Properties

Label 1344.2.p.c.223.1
Level $1344$
Weight $2$
Character 1344.223
Analytic conductor $10.732$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1344,2,Mod(223,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(10.7318940317\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.1
Root \(1.75780 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1344.223
Dual form 1344.2.p.c.223.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -3.88448 q^{5} +(-2.62383 - 0.339877i) q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} -3.88448 q^{5} +(-2.62383 - 0.339877i) q^{7} -1.00000 q^{9} +1.48046 q^{11} -2.00000 q^{13} +3.88448i q^{15} +3.76720i q^{17} -0.679754i q^{19} +(-0.339877 + 2.62383i) q^{21} +5.20473i q^{23} +10.0892 q^{25} +1.00000i q^{27} -7.03122i q^{29} +6.42503 q^{31} -1.48046i q^{33} +(10.1922 + 1.32025i) q^{35} -8.71176i q^{37} +2.00000i q^{39} -3.16101i q^{41} -8.20859 q^{43} +3.88448 q^{45} +9.88913 q^{47} +(6.76897 + 1.78356i) q^{49} +3.76720 q^{51} +6.42503i q^{53} -5.75084 q^{55} -0.679754 q^{57} +7.76897i q^{59} +12.4095 q^{61} +(2.62383 + 0.339877i) q^{63} +7.76897 q^{65} +8.81478 q^{67} +5.20473 q^{69} +12.9737i q^{71} -13.4562i q^{73} -10.0892i q^{75} +(-3.88448 - 0.503175i) q^{77} +4.44872i q^{79} +1.00000 q^{81} +10.4095i q^{83} -14.6336i q^{85} -7.03122 q^{87} +10.6954i q^{89} +(5.24766 + 0.679754i) q^{91} -6.42503i q^{93} +2.64049i q^{95} -9.88913i q^{97} -1.48046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{9} - 24 q^{13} + 36 q^{25} - 12 q^{49} + 72 q^{61} + 24 q^{69} + 12 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) −3.88448 −1.73719 −0.868597 0.495519i \(-0.834978\pi\)
−0.868597 + 0.495519i \(0.834978\pi\)
\(6\) 0 0
\(7\) −2.62383 0.339877i −0.991715 0.128461i
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.48046 0.446376 0.223188 0.974775i \(-0.428354\pi\)
0.223188 + 0.974775i \(0.428354\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 3.88448i 1.00297i
\(16\) 0 0
\(17\) 3.76720i 0.913679i 0.889549 + 0.456840i \(0.151019\pi\)
−0.889549 + 0.456840i \(0.848981\pi\)
\(18\) 0 0
\(19\) 0.679754i 0.155946i −0.996955 0.0779731i \(-0.975155\pi\)
0.996955 0.0779731i \(-0.0248448\pi\)
\(20\) 0 0
\(21\) −0.339877 + 2.62383i −0.0741672 + 0.572567i
\(22\) 0 0
\(23\) 5.20473i 1.08526i 0.839971 + 0.542631i \(0.182572\pi\)
−0.839971 + 0.542631i \(0.817428\pi\)
\(24\) 0 0
\(25\) 10.0892 2.01784
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.03122i 1.30566i −0.757503 0.652832i \(-0.773581\pi\)
0.757503 0.652832i \(-0.226419\pi\)
\(30\) 0 0
\(31\) 6.42503 1.15397 0.576985 0.816755i \(-0.304229\pi\)
0.576985 + 0.816755i \(0.304229\pi\)
\(32\) 0 0
\(33\) 1.48046i 0.257716i
\(34\) 0 0
\(35\) 10.1922 + 1.32025i 1.72280 + 0.223162i
\(36\) 0 0
\(37\) 8.71176i 1.43220i −0.697995 0.716102i \(-0.745924\pi\)
0.697995 0.716102i \(-0.254076\pi\)
\(38\) 0 0
\(39\) 2.00000i 0.320256i
\(40\) 0 0
\(41\) 3.16101i 0.493666i −0.969058 0.246833i \(-0.920610\pi\)
0.969058 0.246833i \(-0.0793900\pi\)
\(42\) 0 0
\(43\) −8.20859 −1.25180 −0.625899 0.779904i \(-0.715268\pi\)
−0.625899 + 0.779904i \(0.715268\pi\)
\(44\) 0 0
\(45\) 3.88448 0.579065
\(46\) 0 0
\(47\) 9.88913 1.44248 0.721239 0.692686i \(-0.243573\pi\)
0.721239 + 0.692686i \(0.243573\pi\)
\(48\) 0 0
\(49\) 6.76897 + 1.78356i 0.966995 + 0.254794i
\(50\) 0 0
\(51\) 3.76720 0.527513
\(52\) 0 0
\(53\) 6.42503i 0.882545i 0.897373 + 0.441273i \(0.145473\pi\)
−0.897373 + 0.441273i \(0.854527\pi\)
\(54\) 0 0
\(55\) −5.75084 −0.775442
\(56\) 0 0
\(57\) −0.679754 −0.0900356
\(58\) 0 0
\(59\) 7.76897i 1.01143i 0.862700 + 0.505717i \(0.168772\pi\)
−0.862700 + 0.505717i \(0.831228\pi\)
\(60\) 0 0
\(61\) 12.4095 1.58887 0.794434 0.607350i \(-0.207767\pi\)
0.794434 + 0.607350i \(0.207767\pi\)
\(62\) 0 0
\(63\) 2.62383 + 0.339877i 0.330572 + 0.0428205i
\(64\) 0 0
\(65\) 7.76897 0.963622
\(66\) 0 0
\(67\) 8.81478 1.07690 0.538448 0.842659i \(-0.319011\pi\)
0.538448 + 0.842659i \(0.319011\pi\)
\(68\) 0 0
\(69\) 5.20473 0.626576
\(70\) 0 0
\(71\) 12.9737i 1.53969i 0.638228 + 0.769847i \(0.279668\pi\)
−0.638228 + 0.769847i \(0.720332\pi\)
\(72\) 0 0
\(73\) 13.4562i 1.57493i −0.616356 0.787467i \(-0.711392\pi\)
0.616356 0.787467i \(-0.288608\pi\)
\(74\) 0 0
\(75\) 10.0892i 1.16500i
\(76\) 0 0
\(77\) −3.88448 0.503175i −0.442678 0.0573421i
\(78\) 0 0
\(79\) 4.44872i 0.500520i 0.968179 + 0.250260i \(0.0805161\pi\)
