) 4.07010 0.187540
\(472\) 0 0
\(473\) 6.26919 0.288258
\(474\) 0 0
\(475\) −0.0396114 −0.00181750
\(476\) 0 0
\(477\) −0.172258 −0.00788717
\(478\) 0 0
\(479\) 39.0255 1.78312 0.891561 0.452901i \(-0.149611\pi\)
0.891561 + 0.452901i \(0.149611\pi\)
\(480\) 0 0
\(481\) 53.6651 2.44692
\(482\) 0 0
\(483\) −22.3479 −1.01687
\(484\) 0 0
\(485\) −26.8803 −1.22057
\(486\) 0 0
\(487\) 29.3964 1.33208 0.666039 0.745917i \(-0.267989\pi\)
0.666039 + 0.745917i \(0.267989\pi\)
\(488\) 0 0
\(489\) −33.0725 −1.49559
\(490\) 0 0
\(491\) −38.5585 −1.74012 −0.870061 0.492943i \(-0.835921\pi\)
−0.870061 + 0.492943i \(0.835921\pi\)
\(492\) 0 0
\(493\) 47.5623 2.14210
\(494\) 0 0
\(495\) −0.0882223 −0.00396530
\(496\) 0 0
\(497\) −2.25711 −0.101245
\(498\) 0 0
\(499\) −13.4393 −0.601626 −0.300813 0.953683i \(-0.597258\pi\)
−0.300813 + 0.953683i \(0.597258\pi\)
\(500\) 0 0
\(501\) 22.4272 1.00197
\(502\) 0 0
\(503\) 16.4840 0.734984 0.367492 0.930027i \(-0.380217\pi\)
0.367492 + 0.930027i \(0.380217\pi\)
\(504\) 0 0
\(505\) −38.1084 −1.69580
\(506\) 0 0
\(507\) −63.0898 −2.80192
\(508\) 0 0
\(509\) −20.6969 −0.917374 −0.458687 0.888598i \(-0.651680\pi\)
−0.458687 + 0.888598i \(0.651680\pi\)
\(510\) 0 0
\(511\) −11.8943 −0.526173
\(512\) 0 0
\(513\) 5.22989 0.230905
\(514\) 0 0
\(515\) −25.4017 −1.11933
\(516\) 0 0
\(517\) −10.4279 −0.458620
\(518\) 0 0
\(519\) −22.2372 −0.976105
\(520\) 0 0
\(521\) 12.5616 0.550334 0.275167 0.961396i \(-0.411267\pi\)
0.275167 + 0.961396i \(0.411267\pi\)
\(522\) 0 0
\(523\) −40.5981 −1.77523 −0.887616 0.460584i \(-0.847640\pi\)
−0.887616 + 0.460584i \(0.847640\pi\)
\(524\) 0 0
\(525\) 0.134109 0.00585301
\(526\) 0 0
\(527\) 40.8530 1.77959
\(528\) 0 0
\(529\) 20.5711 0.894396
\(530\) 0 0
\(531\) 0.0833891 0.00361878
\(532\) 0 0
\(533\) −66.2810 −2.87095
\(534\) 0 0
\(535\) −14.6105 −0.631669
\(536\) 0 0
\(537\) −12.4796 −0.538535
\(538\) 0 0
\(539\) 3.12807 0.134736
\(540\) 0 0
\(541\) −16.6583 −0.716197 −0.358099 0.933684i \(-0.616575\pi\)
−0.358099 + 0.933684i \(0.616575\pi\)
\(542\) 0 0
\(543\) −38.6941 −1.66052
\(544\) 0 0
\(545\) 3.33285 0.142764
\(546\) 0 0
\(547\) −24.5756 −1.05078 −0.525389 0.850862i \(-0.676080\pi\)
−0.525389 + 0.850862i \(0.676080\pi\)
\(548\) 0 0
\(549\) 0.347567 0.0148338
\(550\) 0 0
\(551\) −6.25147 −0.266322
