Properties

Label 4012.2.b.b.237.27
Level $4012$
Weight $2$
Character 4012.237
Analytic conductor $32.036$
Analytic rank $0$
Dimension $46$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4012,2,Mod(237,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4012.237");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.b (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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.27
Character \(\chi\) \(=\) 4012.237
Dual form 4012.2.b.b.237.20

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.405081i q^{3} -1.20499i q^{5} +0.122460i q^{7} +2.83591 q^{9} +O(q^{10})\) \(q+0.405081i q^{3} -1.20499i q^{5} +0.122460i q^{7} +2.83591 q^{9} -1.65371i q^{11} -1.21607 q^{13} +0.488118 q^{15} +(-4.09888 - 0.446336i) q^{17} +5.49834 q^{19} -0.0496062 q^{21} +1.29315i q^{23} +3.54801 q^{25} +2.36402i q^{27} +9.31434i q^{29} +6.06362i q^{31} +0.669888 q^{33} +0.147562 q^{35} -4.07227i q^{37} -0.492607i q^{39} -1.98151i q^{41} +7.44725 q^{43} -3.41723i q^{45} -10.1355 q^{47} +6.98500 q^{49} +(0.180802 - 1.66038i) q^{51} +4.71660 q^{53} -1.99270 q^{55} +2.22728i q^{57} +1.00000 q^{59} -3.86956i q^{61} +0.347285i q^{63} +1.46535i q^{65} +2.82254 q^{67} -0.523832 q^{69} -12.5130i q^{71} -14.7913i q^{73} +1.43723i q^{75} +0.202513 q^{77} +0.917389i q^{79} +7.55011 q^{81} +12.7272 q^{83} +(-0.537829 + 4.93909i) q^{85} -3.77307 q^{87} -2.24107 q^{89} -0.148920i q^{91} -2.45626 q^{93} -6.62543i q^{95} +7.92714i q^{97} -4.68978i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 46 q - 54 q^{9} + 8 q^{13} - 10 q^{15} + q^{17} - 20 q^{19} - 24 q^{21} - 54 q^{25} + 2 q^{33} + 26 q^{35} - 38 q^{43} + 6 q^{47} - 66 q^{49} + 26 q^{51} + 18 q^{53} - 20 q^{55} + 46 q^{59} + 48 q^{67} + 28 q^{69} + 22 q^{77} + 70 q^{81} - 52 q^{83} - 2 q^{85} + 44 q^{87} - 76 q^{89} - 26 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0.405081i 0.233874i 0.993139 + 0.116937i \(0.0373075\pi\)
−0.993139 + 0.116937i \(0.962692\pi\)
\(4\) 0 0
\(5\) 1.20499i 0.538887i −0.963016 0.269443i \(-0.913160\pi\)
0.963016 0.269443i \(-0.0868397\pi\)
\(6\) 0 0
\(7\) 0.122460i 0.0462854i 0.999732 + 0.0231427i \(0.00736722\pi\)
−0.999732 + 0.0231427i \(0.992633\pi\)
\(8\) 0 0
\(9\) 2.83591 0.945303
\(10\) 0 0
\(11\) 1.65371i 0.498613i −0.968425 0.249306i \(-0.919797\pi\)
0.968425 0.249306i \(-0.0802026\pi\)
\(12\) 0 0
\(13\) −1.21607 −0.337277 −0.168638 0.985678i \(-0.553937\pi\)
−0.168638 + 0.985678i \(0.553937\pi\)
\(14\) 0 0
\(15\) 0.488118 0.126032
\(16\) 0 0
\(17\) −4.09888 0.446336i −0.994123 0.108252i
\(18\) 0 0
\(19\) 5.49834 1.26141 0.630703 0.776025i \(-0.282767\pi\)