−0.968179 + 0.250260i \(0.919484\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.4095i 1.14259i 0.820746 + 0.571293i \(0.193558\pi\)
−0.820746 + 0.571293i \(0.806442\pi\)
\(84\) 0 0
\(85\) 14.6336i 1.58724i
\(86\) 0 0
\(87\) −7.03122 −0.753826
\(88\) 0 0
\(89\) 10.6954i 1.13371i 0.823818 + 0.566855i \(0.191840\pi\)
−0.823818 + 0.566855i \(0.808160\pi\)
\(90\) 0 0
\(91\) 5.24766 + 0.679754i 0.550104 + 0.0712576i
\(92\) 0 0
\(93\) 6.42503i 0.666244i
\(94\) 0 0
\(95\) 2.64049i 0.270909i
\(96\) 0 0
\(97\) 9.88913i 1.00409i −0.864842 0.502044i \(-0.832581\pi\)
0.864842 0.502044i \(-0.167419\pi\)
\(98\) 0 0
\(99\) −1.48046 −0.148792
\(100\) 0 0
\(101\) −3.88448 −0.386521 −0.193260 0.981148i \(-0.561906\pi\)
−0.193260 + 0.981148i \(0.561906\pi\)
\(102\) 0 0
\(103\) −3.06394 −0.301899 −0.150950 0.988541i \(-0.548233\pi\)
−0.150950 + 0.988541i \(0.548233\pi\)
\(104\) 0 0
\(105\) 1.32025 10.1922i 0.128843 0.994659i
\(106\) 0 0
\(107\) −2.08665 −0.201724 −0.100862 0.994900i \(-0.532160\pi\)
−0.100862 + 0.994900i \(0.532160\pi\)
\(108\) 0 0
\(109\) 17.4235i 1.66887i 0.551106 + 0.834435i \(0.314206\pi\)
−0.551106 + 0.834435i \(0.685794\pi\)
\(110\) 0 0
\(111\) −8.71176 −0.826884
\(112\) 0 0
\(113\) −1.76897 −0.166410 −0.0832052 0.996532i \(-0.526516\pi\)
−0.0832052 + 0.996532i \(0.526516\pi\)
\(114\) 0 0
\(115\) 20.2177i 1.88531i
\(116\) 0 0
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) 1.28038 9.88448i 0.117373 0.906109i
\(120\) 0 0
\(121\) −8.80823 −0.800748
\(122\) 0 0
\(123\) −3.16101 −0.285018
\(124\) 0 0
\(125\) −19.7690 −1.76819
\(126\) 0 0
\(127\) 11.0892i 0.984009i 0.870593 + 0.492004i \(0.163736\pi\)
−0.870593 + 0.492004i \(0.836264\pi\)
\(128\) 0 0
\(129\) 8.20859i 0.722726i
\(130\) 0 0
\(131\) 1.59054i 0.138966i −0.997583 0.0694831i \(-0.977865\pi\)
0.997583 0.0694831i \(-0.0221350\pi\)
\(132\) 0 0
\(133\) −0.231033 + 1.78356i −0.0200331 + 0.154654i
\(134\) 0 0
\(135\) 3.88448i 0.334323i
\(136\) 0 0
\(137\) 18.8974 1.61452 0.807259 0.590198i \(-0.200950\pi\)
0.807259 + 0.590198i \(0.200950\pi\)
\(138\) 0 0
\(139\) 10.1784i 0.863323i −0.902036 0.431661i \(-0.857928\pi\)
0.902036 0.431661i \(-0.142072\pi\)
\(140\) 0 0
\(141\) 9.88913i 0.832815i
\(142\) 0 0
\(143\) −2.96093 −0.247605
\(144\) 0 0
\(145\) 27.3127i 2.26819i
\(146\) 0 0
\(147\) 1.78356 6.76897i 0.147105 0.558295i
\(148\) 0 0
\(149\) 4.07029i 0.333451i −0.986003 0.166726i \(-0.946681\pi\)
0.986003 0.166726i \(-0.0533194\pi\)
\(150\) 0 0
\(151\) 8.67975i 0.706348i 0.935558 + 0.353174i \(0.114898\pi\)
−0.935558 + 0.353174i \(0.885102\pi\)
\(152\) 0 0
\(153\) 3.76720i 0.304560i
\(154\) 0 0
\(155\) −24.9579 −2.00467
\(156\) 0 0
\(157\) −3.12847 −0.249679 −0.124840 0.992177i \(-0.539842\pi\)
−0.124840 + 0.992177i \(0.539842\pi\)
\(158\) 0 0
\(159\) 6.42503 0.509538
\(160\) 0 0
\(161\) 1.76897 13.6563i 0.139414 1.07627i
\(162\) 0 0
\(163\) 12.1759 0.953687 0.476844 0.878988i \(-0.341781\pi\)
0.476844 + 0.878988i \(0.341781\pi\)
\(164\) 0 0
\(165\) 5.75084i 0.447702i
\(166\) 0 0
\(167\) 3.56712 0.276032 0.138016 0.990430i \(-0.455927\pi\)
0.138016 + 0.990430i \(0.455927\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0.679754i 0.0519821i
\(172\) 0 0
\(173\) −4.93444 −0.375158 −0.187579 0.982249i \(-0.560064\pi\)
−0.187579 + 0.982249i \(0.560064\pi\)
\(174\) 0 0
\(175\) −26.4724 3.42909i −2.00112 0.259215i
\(176\) 0 0
\(177\) 7.76897 0.583951
\(178\) 0 0
\(179\) 11.3696 0.849803 0.424902 0.905240i \(-0.360309\pi\)
0.424902 + 0.905240i \(0.360309\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 12.4095i 0.917334i
\(184\) 0 0
\(185\) 33.8407i 2.48802i
\(186\) 0 0
\(187\) 5.57720i 0.407845i
\(188\) 0 0
\(189\) 0.339877 2.62383i 0.0247224 0.190856i
\(190\) 0 0
\(191\) 22.3332i 1.61597i −0.589200 0.807987i \(-0.700557\pi\)
0.589200 0.807987i \(-0.299443\pi\)
\(192\) 0 0
\(193\) −20.6271 −1.48477 −0.742387 0.669971i \(-0.766307\pi\)
−0.742387 + 0.669971i \(0.766307\pi\)
\(194\) 0 0
\(195\) 7.76897i 0.556347i
\(196\) 0 0
\(197\) 23.2424i 1.65595i 0.560765 + 0.827975i \(0.310507\pi\)
−0.560765 + 0.827975i \(0.689493\pi\)
\(198\) 0 0
\(199\) 26.2383 1.85998 0.929992 0.367580i \(-0.119814\pi\)
0.929992 + 0.367580i \(0.119814\pi\)
\(200\) 0 0
\(201\) 8.81478i 0.621746i
\(202\) 0 0
\(203\) −2.38975 + 18.4487i −0.167727 + 1.29485i
\(204\) 0 0
\(205\) 12.2789i 0.857594i
\(206\) 0 0
\(207\) 5.20473i 0.361754i
\(208\) 0 0
\(209\) 1.00635i 0.0696107i
\(210\) 0 0