\(552\) 0 0
\(553\) 11.1912 0.475899
\(554\) 0 0
\(555\) 29.1801 1.23863
\(556\) 0 0
\(557\) −21.4752 −0.909934 −0.454967 0.890508i \(-0.650349\pi\)
−0.454967 + 0.890508i \(0.650349\pi\)
\(558\) 0 0
\(559\) −44.1823 −1.86871
\(560\) 0 0
\(561\) −13.0905 −0.552680
\(562\) 0 0
\(563\) −20.9440 −0.882686 −0.441343 0.897338i \(-0.645498\pi\)
−0.441343 + 0.897338i \(0.645498\pi\)
\(564\) 0 0
\(565\) −7.89837 −0.332287
\(566\) 0 0
\(567\) −17.4726 −0.733779
\(568\) 0 0
\(569\) −21.1031 −0.884687 −0.442343 0.896846i \(-0.645853\pi\)
−0.442343 + 0.896846i \(0.645853\pi\)
\(570\) 0 0
\(571\) 27.6827 1.15848 0.579242 0.815156i \(-0.303349\pi\)
0.579242 + 0.815156i \(0.303349\pi\)
\(572\) 0 0
\(573\) 2.81622 0.117649
\(574\) 0 0
\(575\) −0.261469 −0.0109040
\(576\) 0 0
\(577\) 15.9430 0.663714 0.331857 0.943330i \(-0.392325\pi\)
0.331857 + 0.943330i \(0.392325\pi\)
\(578\) 0 0
\(579\) −14.9232 −0.620185
\(580\) 0 0
\(581\) −17.7575 −0.736704
\(582\) 0 0
\(583\) −4.34870 −0.180105
\(584\) 0 0
\(585\) 0.621750 0.0257062
\(586\) 0 0
\(587\) −6.93504 −0.286240 −0.143120 0.989705i \(-0.545713\pi\)
−0.143120 + 0.989705i \(0.545713\pi\)
\(588\) 0 0
\(589\) −5.36962 −0.221251
\(590\) 0 0
\(591\) −24.8929 −1.02396
\(592\) 0 0
\(593\) −13.0536 −0.536049 −0.268024 0.963412i \(-0.586371\pi\)
−0.268024 + 0.963412i \(0.586371\pi\)
\(594\) 0 0
\(595\) 33.3428 1.36692
\(596\) 0 0
\(597\) −15.3627 −0.628755
\(598\) 0 0
\(599\) −21.5165 −0.879142 −0.439571 0.898208i \(-0.644870\pi\)
−0.439571 + 0.898208i \(0.644870\pi\)
\(600\) 0 0
\(601\) −16.9862 −0.692883 −0.346442 0.938072i \(-0.612610\pi\)
−0.346442 + 0.938072i \(0.612610\pi\)
\(602\) 0 0
\(603\) −0.545351 −0.0222084
\(604\) 0 0
\(605\) −2.22719 −0.0905483
\(606\) 0 0
\(607\) 8.55167 0.347102 0.173551 0.984825i \(-0.444476\pi\)
0.173551 + 0.984825i \(0.444476\pi\)
\(608\) 0 0
\(609\) 21.1651 0.857653
\(610\) 0 0
\(611\) 73.4912 2.97314
\(612\) 0 0
\(613\) 2.17578 0.0878790 0.0439395 0.999034i \(-0.486009\pi\)
0.0439395 + 0.999034i \(0.486009\pi\)
\(614\) 0 0
\(615\) −36.0400 −1.45327
\(616\) 0 0
\(617\) 39.5625 1.59273 0.796364 0.604818i \(-0.206754\pi\)
0.796364 + 0.604818i \(0.206754\pi\)
\(618\) 0 0
\(619\) 2.42787 0.0975844 0.0487922 0.998809i \(-0.484463\pi\)
0.0487922 + 0.998809i \(0.484463\pi\)
\(620\) 0 0