0.630703 + 0.776025i \(0.282767\pi\)
\(20\) 0 0
\(21\) −0.0496062 −0.0108250
\(22\) 0 0
\(23\) 1.29315i 0.269641i 0.990870 + 0.134820i \(0.0430458\pi\)
−0.990870 + 0.134820i \(0.956954\pi\)
\(24\) 0 0
\(25\) 3.54801 0.709601
\(26\) 0 0
\(27\) 2.36402i 0.454956i
\(28\) 0 0
\(29\) 9.31434i 1.72963i 0.502091 + 0.864815i \(0.332564\pi\)
−0.502091 + 0.864815i \(0.667436\pi\)
\(30\) 0 0
\(31\) 6.06362i 1.08906i 0.838742 + 0.544530i \(0.183292\pi\)
−0.838742 + 0.544530i \(0.816708\pi\)
\(32\) 0 0
\(33\) 0.669888 0.116613
\(34\) 0 0
\(35\) 0.147562 0.0249426
\(36\) 0 0
\(37\) 4.07227i 0.669478i −0.942311 0.334739i \(-0.891352\pi\)
0.942311 0.334739i \(-0.108648\pi\)
\(38\) 0 0
\(39\) 0.492607i 0.0788803i
\(40\) 0 0
\(41\) 1.98151i 0.309460i −0.987957 0.154730i \(-0.950549\pi\)
0.987957 0.154730i \(-0.0494508\pi\)
\(42\) 0 0
\(43\) 7.44725 1.13570 0.567848 0.823134i \(-0.307776\pi\)
0.567848 + 0.823134i \(0.307776\pi\)
\(44\) 0 0
\(45\) 3.41723i 0.509411i
\(46\) 0 0
\(47\) −10.1355 −1.47841 −0.739207 0.673478i \(-0.764800\pi\)
−0.739207 + 0.673478i \(0.764800\pi\)
\(48\) 0 0
\(49\) 6.98500 0.997858
\(50\) 0 0
\(51\) 0.180802 1.66038i 0.0253174 0.232500i
\(52\) 0 0
\(53\) 4.71660 0.647874 0.323937 0.946079i \(-0.394993\pi\)
0.323937 + 0.946079i \(0.394993\pi\)
\(54\) 0 0
\(55\) −1.99270 −0.268696
\(56\) 0 0
\(57\) 2.22728i 0.295010i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 3.86956i 0.495447i −0.968831 0.247723i \(-0.920318\pi\)
0.968831 0.247723i \(-0.0796824\pi\)
\(62\) 0 0
\(63\) 0.347285i 0.0437538i
\(64\) 0 0
\(65\) 1.46535i 0.181754i
\(66\) 0 0
\(67\) 2.82254 0.344828 0.172414 0.985025i \(-0.444843\pi\)
0.172414 + 0.985025i \(0.444843\pi\)
\(68\) 0 0
\(69\) −0.523832 −0.0630620
\(70\) 0 0
\(71\) 12.5130i 1.48502i −0.669834 0.742511i \(-0.733635\pi\)
0.669834 0.742511i \(-0.266365\pi\)
\(72\) 0 0
\(73\) 14.7913i 1.73119i −0.500740 0.865597i \(-0.666939\pi\)
0.500740 0.865597i \(-0.333061\pi\)
\(74\) 0 0
\(75\) 1.43723i 0.165957i
\(76\) 0 0
\(77\) 0.202513 0.0230785
\(78\) 0 0
\(79\) 0.917389i 0.103214i 0.998667 + 0.0516072i \(0.0164344\pi\)
−0.998667 + 0.0516072i \(0.983566\pi\)
\(80\) 0 0
\(81\) 7.55011 0.838901
\(82\) 0 0
\(83\) 12.7272 1.39699 0.698494 0.715616i \(-0.253854\pi\)
0.698494 + 0.715616i \(0.253854\pi\)
\(84\) 0 0
\(85\) −0.537829 + 4.93909i −0.0583357 + 0.535720i
\(86\) 0 0
\(87\) −3.77307 −0.404515
\(88\) 0 0
\(89\) −2.24107 −0.237553 −0.118776 0.992921i \(-0.537897\pi\)
−0.118776 + 0.992921i \(0.537897\pi\)
\(90\) 0 0
\(91\) 0.148920i 0.0156110i
\(92\) 0 0
\(93\) −2.45626 −0.254702
\(94\) 0 0
\(95\) 6.62543i 0.679755i