\(211\) 12.7821 0.879953 0.439976 0.898009i \(-0.354987\pi\)
0.439976 + 0.898009i \(0.354987\pi\)
\(212\) 0 0
\(213\) 12.9737 0.888943
\(214\) 0 0
\(215\) 31.8861 2.17462
\(216\) 0 0
\(217\) −16.8582 2.18372i −1.14441 0.148240i
\(218\) 0 0
\(219\) −13.4562 −0.909289
\(220\) 0 0
\(221\) 7.53439i 0.506818i
\(222\) 0 0
\(223\) −4.03528 −0.270222 −0.135111 0.990830i \(-0.543139\pi\)
−0.135111 + 0.990830i \(0.543139\pi\)
\(224\) 0 0
\(225\) −10.0892 −0.672614
\(226\) 0 0
\(227\) 20.8189i 1.38180i 0.722950 + 0.690900i \(0.242786\pi\)
−0.722950 + 0.690900i \(0.757214\pi\)
\(228\) 0 0
\(229\) 26.8974 1.77743 0.888717 0.458457i \(-0.151598\pi\)
0.888717 + 0.458457i \(0.151598\pi\)
\(230\) 0 0
\(231\) −0.503175 + 3.88448i −0.0331065 + 0.255580i
\(232\) 0 0
\(233\) −25.2284 −1.65277 −0.826383 0.563108i \(-0.809605\pi\)
−0.826383 + 0.563108i \(0.809605\pi\)
\(234\) 0 0
\(235\) −38.4142 −2.50586
\(236\) 0 0
\(237\) 4.44872 0.288975
\(238\) 0 0
\(239\) 8.74266i 0.565516i −0.959191 0.282758i \(-0.908751\pi\)
0.959191 0.282758i \(-0.0912493\pi\)
\(240\) 0 0
\(241\) 17.0234i 1.09657i −0.836291 0.548286i \(-0.815281\pi\)
0.836291 0.548286i \(-0.184719\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) −26.2939 6.92820i −1.67986 0.442627i
\(246\) 0 0
\(247\) 1.35951i 0.0865034i
\(248\) 0 0
\(249\) 10.4095 0.659673
\(250\) 0 0
\(251\) 24.8974i 1.57151i 0.618536 + 0.785756i \(0.287726\pi\)
−0.618536 + 0.785756i \(0.712274\pi\)
\(252\) 0 0
\(253\) 7.70541i 0.484435i
\(254\) 0 0
\(255\) −14.6336 −0.916392
\(256\) 0 0
\(257\) 3.56117i 0.222139i 0.993813 + 0.111070i \(0.0354277\pi\)
−0.993813 + 0.111070i \(0.964572\pi\)
\(258\) 0 0
\(259\) −2.96093 + 22.8582i −0.183983 + 1.42034i
\(260\) 0 0
\(261\) 7.03122i 0.435221i
\(262\) 0 0
\(263\) 24.9737i 1.53994i −0.638078 0.769972i \(-0.720270\pi\)
0.638078 0.769972i \(-0.279730\pi\)
\(264\) 0 0
\(265\) 24.9579i 1.53315i
\(266\) 0 0
\(267\) 10.6954 0.654548
\(268\) 0 0
\(269\) 2.83453 0.172824 0.0864122 0.996259i \(-0.472460\pi\)
0.0864122 + 0.996259i \(0.472460\pi\)
\(270\) 0 0
\(271\) 3.86426 0.234737 0.117369 0.993088i \(-0.462554\pi\)
0.117369 + 0.993088i \(0.462554\pi\)
\(272\) 0 0
\(273\) 0.679754 5.24766i 0.0411406 0.317603i
\(274\) 0 0
\(275\) 14.9367 0.900717
\(276\) 0 0
\(277\) 1.78356i 0.107164i −0.998563 0.0535818i \(-0.982936\pi\)
0.998563 0.0535818i \(-0.0170638\pi\)
\(278\) 0 0
\(279\) −6.42503 −0.384656
\(280\) 0 0
\(281\) 20.4880 1.22221 0.611105 0.791549i \(-0.290725\pi\)
0.611105 + 0.791549i \(0.290725\pi\)
\(282\) 0 0
\(283\) 12.1392i 0.721599i −0.932643 0.360799i \(-0.882504\pi\)
0.932643 0.360799i \(-0.117496\pi\)
\(284\) 0 0
\(285\) 2.64049 0.156409
\(286\) 0 0
\(287\) −1.07435 + 8.29394i −0.0634171 + 0.489576i
\(288\) 0 0
\(289\) 2.80823 0.165190
\(290\) 0 0
\(291\) −9.88913 −0.579711
\(292\) 0 0
\(293\) −4.93444 −0.288273 −0.144136 0.989558i \(-0.546040\pi\)
−0.144136 + 0.989558i \(0.546040\pi\)
\(294\) 0 0
\(295\) 30.1784i 1.75706i
\(296\) 0 0
\(297\) 1.48046i 0.0859052i
\(298\) 0 0
\(299\) 10.4095i 0.601995i
\(300\) 0 0
\(301\) 21.5379 + 2.78991i 1.24143 + 0.160808i
\(302\) 0 0
\(303\) 3.88448i 0.223158i
\(304\) 0 0
\(305\) −48.2043 −2.76017
\(306\) 0 0
\(307\) 21.4987i 1.22699i 0.789697 + 0.613497i \(0.210238\pi\)
−0.789697 + 0.613497i \(0.789762\pi\)
\(308\) 0 0
\(309\) 3.06394i 0.174302i
\(310\) 0 0
\(311\) −6.52804 −0.370171 −0.185086 0.982722i \(-0.559256\pi\)
−0.185086 + 0.982722i \(0.559256\pi\)
\(312\) 0 0
\(313\) 33.6347i 1.90114i 0.310505 + 0.950572i \(0.399502\pi\)
−0.310505 + 0.950572i \(0.600498\pi\)
\(314\) 0 0
\(315\) −10.1922 1.32025i −0.574267 0.0743874i
\(316\) 0 0
\(317\) 4.07029i 0.228610i −0.993446 0.114305i \(-0.963536\pi\)
0.993446 0.114305i \(-0.0364642\pi\)
\(318\) 0 0
\(319\) 10.4095i 0.582818i
\(320\) 0 0
\(321\) 2.08665i 0.116466i
\(322\) 0 0
\(323\) 2.56077 0.142485
\(324\) 0 0
\(325\) −20.1784 −1.11930
\(326\) 0 0
\(327\) 17.4235 0.963523
\(328\) 0 0
\(329\) −25.9474 3.36109i −1.43053 0.185303i
\(330\) 0 0
\(331\) −15.1368 −0.831993 −0.415997 0.909366i \(-0.636567\pi\)
−0.415997 + 0.909366i \(0.636567\pi\)
\(332\) 0 0
\(333\) 8.71176i 0.477402i
\(334\) 0 0
\(335\) −34.2409 −1.87078
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 1.76897i 0.0960771i
\(340\) 0 0
\(341\) 9.51202 0.515105
\(342\) 0 0
\(343\) −17.1544 6.98037i −0.926252 0.376905i
\(344\) 0 0
\(345\) −20.2177 −1.08848
\(346\) 0 0
\(347\) −8.61470 −0.462461 −0.231231 0.972899i \(-0.574275\pi\)