\(621\) 34.5217 1.38531
\(622\) 0 0
\(623\) −9.50745 −0.380908
\(624\) 0 0
\(625\) −24.8004 −0.992015
\(626\) 0 0
\(627\) 1.72058 0.0687133
\(628\) 0 0
\(629\) −57.9342 −2.30999
\(630\) 0 0
\(631\) 11.9436 0.475467 0.237733 0.971330i \(-0.423596\pi\)
0.237733 + 0.971330i \(0.423596\pi\)
\(632\) 0 0
\(633\) −23.9670 −0.952601
\(634\) 0 0
\(635\) −10.4778 −0.415798
\(636\) 0 0
\(637\) −22.0452 −0.873462
\(638\) 0 0
\(639\) 0.0454370 0.00179746
\(640\) 0 0
\(641\) 26.0294 1.02810 0.514050 0.857760i \(-0.328145\pi\)
0.514050 + 0.857760i \(0.328145\pi\)
\(642\) 0 0
\(643\) 37.6765 1.48582 0.742909 0.669393i \(-0.233446\pi\)
0.742909 + 0.669393i \(0.233446\pi\)
\(644\) 0 0
\(645\) −24.0239 −0.945940
\(646\) 0 0
\(647\) −37.3030 −1.46653 −0.733266 0.679941i \(-0.762005\pi\)
−0.733266 + 0.679941i \(0.762005\pi\)
\(648\) 0 0
\(649\) 2.10518 0.0826354
\(650\) 0 0
\(651\) 18.1795 0.712511
\(652\) 0 0
\(653\) 2.67401 0.104642 0.0523210 0.998630i \(-0.483338\pi\)
0.0523210 + 0.998630i \(0.483338\pi\)
\(654\) 0 0
\(655\) −3.41033 −0.133253
\(656\) 0 0
\(657\) 0.239440 0.00934143
\(658\) 0 0
\(659\) 43.7451 1.70407 0.852034 0.523486i \(-0.175369\pi\)
0.852034 + 0.523486i \(0.175369\pi\)
\(660\) 0 0
\(661\) −43.6277 −1.69692 −0.848460 0.529260i \(-0.822470\pi\)
−0.848460 + 0.529260i \(0.822470\pi\)
\(662\) 0 0
\(663\) 92.2555 3.58291
\(664\) 0 0
\(665\) −4.38250 −0.169946
\(666\) 0 0
\(667\) −41.2650 −1.59779
\(668\) 0 0
\(669\) −2.12927 −0.0823224
\(670\) 0 0
\(671\) 8.77442 0.338733
\(672\) 0 0
\(673\) −35.3184 −1.36142 −0.680712 0.732551i \(-0.738329\pi\)
−0.680712 + 0.732551i \(0.738329\pi\)
\(674\) 0 0
\(675\) −0.207163 −0.00797372
\(676\) 0 0
\(677\) −17.8219 −0.684952 −0.342476 0.939527i \(-0.611265\pi\)
−0.342476 + 0.939527i \(0.611265\pi\)
\(678\) 0 0
\(679\) 23.7487 0.911390
\(680\) 0 0
\(681\) −33.3938 −1.27965
\(682\) 0 0
\(683\) 18.3781 0.703220 0.351610 0.936147i \(-0.385634\pi\)
0.351610 + 0.936147i \(0.385634\pi\)
\(684\) 0 0
\(685\) 36.6999 1.40223
\(686\) 0 0
\(687\) 23.4509 0.894709
\(688\) 0 0
\(689\) 30.6476 1.16758
\(690\) 0 0
\(691\) 2.93938 0.111819 0.0559097 0.998436i \(-0.482194\pi\)
0.0559097 + 0.998436i \(0.482194\pi\)
\(692\) 0 0
\(693\) 0.0779443 0.00296086
\(694\) 0 0
\(695\) 5.83500 0.221334
\(696\) 0 0
\(697\) 71.5538 2.71029