\(96\) 0 0
\(97\) 7.92714i 0.804879i 0.915446 + 0.402440i \(0.131838\pi\)
−0.915446 + 0.402440i \(0.868162\pi\)
\(98\) 0 0
\(99\) 4.68978i 0.471340i
\(100\) 0 0
\(101\) −4.46389 −0.444174 −0.222087 0.975027i \(-0.571287\pi\)
−0.222087 + 0.975027i \(0.571287\pi\)
\(102\) 0 0
\(103\) 8.70518 0.857746 0.428873 0.903365i \(-0.358911\pi\)
0.428873 + 0.903365i \(0.358911\pi\)
\(104\) 0 0
\(105\) 0.0597748i 0.00583343i
\(106\) 0 0
\(107\) 0.613356i 0.0592953i −0.999560 0.0296477i \(-0.990561\pi\)
0.999560 0.0296477i \(-0.00943853\pi\)
\(108\) 0 0
\(109\) 9.44960i 0.905108i −0.891737 0.452554i \(-0.850513\pi\)
0.891737 0.452554i \(-0.149487\pi\)
\(110\) 0 0
\(111\) 1.64960 0.156573
\(112\) 0 0
\(113\) 15.9263i 1.49822i 0.662447 + 0.749109i \(0.269518\pi\)
−0.662447 + 0.749109i \(0.730482\pi\)
\(114\) 0 0
\(115\) 1.55823 0.145306
\(116\) 0 0
\(117\) −3.44866 −0.318829
\(118\) 0 0
\(119\) 0.0546582 0.501947i 0.00501051 0.0460134i
\(120\) 0 0
\(121\) 8.26524 0.751385
\(122\) 0 0
\(123\) 0.802673 0.0723746
\(124\) 0 0
\(125\) 10.3002i 0.921281i
\(126\) 0 0
\(127\) 0.0633634 0.00562259 0.00281130 0.999996i \(-0.499105\pi\)
0.00281130 + 0.999996i \(0.499105\pi\)
\(128\) 0 0
\(129\) 3.01675i 0.265610i
\(130\) 0 0
\(131\) 7.72225i 0.674697i 0.941380 + 0.337348i \(0.109530\pi\)
−0.941380 + 0.337348i \(0.890470\pi\)
\(132\) 0 0
\(133\) 0.673325i 0.0583847i
\(134\) 0 0
\(135\) 2.84861 0.245170
\(136\) 0 0
\(137\) −5.92432 −0.506149 −0.253074 0.967447i \(-0.581442\pi\)
−0.253074 + 0.967447i \(0.581442\pi\)
\(138\) 0 0
\(139\) 16.4843i 1.39818i 0.715034 + 0.699090i \(0.246411\pi\)
−0.715034 + 0.699090i \(0.753589\pi\)
\(140\) 0 0
\(141\) 4.10570i 0.345763i
\(142\) 0 0
\(143\) 2.01103i 0.168171i
\(144\) 0 0
\(145\) 11.2237 0.932074
\(146\) 0 0
\(147\) 2.82950i 0.233373i
\(148\) 0 0
\(149\) −0.397789 −0.0325881 −0.0162941 0.999867i \(-0.505187\pi\)
−0.0162941 + 0.999867i \(0.505187\pi\)
\(150\) 0 0
\(151\) 19.2318 1.56506 0.782532 0.622611i \(-0.213928\pi\)
0.782532 + 0.622611i \(0.213928\pi\)
\(152\) 0 0
\(153\) −11.6240 1.26577i −0.939748 0.102331i
\(154\) 0 0
\(155\) 7.30659 0.586879
\(156\) 0 0
\(157\) −7.26619 −0.579905 −0.289953 0.957041i \(-0.593640\pi\)
−0.289953 + 0.957041i \(0.593640\pi\)
\(158\) 0 0
\(159\) 1.91061i 0.151521i
\(160\) 0 0
\(161\) −0.158359 −0.0124805
\(162\) 0 0
\(163\) 10.3754i 0.812662i −0.913726 0.406331i \(-0.866808\pi\)
0.913726 0.406331i \(-0.133192\pi\)
\(164\) 0 0
\(165\) 0.807207i 0.0628409i
\(166\) 0 0
\(167\) 14.1828i 1.09750i 0.835987 + 0.548750i \(0.184896\pi\)
−0.835987 + 0.548750i \(0.815104\pi\)
\(168\) 0 0
\(169\) −11.5212 −0.886244
\(170\) 0 0