−0.231231 + 0.972899i \(0.574275\pi\)
\(348\) 0 0
\(349\) −15.5905 −0.834542 −0.417271 0.908782i \(-0.637013\pi\)
−0.417271 + 0.908782i \(0.637013\pi\)
\(350\) 0 0
\(351\) 2.00000i 0.106752i
\(352\) 0 0
\(353\) 11.9078i 0.633787i −0.948461 0.316894i \(-0.897360\pi\)
0.948461 0.316894i \(-0.102640\pi\)
\(354\) 0 0
\(355\) 50.3961i 2.67475i
\(356\) 0 0
\(357\) −9.88448 1.28038i −0.523142 0.0677651i
\(358\) 0 0
\(359\) 11.0263i 0.581946i 0.956731 + 0.290973i \(0.0939790\pi\)
−0.956731 + 0.290973i \(0.906021\pi\)
\(360\) 0 0
\(361\) 18.5379 0.975681
\(362\) 0 0
\(363\) 8.80823i 0.462312i
\(364\) 0 0
\(365\) 52.2706i 2.73597i
\(366\) 0 0
\(367\) 11.1695 0.583044 0.291522 0.956564i \(-0.405838\pi\)
0.291522 + 0.956564i \(0.405838\pi\)
\(368\) 0 0
\(369\) 3.16101i 0.164555i
\(370\) 0 0
\(371\) 2.18372 16.8582i 0.113373 0.875233i
\(372\) 0 0
\(373\) 8.14058i 0.421503i 0.977540 + 0.210752i \(0.0675912\pi\)
−0.977540 + 0.210752i \(0.932409\pi\)
\(374\) 0 0
\(375\) 19.7690i 1.02087i
\(376\) 0 0
\(377\) 14.0624i 0.724252i
\(378\) 0 0
\(379\) 17.4915 0.898479 0.449240 0.893411i \(-0.351695\pi\)
0.449240 + 0.893411i \(0.351695\pi\)
\(380\) 0 0
\(381\) 11.0892 0.568118
\(382\) 0 0
\(383\) 31.8861 1.62930 0.814652 0.579950i \(-0.196928\pi\)
0.814652 + 0.579950i \(0.196928\pi\)
\(384\) 0 0
\(385\) 15.0892 + 1.95458i 0.769018 + 0.0996144i
\(386\) 0 0
\(387\) 8.20859 0.417266
\(388\) 0 0
\(389\) 23.2424i 1.17843i −0.807975 0.589217i \(-0.799436\pi\)
0.807975 0.589217i \(-0.200564\pi\)
\(390\) 0 0
\(391\) −19.6072 −0.991581
\(392\) 0 0
\(393\) −1.59054 −0.0802321
\(394\) 0 0
\(395\) 17.2810i 0.869501i
\(396\) 0 0
\(397\) −3.04995 −0.153073 −0.0765364 0.997067i \(-0.524386\pi\)
−0.0765364 + 0.997067i \(0.524386\pi\)
\(398\) 0 0
\(399\) 1.78356 + 0.231033i 0.0892896 + 0.0115661i
\(400\) 0 0
\(401\) 17.4594 0.871881 0.435941 0.899975i \(-0.356416\pi\)
0.435941 + 0.899975i \(0.356416\pi\)
\(402\) 0 0
\(403\) −12.8501 −0.640107
\(404\) 0 0
\(405\) −3.88448 −0.193022
\(406\) 0 0
\(407\) 12.8974i 0.639302i
\(408\) 0 0
\(409\) 17.0234i 0.841751i −0.907118 0.420876i \(-0.861723\pi\)
0.907118 0.420876i \(-0.138277\pi\)
\(410\) 0 0
\(411\) 18.8974i 0.932142i
\(412\) 0 0
\(413\) 2.64049 20.3844i 0.129930 1.00305i
\(414\) 0 0
\(415\) 40.4354i 1.98489i
\(416\) 0 0
\(417\) −10.1784 −0.498440
\(418\) 0 0
\(419\) 11.3069i 0.552378i −0.961103 0.276189i \(-0.910928\pi\)
0.961103 0.276189i \(-0.0890716\pi\)
\(420\) 0 0
\(421\) 8.91779i 0.434627i 0.976102 + 0.217313i \(0.0697293\pi\)
−0.976102 + 0.217313i \(0.930271\pi\)
\(422\) 0 0
\(423\) −9.88913 −0.480826
\(424\) 0 0
\(425\) 38.0081i 1.84366i
\(426\) 0 0
\(427\) −32.5603 4.21769i −1.57570 0.204108i
\(428\) 0 0
\(429\) 2.96093i 0.142955i
\(430\) 0 0
\(431\) 15.4617i 0.744763i 0.928080 + 0.372381i \(0.121459\pi\)
−0.928080 + 0.372381i \(0.878541\pi\)
\(432\) 0 0
\(433\) 7.93455i 0.381310i −0.981657 0.190655i \(-0.938939\pi\)
0.981657 0.190655i \(-0.0610612\pi\)
\(434\) 0 0
\(435\) 27.3127 1.30954
\(436\) 0 0
\(437\) 3.53793 0.169242
\(438\) 0 0
\(439\) −1.88657 −0.0900412 −0.0450206 0.998986i \(-0.514335\pi\)
−0.0450206 + 0.998986i \(0.514335\pi\)
\(440\) 0 0
\(441\) −6.76897 1.78356i −0.322332 0.0849313i
\(442\) 0 0
\(443\) −22.8713 −1.08665 −0.543323 0.839524i \(-0.682834\pi\)
−0.543323 + 0.839524i \(0.682834\pi\)
\(444\) 0 0
\(445\) 41.5461i 1.96947i
\(446\) 0 0
\(447\) −4.07029 −0.192518
\(448\) 0 0
\(449\) 7.04995 0.332708 0.166354 0.986066i \(-0.446801\pi\)
0.166354 + 0.986066i \(0.446801\pi\)
\(450\) 0 0
\(451\) 4.67975i 0.220361i
\(452\) 0 0
\(453\) 8.67975 0.407810
\(454\) 0 0
\(455\) −20.3844 2.64049i −0.955638 0.123788i
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) −3.76720 −0.175838
\(460\) 0 0
\(461\) −4.57758 −0.213199 −0.106600 0.994302i \(-0.533996\pi\)
−0.106600 + 0.994302i \(0.533996\pi\)
\(462\) 0 0
\(463\) 25.2676i 1.17429i −0.809483 0.587143i \(-0.800252\pi\)
0.809483 0.587143i \(-0.199748\pi\)
\(464\) 0 0
\(465\) 24.9579i 1.15740i
\(466\) 0 0
\(467\) 6.17843i 0.285904i 0.989730 + 0.142952i \(0.0456594\pi\)
−0.989730 + 0.142952i \(0.954341\pi\)
\(468\) 0 0
\(469\) −23.1285 2.99594i −1.06797 0.138340i
\(470\) 0 0
\(471\) 3.12847i 0.144153i
\(472\) 0 0
\(473\) −12.1525 −0.558773
\(474\) 0 0
\(475\) 6.85818i 0.314675i
\(476\) 0 0
\(477\) 6.42503i 0.294182i
\(478\) 0 0
\(479\) 27.3127 1.24795 0.623973 0.781446i \(-0.285517\pi\)
0.623973 + 0.781446i \(0.285517\pi\)