\(698\) 0 0
\(699\) −11.8267 −0.447328
\(700\) 0 0
\(701\) 46.4261 1.75349 0.876744 0.480956i \(-0.159710\pi\)
0.876744 + 0.480956i \(0.159710\pi\)
\(702\) 0 0
\(703\) 7.61473 0.287195
\(704\) 0 0
\(705\) 39.9604 1.50500
\(706\) 0 0
\(707\) 33.6687 1.26624
\(708\) 0 0
\(709\) 8.55669 0.321353 0.160677 0.987007i \(-0.448632\pi\)
0.160677 + 0.987007i \(0.448632\pi\)
\(710\) 0 0
\(711\) −0.225286 −0.00844888
\(712\) 0 0
\(713\) −35.4440 −1.32739
\(714\) 0 0
\(715\) 15.6962 0.587006
\(716\) 0 0
\(717\) 46.2871 1.72862
\(718\) 0 0
\(719\) −1.29599 −0.0483324 −0.0241662 0.999708i \(-0.507693\pi\)
−0.0241662 + 0.999708i \(0.507693\pi\)
\(720\) 0 0
\(721\) 22.4424 0.835798
\(722\) 0 0
\(723\) −17.6144 −0.655085
\(724\) 0 0
\(725\) 0.247630 0.00919674
\(726\) 0 0
\(727\) −12.2174 −0.453119 −0.226559 0.973997i \(-0.572748\pi\)
−0.226559 + 0.973997i \(0.572748\pi\)
\(728\) 0 0
\(729\) 27.3470 1.01285
\(730\) 0 0
\(731\) 47.6971 1.76414
\(732\) 0 0
\(733\) −19.2105 −0.709556 −0.354778 0.934951i \(-0.615444\pi\)
−0.354778 + 0.934951i \(0.615444\pi\)
\(734\) 0 0
\(735\) −11.9870 −0.442145
\(736\) 0 0
\(737\) −13.7675 −0.507132
\(738\) 0 0
\(739\) 29.7110 1.09294 0.546469 0.837480i \(-0.315972\pi\)
0.546469 + 0.837480i \(0.315972\pi\)
\(740\) 0 0
\(741\) −12.1258 −0.445454
\(742\) 0 0
\(743\) 7.71120 0.282896 0.141448 0.989946i \(-0.454824\pi\)
0.141448 + 0.989946i \(0.454824\pi\)
\(744\) 0 0
\(745\) −24.6814 −0.904257
\(746\) 0 0
\(747\) 0.357469 0.0130791
\(748\) 0 0
\(749\) 12.9084 0.471662
\(750\) 0 0
\(751\) 33.3681 1.21762 0.608809 0.793317i \(-0.291647\pi\)
0.608809 + 0.793317i \(0.291647\pi\)
\(752\) 0 0
\(753\) −25.5281 −0.930295
\(754\) 0 0
\(755\) −4.09589 −0.149065
\(756\) 0 0
\(757\) −45.7593 −1.66315 −0.831575 0.555412i \(-0.812561\pi\)
−0.831575 + 0.555412i \(0.812561\pi\)
\(758\) 0 0
\(759\) 11.3573 0.412242
\(760\) 0 0
\(761\) 35.9488 1.30314 0.651572 0.758587i \(-0.274110\pi\)
0.651572 + 0.758587i \(0.274110\pi\)
\(762\) 0 0
\(763\) −2.94457 −0.106601
\(764\) 0 0
\(765\) −0.671211 −0.0242677
\(766\) 0 0
\(767\) −14.8363 −0.535708
\(768\) 0 0
\(769\) −20.5476 −0.740964 −0.370482 0.928840i \(-0.620807\pi\)
−0.370482 + 0.928840i \(0.620807\pi\)
\(770\) 0 0
\(771\) −10.1324 −0.364908
\(772\) 0 0
\(773\) −11.1888 −0.402434 −0.201217 0.979547i \(-0.564490\pi\)