\(171\) 15.5928 1.19241
\(172\) 0 0
\(173\) 6.69164i 0.508756i 0.967105 + 0.254378i \(0.0818708\pi\)
−0.967105 + 0.254378i \(0.918129\pi\)
\(174\) 0 0
\(175\) 0.434488i 0.0328442i
\(176\) 0 0
\(177\) 0.405081i 0.0304478i
\(178\) 0 0
\(179\) 11.8396 0.884931 0.442465 0.896786i \(-0.354104\pi\)
0.442465 + 0.896786i \(0.354104\pi\)
\(180\) 0 0
\(181\) 15.6430i 1.16274i −0.813641 0.581368i \(-0.802518\pi\)
0.813641 0.581368i \(-0.197482\pi\)
\(182\) 0 0
\(183\) 1.56749 0.115872
\(184\) 0 0
\(185\) −4.90704 −0.360773
\(186\) 0 0
\(187\) −0.738111 + 6.77836i −0.0539760 + 0.495683i
\(188\) 0 0
\(189\) −0.289497 −0.0210578
\(190\) 0 0
\(191\) 8.26989 0.598388 0.299194 0.954192i \(-0.403282\pi\)
0.299194 + 0.954192i \(0.403282\pi\)
\(192\) 0 0
\(193\) 16.7443i 1.20528i −0.798012 0.602642i \(-0.794115\pi\)
0.798012 0.602642i \(-0.205885\pi\)
\(194\) 0 0
\(195\) −0.593585 −0.0425075
\(196\) 0 0
\(197\) 8.46212i 0.602901i 0.953482 + 0.301450i \(0.0974708\pi\)
−0.953482 + 0.301450i \(0.902529\pi\)
\(198\) 0 0
\(199\) 27.1755i 1.92642i −0.268756 0.963208i \(-0.586612\pi\)
0.268756 0.963208i \(-0.413388\pi\)
\(200\) 0 0
\(201\) 1.14336i 0.0806463i
\(202\) 0 0
\(203\) −1.14063 −0.0800567
\(204\) 0 0
\(205\) −2.38770 −0.166764
\(206\) 0 0
\(207\) 3.66726i 0.254892i
\(208\) 0 0
\(209\) 9.09267i 0.628953i
\(210\) 0 0
\(211\) 8.91458i 0.613705i −0.951757 0.306853i \(-0.900724\pi\)
0.951757 0.306853i \(-0.0992759\pi\)
\(212\) 0 0
\(213\) 5.06879 0.347308
\(214\) 0 0
\(215\) 8.97385i 0.612011i
\(216\) 0 0
\(217\) −0.742550 −0.0504076
\(218\) 0 0
\(219\) 5.99170 0.404881
\(220\) 0 0
\(221\) 4.98452 + 0.542775i 0.335295 + 0.0365110i
\(222\) 0 0
\(223\) 11.9201 0.798232 0.399116 0.916900i \(-0.369317\pi\)
0.399116 + 0.916900i \(0.369317\pi\)
\(224\) 0 0
\(225\) 10.0618 0.670788
\(226\) 0 0
\(227\) 13.2247i 0.877751i −0.898548 0.438876i \(-0.855377\pi\)
0.898548 0.438876i \(-0.144623\pi\)
\(228\) 0 0
\(229\) −8.47499 −0.560043 −0.280022 0.959994i \(-0.590342\pi\)
−0.280022 + 0.959994i \(0.590342\pi\)
\(230\) 0 0
\(231\) 0.0820343i 0.00539746i
\(232\) 0 0
\(233\) 4.37200i 0.286419i −0.989692 0.143210i \(-0.954258\pi\)
0.989692 0.143210i \(-0.0457423\pi\)
\(234\) 0 0
\(235\) 12.2132i 0.796698i
\(236\) 0 0
\(237\) −0.371617 −0.0241391
\(238\) 0 0
\(239\) 29.9809 1.93930 0.969651 0.244491i \(-0.0786210\pi\)
0.969651 + 0.244491i \(0.0786210\pi\)
\(240\) 0 0
\(241\) 20.5142i 1.32144i −0.750634 0.660718i \(-0.770252\pi\)
0.750634 0.660718i \(-0.229748\pi\)
\(242\) 0 0
\(243\) 10.1505i 0.651153i
\(244\) 0 0
\(245\) 8.41684i 0.537732i
\(246\) 0 0
\(247\) −6.68636 −0.425443
\(248\) 0 0