\(480\) 0 0
\(481\) 17.4235i 0.794444i
\(482\) 0 0
\(483\) −13.6563 1.76897i −0.621384 0.0804908i
\(484\) 0 0
\(485\) 38.4142i 1.74430i
\(486\) 0 0
\(487\) 20.1392i 0.912593i 0.889828 + 0.456296i \(0.150824\pi\)
−0.889828 + 0.456296i \(0.849176\pi\)
\(488\) 0 0
\(489\) 12.1759i 0.550612i
\(490\) 0 0
\(491\) 9.22089 0.416133 0.208066 0.978115i \(-0.433283\pi\)
0.208066 + 0.978115i \(0.433283\pi\)
\(492\) 0 0
\(493\) 26.4880 1.19296
\(494\) 0 0
\(495\) 5.75084 0.258481
\(496\) 0 0
\(497\) 4.40946 34.0408i 0.197791 1.52694i
\(498\) 0 0
\(499\) −4.64147 −0.207781 −0.103890 0.994589i \(-0.533129\pi\)
−0.103890 + 0.994589i \(0.533129\pi\)
\(500\) 0 0
\(501\) 3.56712i 0.159367i
\(502\) 0 0
\(503\) −23.7455 −1.05876 −0.529381 0.848384i \(-0.677576\pi\)
−0.529381 + 0.848384i \(0.677576\pi\)
\(504\) 0 0
\(505\) 15.0892 0.671461
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 9.16547 0.406252 0.203126 0.979153i \(-0.434890\pi\)
0.203126 + 0.979153i \(0.434890\pi\)
\(510\) 0 0
\(511\) −4.57347 + 35.3069i −0.202318 + 1.56189i
\(512\) 0 0
\(513\) 0.679754 0.0300119
\(514\) 0 0
\(515\) 11.9018 0.524457
\(516\) 0 0
\(517\) 14.6405 0.643888
\(518\) 0 0
\(519\) 4.93444i 0.216598i
\(520\) 0 0
\(521\) 18.8360i 0.825219i 0.910908 + 0.412610i \(0.135383\pi\)
−0.910908 + 0.412610i \(0.864617\pi\)
\(522\) 0 0
\(523\) 41.2543i 1.80392i 0.431815 + 0.901962i \(0.357873\pi\)
−0.431815 + 0.901962i \(0.642127\pi\)
\(524\) 0 0
\(525\) −3.42909 + 26.4724i −0.149658 + 1.15535i
\(526\) 0 0
\(527\) 24.2043i 1.05436i
\(528\) 0 0
\(529\) −4.08921 −0.177792
\(530\) 0 0
\(531\) 7.76897i 0.337144i
\(532\) 0 0
\(533\) 6.32201i 0.273837i
\(534\) 0 0
\(535\) 8.10557 0.350434
\(536\) 0 0
\(537\) 11.3696i 0.490634i
\(538\) 0 0
\(539\) 10.0212 + 2.64049i 0.431644 + 0.113734i
\(540\) 0 0
\(541\) 37.8430i 1.62700i −0.581567 0.813499i \(-0.697560\pi\)
0.581567 0.813499i \(-0.302440\pi\)
\(542\) 0 0
\(543\) 2.00000i 0.0858282i
\(544\) 0 0
\(545\) 67.6814i 2.89915i
\(546\) 0 0
\(547\) 31.9541 1.36626 0.683130 0.730297i \(-0.260618\pi\)
0.683130 + 0.730297i \(0.260618\pi\)
\(548\) 0 0
\(549\) −12.4095 −0.529623
\(550\) 0 0
\(551\) −4.77950 −0.203613
\(552\) 0 0
\(553\) 1.51202 11.6727i 0.0642975 0.496373i
\(554\) 0 0
\(555\) 33.8407 1.43646
\(556\) 0 0
\(557\) 17.7207i 0.750849i −0.926853 0.375424i \(-0.877497\pi\)
0.926853 0.375424i \(-0.122503\pi\)
\(558\) 0 0
\(559\) 16.4172 0.694372
\(560\) 0 0
\(561\) 5.57720 0.235469
\(562\) 0 0
\(563\) 5.12847i 0.216139i 0.994143 + 0.108070i \(0.0344670\pi\)
−0.994143 + 0.108070i \(0.965533\pi\)
\(564\) 0 0
\(565\) 6.87153 0.289087
\(566\) 0 0
\(567\) −2.62383 0.339877i −0.110191 0.0142735i
\(568\) 0 0
\(569\) 28.4095 1.19099 0.595493 0.803360i \(-0.296957\pi\)
0.595493 + 0.803360i \(0.296957\pi\)
\(570\) 0 0
\(571\) 30.8118 1.28943 0.644716 0.764422i \(-0.276976\pi\)
0.644716 + 0.764422i \(0.276976\pi\)
\(572\) 0 0
\(573\) −22.3332 −0.932983
\(574\) 0 0
\(575\) 52.5116i 2.18989i
\(576\) 0 0
\(577\) 36.0594i 1.50117i 0.660772 + 0.750587i \(0.270229\pi\)
−0.660772 + 0.750587i \(0.729771\pi\)
\(578\) 0 0
\(579\) 20.6271i 0.857235i
\(580\) 0 0
\(581\) 3.53793 27.3127i 0.146778 1.13312i
\(582\) 0 0
\(583\) 9.51202i 0.393948i
\(584\) 0 0
\(585\) −7.76897 −0.321207
\(586\) 0 0
\(587\) 18.1784i 0.750304i 0.926963 + 0.375152i \(0.122409\pi\)
−0.926963 + 0.375152i \(0.877591\pi\)
\(588\) 0 0
\(589\) 4.36744i 0.179957i
\(590\) 0 0
\(591\) 23.2424 0.956063
\(592\) 0 0
\(593\) 10.9014i 0.447668i −0.974627 0.223834i \(-0.928143\pi\)
0.974627 0.223834i \(-0.0718574\pi\)
\(594\) 0 0
\(595\) −4.97363 + 38.3961i −0.203899 + 1.57409i
\(596\) 0 0
\(597\) 26.2383i 1.07386i
\(598\) 0 0
\(599\) 5.74532i 0.234747i −0.993088 0.117374i \(-0.962552\pi\)
0.993088 0.117374i \(-0.0374475\pi\)
\(600\) 0 0
\(601\) 35.9894i 1.46804i −0.679129 0.734019i \(-0.737642\pi\)
0.679129 0.734019i \(-0.262358\pi\)
\(602\) 0 0
\(603\) −8.81478 −0.358965
\(604\) 0 0
\(605\) 34.2154 1.39105
\(606\) 0 0
\(607\) 22.4652 0.911832 0.455916 0.890023i \(-0.349312\pi\)
0.455916 + 0.890023i \(0.349312\pi\)
\(608\) 0 0
\(609\) 18.4487 + 2.38975i 0.747580 + 0.0968375i
\(610\) 0 0
\(611\) −19.7783 −0.800143
\(612\) 0 0
\(613\) 36.1954i 1.46192i 0.682420 + 0.730960i \(0.260927\pi\)
−0.682420 + 0.730960i \(0.739073\pi\)
\(614\) 0 0
\(615\) 12.2789 0.495132
\(616\) 0 0
\(617\) 30.8974 1.24388 0.621942 0.783063i \(-0.286344\pi\)
0.621942 + 0.783063i \(0.286344\pi\)