−0.201217 + 0.979547i \(0.564490\pi\)
\(774\) 0 0
\(775\) 0.212698 0.00764036
\(776\) 0 0
\(777\) −25.7806 −0.924874
\(778\) 0 0
\(779\) −9.40485 −0.336964
\(780\) 0 0
\(781\) 1.14707 0.0410453
\(782\) 0 0
\(783\) −32.6945 −1.16841
\(784\) 0 0
\(785\) 5.26852 0.188041
\(786\) 0 0
\(787\) 24.3619 0.868409 0.434205 0.900814i \(-0.357029\pi\)
0.434205 + 0.900814i \(0.357029\pi\)
\(788\) 0 0
\(789\) 28.8061 1.02552
\(790\) 0 0
\(791\) 6.97820 0.248116
\(792\) 0 0
\(793\) −61.8380 −2.19593
\(794\) 0 0
\(795\) 16.6645 0.591028
\(796\) 0 0
\(797\) −20.6680 −0.732097 −0.366048 0.930596i \(-0.619290\pi\)
−0.366048 + 0.930596i \(0.619290\pi\)
\(798\) 0 0
\(799\) −79.3375 −2.80676
\(800\) 0 0
\(801\) 0.191391 0.00676246
\(802\) 0 0
\(803\) 6.04471 0.213313
\(804\) 0 0
\(805\) −28.9282 −1.01958
\(806\) 0 0
\(807\) −16.5762 −0.583511
\(808\) 0 0
\(809\) −16.4564 −0.578577 −0.289289 0.957242i \(-0.593419\pi\)
−0.289289 + 0.957242i \(0.593419\pi\)
\(810\) 0 0
\(811\) 21.7377 0.763312 0.381656 0.924304i \(-0.375354\pi\)
0.381656 + 0.924304i \(0.375354\pi\)
\(812\) 0 0
\(813\) 45.1867 1.58477
\(814\) 0 0
\(815\) −42.8105 −1.49959
\(816\) 0 0
\(817\) −6.26919 −0.219331
\(818\) 0 0
\(819\) −0.549315 −0.0191946
\(820\) 0 0
\(821\) 2.84811 0.0993997 0.0496998 0.998764i \(-0.484174\pi\)
0.0496998 + 0.998764i \(0.484174\pi\)
\(822\) 0 0
\(823\) −2.30793 −0.0804494 −0.0402247 0.999191i \(-0.512807\pi\)
−0.0402247 + 0.999191i \(0.512807\pi\)
\(824\) 0 0
\(825\) −0.0681546 −0.00237284
\(826\) 0 0
\(827\) −40.2836 −1.40080 −0.700398 0.713752i \(-0.746994\pi\)
−0.700398 + 0.713752i \(0.746994\pi\)
\(828\) 0 0
\(829\) −12.6009 −0.437647 −0.218824 0.975764i \(-0.570222\pi\)
−0.218824 + 0.975764i \(0.570222\pi\)
\(830\) 0 0
\(831\) −46.6388 −1.61788
\(832\) 0 0
\(833\) 23.7989 0.824583
\(834\) 0 0
\(835\) 29.0307 1.00465
\(836\) 0 0
\(837\) −28.0825 −0.970674
\(838\) 0 0
\(839\) −10.5744 −0.365068 −0.182534 0.983200i \(-0.558430\pi\)
−0.182534 + 0.983200i \(0.558430\pi\)
\(840\) 0 0
\(841\) 10.0809 0.347617
\(842\) 0 0
\(843\) 43.4319 1.49588
\(844\) 0 0
\(845\) −81.6662 −2.80940
\(846\) 0 0
\(847\) 1.96772 0.0676117
\(848\) 0 0
\(849\) 6.67802 0.229189
\(850\) 0 0
\(851\) 50.2636 1.72302
\(852\) 0 0
\(853\) −4.84636 −0.165936 −0.0829681 0.996552i \(-0.526440\pi\)
−0.0829681 + 0.996552i \(0.526440\pi\)
\(854\) 0 0
\(855\) 0.0882223 0.00301714
\(856\) 0 0
\(857\) −40.1185 −1.37042 −0.685211 0.728344i \(-0.740290\pi\)
−0.685211 + 0.728344i \(0.740290\pi\)
\(858\) 0 0
\(859\) −35.7826 −1.22089 −0.610443 0.792060i \(-0.709008\pi\)
−0.610443 + 0.792060i \(0.709008\pi\)
\(860\) 0 0
\(861\) 31.8412 1.08515
\(862\) 0 0
\(863\) 37.8507 1.28845 0.644226 0.764835i \(-0.277180\pi\)
0.644226 + 0.764835i \(0.277180\pi\)
\(864\) 0 0
\(865\) −28.7848 −0.978714
\(866\) 0 0
\(867\) −70.3446 −2.38903
\(868\) 0 0
\(869\) −5.68740 −0.192932
\(870\) 0 0
\(871\) 97.0270 3.28763
\(872\) 0 0
\(873\) −0.478075 −0.0161804
\(874\) 0 0
\(875\) 22.0861 0.746646
\(876\) 0 0
\(877\) 19.9970 0.675250 0.337625 0.941281i \(-0.390377\pi\)
0.337625 + 0.941281i \(0.390377\pi\)
\(878\) 0 0
\(879\) −15.7616 −0.531626
\(880\) 0 0
\(881\) 29.6003 0.997261 0.498631 0.866815i \(-0.333836\pi\)
0.498631 + 0.866815i \(0.333836\pi\)
\(882\) 0 0
\(883\) 13.3489 0.449225 0.224612 0.974448i \(-0.427888\pi\)
0.224612 + 0.974448i \(0.427888\pi\)
\(884\) 0 0
\(885\) −8.06716 −0.271175
\(886\) 0 0
\(887\) 25.8492 0.867931 0.433966 0.900929i \(-0.357114\pi\)
0.433966 + 0.900929i \(0.357114\pi\)
\(888\) 0 0
\(889\) 9.25709 0.310473
\(890\) 0 0
\(891\) 8.87960 0.297478
\(892\) 0 0
\(893\) 10.4279 0.348957
\(894\) 0 0
\(895\) −16.1542 −0.539975
\(896\) 0 0
\(897\) −80.0407 −2.67248
\(898\) 0 0
\(899\) 33.5680 1.11956
\(900\) 0 0
\(901\) −33.0857 −1.10224
\(902\) 0 0
\(903\) 21.2251 0.706327
\(904\) 0 0
\(905\) −50.0873 −1.66496
\(906\) 0 0
\(907\) 37.2165 1.23575 0.617876 0.786276i \(-0.287993\pi\)
0.617876 + 0.786276i \(0.287993\pi\)
\(908\) 0 0
\(909\) −0.677772 −0.0224803
\(910\) 0 0
\(911\) −26.0666 −0.863624 −0.431812 0.901964i \(-0.642126\pi\)
−0.431812 + 0.901964i \(0.642126\pi\)
\(912\) 0 0
\(913\) 9.02439 0.298664
\(914\) 0 0
\(915\) −33.6241 −1.11158
\(916\) 0 0
\(917\) 3.01302 0.0994988
\(918\) 0 0
\(919\) −23.4079 −0.772154 −0.386077 0.922467i \(-0.626170\pi\)
−0.386077 + 0.922467i \(0.626170\pi\)
\(920\) 0 0
\(921\) −12.8202 −0.422440
\(922\) 0 0
\(923\) −8.08400 −0.266088
\(924\) 0 0
\(925\) −0.301631 −0.00991755
\(926\) 0 0
\(927\) −0.451779 −0.0148384
\(928\) 0 0
\(929\) −25.4027 −0.833436 −0.416718 0.909036i \(-0.636820\pi\)
−0.416718 + 0.909036i \(0.636820\pi\)