\(618\) 0 0
\(619\) 7.45941i 0.299819i 0.988700 + 0.149910i \(0.0478982\pi\)
−0.988700 + 0.149910i \(0.952102\pi\)
\(620\) 0 0
\(621\) −5.20473 −0.208859
\(622\) 0 0
\(623\) 3.63512 28.0629i 0.145638 1.12432i
\(624\) 0 0
\(625\) 26.3462 1.05385
\(626\) 0 0
\(627\) −1.00635 −0.0401898
\(628\) 0 0
\(629\) 32.8189 1.30858
\(630\) 0 0
\(631\) 17.2676i 0.687414i 0.939077 + 0.343707i \(0.111683\pi\)
−0.939077 + 0.343707i \(0.888317\pi\)
\(632\) 0 0
\(633\) 12.7821i 0.508041i
\(634\) 0 0
\(635\) 43.0759i 1.70941i
\(636\) 0 0
\(637\) −13.5379 3.56712i −0.536393 0.141334i
\(638\) 0 0
\(639\) 12.9737i 0.513232i
\(640\) 0 0
\(641\) 8.09990 0.319927 0.159963 0.987123i \(-0.448862\pi\)
0.159963 + 0.987123i \(0.448862\pi\)
\(642\) 0 0
\(643\) 24.2177i 0.955052i 0.878618 + 0.477526i \(0.158466\pi\)
−0.878618 + 0.477526i \(0.841534\pi\)
\(644\) 0 0
\(645\) 31.8861i 1.25551i
\(646\) 0 0
\(647\) −8.14058 −0.320039 −0.160020 0.987114i \(-0.551156\pi\)
−0.160020 + 0.987114i \(0.551156\pi\)
\(648\) 0 0
\(649\) 11.5017i 0.451480i
\(650\) 0 0
\(651\) −2.18372 + 16.8582i −0.0855867 + 0.660724i
\(652\) 0 0
\(653\) 35.3502i 1.38336i 0.722204 + 0.691681i \(0.243129\pi\)
−0.722204 + 0.691681i \(0.756871\pi\)
\(654\) 0 0
\(655\) 6.17843i 0.241411i
\(656\) 0 0
\(657\) 13.4562i 0.524978i
\(658\) 0 0
\(659\) −31.5480 −1.22894 −0.614468 0.788942i \(-0.710629\pi\)
−0.614468 + 0.788942i \(0.710629\pi\)
\(660\) 0 0
\(661\) −10.3096 −0.400995 −0.200498 0.979694i \(-0.564256\pi\)
−0.200498 + 0.979694i \(0.564256\pi\)
\(662\) 0 0
\(663\) −7.53439 −0.292612
\(664\) 0 0
\(665\) 0.897442 6.92820i 0.0348013 0.268664i
\(666\) 0 0
\(667\) 36.5956 1.41699
\(668\) 0 0
\(669\) 4.03528i 0.156013i
\(670\) 0 0
\(671\) 18.3717 0.709233
\(672\) 0 0
\(673\) −25.0892 −0.967118 −0.483559 0.875312i \(-0.660656\pi\)
−0.483559 + 0.875312i \(0.660656\pi\)
\(674\) 0 0
\(675\) 10.0892i 0.388334i
\(676\) 0 0
\(677\) 26.1414 1.00470 0.502348 0.864665i \(-0.332469\pi\)
0.502348 + 0.864665i \(0.332469\pi\)
\(678\) 0 0
\(679\) −3.36109 + 25.9474i −0.128987 + 0.995770i
\(680\) 0 0
\(681\) 20.8189 0.797783
\(682\) 0 0
\(683\) 39.6886 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(684\) 0 0
\(685\) −73.4068 −2.80473
\(686\) 0 0
\(687\) 26.8974i 1.02620i
\(688\) 0 0
\(689\) 12.8501i 0.489548i
\(690\) 0 0
\(691\) 20.4354i 0.777398i −0.921365 0.388699i \(-0.872925\pi\)
0.921365 0.388699i \(-0.127075\pi\)
\(692\) 0 0
\(693\) 3.88448 + 0.503175i 0.147559 + 0.0191140i
\(694\) 0 0
\(695\) 39.5379i 1.49976i
\(696\) 0 0
\(697\) 11.9081 0.451053
\(698\) 0 0
\(699\) 25.2284i 0.954225i
\(700\) 0 0
\(701\) 23.0482i 0.870520i −0.900305 0.435260i \(-0.856657\pi\)
0.900305 0.435260i \(-0.143343\pi\)
\(702\) 0 0
\(703\) −5.92185 −0.223347
\(704\) 0 0
\(705\) 38.4142i 1.44676i
\(706\) 0 0
\(707\) 10.1922 + 1.32025i 0.383318 + 0.0496530i
\(708\) 0 0
\(709\) 6.72217i 0.252457i 0.992001 + 0.126228i \(0.0402872\pi\)
−0.992001 + 0.126228i \(0.959713\pi\)
\(710\) 0 0
\(711\) 4.44872i 0.166840i
\(712\) 0 0
\(713\) 33.4405i 1.25236i
\(714\) 0 0
\(715\) 11.5017 0.430138
\(716\) 0 0
\(717\) −8.74266 −0.326501
\(718\) 0 0
\(719\) −14.6686 −0.547047 −0.273524 0.961865i \(-0.588189\pi\)
−0.273524 + 0.961865i \(0.588189\pi\)
\(720\) 0 0
\(721\) 8.03926 + 1.04136i 0.299398 + 0.0387824i
\(722\) 0 0
\(723\) −17.0234 −0.633106
\(724\) 0 0
\(725\) 70.9395i 2.63463i
\(726\) 0 0
\(727\) −9.99214 −0.370588 −0.185294 0.982683i \(-0.559324\pi\)
−0.185294 + 0.982683i \(0.559324\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 30.9234i 1.14374i
\(732\) 0 0
\(733\) 9.45941 0.349391 0.174696 0.984622i \(-0.444106\pi\)
0.174696 + 0.984622i \(0.444106\pi\)
\(734\) 0 0
\(735\) −6.92820 + 26.2939i −0.255551 + 0.969867i
\(736\) 0 0
\(737\) 13.0500 0.480701
\(738\) 0 0
\(739\) 12.3119 0.452899 0.226450 0.974023i \(-0.427288\pi\)
0.226450 + 0.974023i \(0.427288\pi\)
\(740\) 0 0
\(741\) 1.35951 0.0499428
\(742\) 0 0
\(743\) 1.51429i 0.0555537i 0.999614 + 0.0277769i \(0.00884279\pi\)
−0.999614 + 0.0277769i \(0.991157\pi\)
\(744\) 0 0
\(745\) 15.8110i 0.579270i
\(746\) 0 0
\(747\) 10.4095i 0.380862i
\(748\) 0 0
\(749\) 5.47502 + 0.709205i 0.200053 + 0.0259138i
\(750\) 0 0
\(751\) 52.5745i 1.91847i 0.282606 + 0.959236i \(0.408801\pi\)
−0.282606 + 0.959236i \(0.591199\pi\)
\(752\) 0 0
\(753\) 24.8974 0.907313
\(754\) 0 0
\(755\) 33.7164i 1.22706i
\(756\) 0 0
\(757\) 22.2030i 0.806982i −0.914984 0.403491i \(-0.867797\pi\)