\(930\) 0 0
\(931\) −3.12807 −0.102518
\(932\) 0 0
\(933\) −48.3964 −1.58443
\(934\) 0 0
\(935\) −16.9449 −0.554157
\(936\) 0 0
\(937\) 47.2309 1.54297 0.771483 0.636249i \(-0.219515\pi\)
0.771483 + 0.636249i \(0.219515\pi\)
\(938\) 0 0
\(939\) 6.39035 0.208541
\(940\) 0 0
\(941\) −9.11795 −0.297237 −0.148618 0.988895i \(-0.547483\pi\)
−0.148618 + 0.988895i \(0.547483\pi\)
\(942\) 0 0
\(943\) −62.0799 −2.02160
\(944\) 0 0
\(945\) −22.9200 −0.745587
\(946\) 0 0
\(947\) 51.4205 1.67094 0.835471 0.549535i \(-0.185195\pi\)
0.835471 + 0.549535i \(0.185195\pi\)
\(948\) 0 0
\(949\) −42.6003 −1.38287
\(950\) 0 0
\(951\) −8.25517 −0.267692
\(952\) 0 0
\(953\) 20.6292 0.668247 0.334123 0.942529i \(-0.391560\pi\)
0.334123 + 0.942529i \(0.391560\pi\)
\(954\) 0 0
\(955\) 3.64544 0.117964
\(956\) 0 0
\(957\) −10.7561 −0.347697
\(958\) 0 0
\(959\) −32.4243 −1.04704
\(960\) 0 0
\(961\) −2.16716 −0.0699083
\(962\) 0 0
\(963\) −0.259854 −0.00837367
\(964\) 0 0
\(965\) −19.3172 −0.621843
\(966\) 0 0
\(967\) 31.7119 1.01978 0.509892 0.860238i \(-0.329685\pi\)
0.509892 + 0.860238i \(0.329685\pi\)
\(968\) 0 0
\(969\) 13.0905 0.420526
\(970\) 0 0
\(971\) −20.6220 −0.661791 −0.330896 0.943667i \(-0.607351\pi\)
−0.330896 + 0.943667i \(0.607351\pi\)
\(972\) 0 0
\(973\) −5.15521 −0.165269
\(974\) 0 0
\(975\) 0.480322 0.0153826
\(976\) 0 0
\(977\) 39.1819 1.25354 0.626771 0.779204i \(-0.284376\pi\)
0.626771 + 0.779204i \(0.284376\pi\)
\(978\) 0 0
\(979\) 4.83170 0.154422
\(980\) 0 0
\(981\) 0.0592760 0.00189254
\(982\) 0 0
\(983\) 0.917754 0.0292718 0.0146359 0.999893i \(-0.495341\pi\)
0.0146359 + 0.999893i \(0.495341\pi\)
\(984\) 0 0
\(985\) −32.2225 −1.02669
\(986\) 0 0
\(987\) −35.3050 −1.12377
\(988\) 0 0
\(989\) −41.3819 −1.31587
\(990\) 0 0
\(991\) −2.64141 −0.0839070 −0.0419535 0.999120i \(-0.513358\pi\)
−0.0419535 + 0.999120i \(0.513358\pi\)
\(992\) 0 0
\(993\) 16.2567 0.515891
\(994\) 0 0
\(995\) −19.8862 −0.630436
\(996\) 0 0
\(997\) 0.656840 0.0208023 0.0104012 0.999946i \(-0.496689\pi\)
0.0104012 + 0.999946i \(0.496689\pi\)
\(998\) 0 0
\(999\) 39.8242 1.25998
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.y.1.1 6
4.3 odd 2 1672.2.a.h.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1672.2.a.h.1.6 6 4.3 odd 2
3344.2.a.y.1.1 6 1.1 even 1 trivial