0.914984 0.403491i \(-0.132203\pi\)
\(758\) 0 0
\(759\) 7.70541 0.279689
\(760\) 0 0
\(761\) 11.3016i 0.409682i 0.978795 + 0.204841i \(0.0656678\pi\)
−0.978795 + 0.204841i \(0.934332\pi\)
\(762\) 0 0
\(763\) 5.92185 45.7164i 0.214385 1.65504i
\(764\) 0 0
\(765\) 14.6336i 0.529079i
\(766\) 0 0
\(767\) 15.5379i 0.561042i
\(768\) 0 0
\(769\) 32.4923i 1.17170i 0.810419 + 0.585851i \(0.199240\pi\)
−0.810419 + 0.585851i \(0.800760\pi\)
\(770\) 0 0
\(771\) 3.56117 0.128252
\(772\) 0 0
\(773\) −35.6535 −1.28237 −0.641183 0.767388i \(-0.721556\pi\)
−0.641183 + 0.767388i \(0.721556\pi\)
\(774\) 0 0
\(775\) 64.8235 2.32853
\(776\) 0 0
\(777\) 22.8582 + 2.96093i 0.820033 + 0.106223i
\(778\) 0 0
\(779\) −2.14871 −0.0769854
\(780\) 0 0
\(781\) 19.2071i 0.687283i
\(782\) 0 0
\(783\) 7.03122 0.251275
\(784\) 0 0
\(785\) 12.1525 0.433742
\(786\) 0 0
\(787\) 4.00000i 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 0 0
\(789\) −24.9737 −0.889087
\(790\) 0 0
\(791\) 4.64147 + 0.601231i 0.165032 + 0.0213773i
\(792\) 0 0
\(793\) −24.8189 −0.881346
\(794\) 0 0
\(795\) −24.9579 −0.885166
\(796\) 0 0
\(797\) −44.8604 −1.58904 −0.794519 0.607239i \(-0.792277\pi\)
−0.794519 + 0.607239i \(0.792277\pi\)
\(798\) 0 0
\(799\) 37.2543i 1.31796i
\(800\) 0 0
\(801\) 10.6954i 0.377903i
\(802\) 0 0
\(803\) 19.9215i 0.703014i
\(804\) 0 0
\(805\) −6.87153 + 53.0478i −0.242189 + 1.86969i
\(806\) 0 0
\(807\) 2.83453i 0.0997802i
\(808\) 0 0
\(809\) 25.2284 0.886983 0.443491 0.896279i \(-0.353740\pi\)
0.443491 + 0.896279i \(0.353740\pi\)
\(810\) 0 0
\(811\) 0.0785226i 0.00275730i −0.999999 0.00137865i \(-0.999561\pi\)
0.999999 0.00137865i \(-0.000438839\pi\)
\(812\) 0 0
\(813\) 3.86426i 0.135526i
\(814\) 0 0
\(815\) −47.2969 −1.65674
\(816\) 0 0
\(817\) 5.57982i 0.195213i
\(818\) 0 0
\(819\) −5.24766 0.679754i −0.183368 0.0237525i
\(820\) 0 0
\(821\) 26.4093i 0.921692i −0.887480 0.460846i \(-0.847546\pi\)
0.887480 0.460846i \(-0.152454\pi\)
\(822\) 0 0
\(823\) 6.54863i 0.228271i −0.993465 0.114135i \(-0.963590\pi\)
0.993465 0.114135i \(-0.0364098\pi\)
\(824\) 0 0
\(825\) 14.9367i 0.520029i
\(826\) 0 0
\(827\) −45.0043 −1.56495 −0.782476 0.622681i \(-0.786043\pi\)
−0.782476 + 0.622681i \(0.786043\pi\)
\(828\) 0 0
\(829\) −14.4354 −0.501361 −0.250681 0.968070i \(-0.580654\pi\)
−0.250681 + 0.968070i \(0.580654\pi\)
\(830\) 0 0
\(831\) −1.78356 −0.0618710
\(832\) 0 0
\(833\) −6.71902 + 25.5000i −0.232800 + 0.883524i
\(834\) 0 0
\(835\) −13.8564 −0.479521
\(836\) 0 0
\(837\) 6.42503i 0.222081i
\(838\) 0 0
\(839\) −44.5420 −1.53776 −0.768881 0.639392i \(-0.779186\pi\)
−0.768881 + 0.639392i \(0.779186\pi\)
\(840\) 0 0
\(841\) −20.4380 −0.704760
\(842\) 0 0
\(843\) 20.4880i 0.705644i
\(844\) 0 0
\(845\) 34.9604 1.20267
\(846\) 0 0
\(847\) 23.1113 + 2.99371i 0.794113 + 0.102865i
\(848\) 0 0
\(849\) −12.1392 −0.416615
\(850\) 0 0
\(851\) 45.3424 1.55432
\(852\) 0 0
\(853\) 36.4095 1.24664 0.623318 0.781968i \(-0.285784\pi\)
0.623318 + 0.781968i \(0.285784\pi\)
\(854\) 0 0
\(855\) 2.64049i 0.0903029i
\(856\) 0 0
\(857\) 50.4579i 1.72361i −0.507239 0.861805i \(-0.669334\pi\)
0.507239 0.861805i \(-0.330666\pi\)
\(858\) 0 0
\(859\) 13.0366i 0.444803i 0.974955 + 0.222402i \(0.0713896\pi\)
−0.974955 + 0.222402i \(0.928610\pi\)
\(860\) 0 0
\(861\) 8.29394 + 1.07435i 0.282657 + 0.0366139i
\(862\) 0 0
\(863\) 53.4091i 1.81807i −0.416724 0.909033i \(-0.636822\pi\)
0.416724 0.909033i \(-0.363178\pi\)
\(864\) 0 0
\(865\) 19.1677 0.651723
\(866\) 0 0
\(867\) 2.80823i 0.0953724i
\(868\) 0 0
\(869\) 6.58617i 0.223420i
\(870\) 0 0
\(871\) −17.6296 −0.597355
\(872\) 0 0
\(873\) 9.88913i 0.334696i
\(874\) 0 0
\(875\) 51.8704 + 6.71902i 1.75354 + 0.227144i
\(876\) 0 0
\(877\) 11.7077i 0.395341i 0.980269 + 0.197670i \(0.0633376\pi\)
−0.980269 + 0.197670i \(0.936662\pi\)
\(878\) 0 0
\(879\) 4.93444i 0.166434i
\(880\) 0 0
\(881\) 31.0798i 1.04711i −0.851993 0.523553i \(-0.824606\pi\)
0.851993 0.523553i \(-0.175394\pi\)
\(882\) 0 0
\(883\) −23.4834 −0.790279 −0.395140 0.918621i \(-0.629304\pi\)
−0.395140 + 0.918621i \(0.629304\pi\)
\(884\) 0 0
\(885\) −30.1784 −1.01444
\(886\) 0 0
\(887\) 8.27659 0.277901 0.138950 0.990299i \(-0.455627\pi\)
0.138950 + 0.990299i \(0.455627\pi\)
\(888\) 0 0
\(889\) 3.76897 29.0962i 0.126407 0.975856i
\(890\) 0 0
\(891\) 1.48046 0.0495974
\(892\) 0 0
\(893\) 6.72217i 0.224949i
\(894\) 0 0
\(895\) −44.1650 −1.47627
\(896\) 0 0
\(897\) −10.4095 −0.347562
\(898\) 0 0
\(899\) 45.1758i 1.50670i
\(900\) 0 0
\(901\) −24.2043 −0.806364
\(902\) 0 0
\(903\) 2.78991 21.5379i 0.0928424 0.716738i
\(904\) 0 0
\(905\) −7.76897 −0.258249
\(906\) 0 0
\(907\) 31.7600 1.05457 0.527287 0.849687i \(-0.323209\pi\)
0.527287 + 0.849687i \(0.323209\pi\)
\(908\) 0 0
\(909\) 3.88448 0.128840
\(910\) 0 0
\(911\) 48.9737i 1.62257i 0.584650 + 0.811285i \(0.301232\pi\)
−0.584650 + 0.811285i \(0.698768\pi\)
\(912\) 0 0
\(913\) 15.4108i 0.510024i
\(914\) 0 0
\(915\) 48.2043i 1.59359i
\(916\) 0 0
\(917\) −0.540588 + 4.17331i −0.0178518 + 0.137815i
\(918\) 0 0
\(919\) 21.0366i 0.693934i 0.937877 + 0.346967i \(0.112788\pi\)
−0.937877 + 0.346967i \(0.887212\pi\)
\(920\) 0 0
\(921\) 21.4987 0.708405
\(922\) 0 0
\(923\) 25.9474i 0.854069i
\(924\) 0 0
\(925\) 87.8948i 2.88996i
\(926\) 0 0
\(927\) 3.06394 0.100633
\(928\) 0 0
\(929\) 16.6173i 0.545194i −0.962128 0.272597i \(-0.912117\pi\)
0.962128 0.272597i \(-0.0878826\pi\)
\(930\) 0 0
\(931\) 1.21238 4.60123i 0.0397342 0.150799i
\(932\) 0 0
\(933\) 6.52804i 0.213719i
\(934\) 0 0
\(935\) 21.6645i 0.708506i
\(936\) 0 0
\(937\) 37.2018i 1.21533i 0.794194 + 0.607665i \(0.207894\pi\)
−0.794194 + 0.607665i \(0.792106\pi\)
\(938\) 0 0
\(939\) 33.6347 1.09763
\(940\) 0 0
\(941\) 47.6535 1.55346 0.776729 0.629835i \(-0.216877\pi\)
0.776729 + 0.629835i \(0.216877\pi\)
\(942\) 0 0
\(943\) 16.4522 0.535757
\(944\) 0 0
\(945\) −1.32025 + 10.1922i −0.0429476 + 0.331553i
\(946\) 0 0
\(947\) −14.3305 −0.465679 −0.232840 0.972515i \(-0.574802\pi\)
−0.232840 + 0.972515i \(0.574802\pi\)
\(948\) 0 0
\(949\) 26.9125i 0.873616i
\(950\) 0 0
\(951\) −4.07029 −0.131988
\(952\) 0 0
\(953\) −3.71636 −0.120385 −0.0601924 0.998187i \(-0.519171\pi\)
−0.0601924 + 0.998187i \(0.519171\pi\)
\(954\) 0 0
\(955\) 86.7530i 2.80726i
\(956\) 0 0
\(957\) −10.4095 −0.336490
\(958\) 0 0
\(959\) −49.5837 6.42280i −1.60114 0.207403i
\(960\) 0 0
\(961\) 10.2810 0.331645
\(962\) 0 0
\(963\) 2.08665 0.0672415
\(964\) 0 0
\(965\) 80.1258 2.57934
\(966\) 0 0
\(967\) 26.1651i 0.841412i 0.907197 + 0.420706i \(0.138218\pi\)
−0.907197 + 0.420706i \(0.861782\pi\)
\(968\) 0 0
\(969\) 2.56077i 0.0822637i
\(970\) 0 0
\(971\) 25.7949i 0.827797i −0.910323 0.413899i \(-0.864167\pi\)
0.910323 0.413899i \(-0.135833\pi\)
\(972\) 0 0
\(973\) −3.45941 + 26.7065i −0.110904 + 0.856170i
\(974\) 0 0
\(975\) 20.1784i 0.646227i
\(976\) 0 0
\(977\) −6.69310 −0.214131 −0.107066 0.994252i \(-0.534145\pi\)
−0.107066 + 0.994252i \(0.534145\pi\)
\(978\) 0 0
\(979\) 15.8341i 0.506062i
\(980\) 0 0
\(981\) 17.4235i 0.556290i
\(982\) 0 0
\(983\) 52.6826 1.68031 0.840157 0.542343i \(-0.182462\pi\)
0.840157 + 0.542343i \(0.182462\pi\)
\(984\) 0 0
\(985\) 90.2846i 2.87671i
\(986\) 0 0
\(987\) −3.36109 + 25.9474i −0.106985 + 0.825915i
\(988\) 0 0
\(989\) 42.7235i 1.35853i
\(990\) 0 0
\(991\) 18.9367i 0.601544i −0.953696 0.300772i \(-0.902756\pi\)
0.953696 0.300772i \(-0.0972444\pi\)
\(992\) 0 0
\(993\) 15.1368i 0.480351i
\(994\) 0 0
\(995\) −101.922 −3.23115
\(996\) 0 0
\(997\) −1.33359 −0.0422352 −0.0211176 0.999777i \(-0.506722\pi\)
−0.0211176 + 0.999777i \(0.506722\pi\)
\(998\) 0 0
\(999\) 8.71176 0.275628
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 1344.2.p.c.223.1 12
3.2 odd 2 4032.2.p.j.1567.9 12
4.3 odd 2 inner 1344.2.p.c.223.8 yes 12
7.6 odd 2 1344.2.p.d.223.12 yes 12
8.3 odd 2 1344.2.p.d.223.6 yes 12
8.5 even 2 1344.2.p.d.223.11 yes 12
12.11 even 2 4032.2.p.j.1567.12 12
21.20 even 2 4032.2.p.k.1567.3 12
24.5 odd 2 4032.2.p.k.1567.1 12
24.11 even 2 4032.2.p.k.1567.4 12
28.27 even 2 1344.2.p.d.223.5 yes 12
56.13 odd 2 inner 1344.2.p.c.223.2 yes 12
56.27 even 2 inner 1344.2.p.c.223.7 yes 12
84.83 odd 2 4032.2.p.k.1567.2 12
168.83 odd 2 4032.2.p.j.1567.10 12
168.125 even 2 4032.2.p.j.1567.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.2.p.c.223.1 12 1.1 even 1 trivial
1344.2.p.c.223.2 yes 12 56.13 odd 2 inner
1344.2.p.c.223.7 yes 12 56.27 even 2 inner
1344.2.p.c.223.8 yes 12 4.3 odd 2 inner
1344.2.p.d.223.5 yes 12 28.27 even 2
1344.2.p.d.223.6 yes 12 8.3 odd 2
1344.2.p.d.223.11 yes 12 8.5 even 2
1344.2.p.d.223.12 yes 12 7.6 odd 2
4032.2.p.j.1567.9 12 3.2 odd 2
4032.2.p.j.1567.10 12 168.83 odd 2
4032.2.p.j.1567.11 12 168.125 even 2
4032.2.p.j.1567.12 12 12.11 even 2
4032.2.p.k.1567.1 12 24.5 odd 2
4032.2.p.k.1567.2 12 84.83 odd 2
4032.2.p.k.1567.3 12 21.20 even 2
4032.2.p.k.1567.4 12 24.11 even 2