\(249\) 5.15554i 0.326719i
\(250\) 0 0
\(251\) 13.4172 0.846888 0.423444 0.905922i \(-0.360821\pi\)
0.423444 + 0.905922i \(0.360821\pi\)
\(252\) 0 0
\(253\) 2.13850 0.134446
\(254\) 0 0
\(255\) −2.00074 0.217865i −0.125291 0.0136432i
\(256\) 0 0
\(257\) 15.9305 0.993717 0.496859 0.867831i \(-0.334487\pi\)
0.496859 + 0.867831i \(0.334487\pi\)
\(258\) 0 0
\(259\) 0.498690 0.0309871
\(260\) 0 0
\(261\) 26.4146i 1.63502i
\(262\) 0 0
\(263\) 3.30320 0.203684 0.101842 0.994801i \(-0.467526\pi\)
0.101842 + 0.994801i \(0.467526\pi\)
\(264\) 0 0
\(265\) 5.68344i 0.349131i
\(266\) 0 0
\(267\) 0.907815i 0.0555574i
\(268\) 0 0
\(269\) 18.0068i 1.09790i 0.835856 + 0.548948i \(0.184972\pi\)
−0.835856 + 0.548948i \(0.815028\pi\)
\(270\) 0 0
\(271\) 8.49255 0.515886 0.257943 0.966160i \(-0.416955\pi\)
0.257943 + 0.966160i \(0.416955\pi\)
\(272\) 0 0
\(273\) 0.0603246 0.00365101
\(274\) 0 0
\(275\) 5.86738i 0.353816i
\(276\) 0 0
\(277\) 13.1893i 0.792467i 0.918150 + 0.396234i \(0.129683\pi\)
−0.918150 + 0.396234i \(0.870317\pi\)
\(278\) 0 0
\(279\) 17.1959i 1.02949i
\(280\) 0 0
\(281\) 9.65973 0.576251 0.288126 0.957593i \(-0.406968\pi\)
0.288126 + 0.957593i \(0.406968\pi\)
\(282\) 0 0
\(283\) 11.4726i 0.681975i 0.940068 + 0.340987i \(0.110761\pi\)
−0.940068 + 0.340987i \(0.889239\pi\)
\(284\) 0 0
\(285\) 2.68384 0.158977
\(286\) 0 0
\(287\) 0.242655 0.0143235
\(288\) 0 0
\(289\) 16.6016 + 3.65895i 0.976563 + 0.215232i
\(290\) 0 0
\(291\) −3.21114 −0.188240
\(292\) 0 0
\(293\) −2.06902 −0.120873 −0.0604366 0.998172i \(-0.519249\pi\)
−0.0604366 + 0.998172i \(0.519249\pi\)
\(294\) 0 0
\(295\) 1.20499i 0.0701571i
\(296\) 0 0
\(297\) 3.90941 0.226847
\(298\) 0 0
\(299\) 1.57256i 0.0909437i
\(300\) 0 0
\(301\) 0.911989i 0.0525662i
\(302\) 0 0
\(303\) 1.80824i 0.103881i
\(304\) 0 0
\(305\) −4.66278 −0.266990
\(306\) 0 0
\(307\) −19.5503 −1.11580 −0.557898 0.829909i \(-0.688392\pi\)
−0.557898 + 0.829909i \(0.688392\pi\)
\(308\) 0 0
\(309\) 3.52631i 0.200605i
\(310\) 0 0
\(311\) 23.0757i 1.30850i 0.756277 + 0.654252i \(0.227016\pi\)
−0.756277 + 0.654252i \(0.772984\pi\)
\(312\) 0 0
\(313\) 14.2773i 0.807000i 0.914980 + 0.403500i \(0.132207\pi\)
−0.914980 + 0.403500i \(0.867793\pi\)
\(314\) 0 0
\(315\) 0.418474 0.0235783
\(316\) 0 0
\(317\) 17.8592i 1.00307i −0.865137 0.501536i \(-0.832769\pi\)
0.865137 0.501536i \(-0.167231\pi\)
\(318\) 0 0
\(319\) 15.4032 0.862415
\(320\) 0 0
\(321\) 0.248459 0.0138676
\(322\) 0 0
\(323\) −22.5370 2.45411i −1.25399 0.136550i
\(324\) 0 0
\(325\) −4.31462 −0.239332
\(326\) 0 0
\(327\) 3.82786 0.211681
\(328\) 0 0
\(329\) 1.24119i 0.0684291i
\(330\) 0 0
\(331\) 22.6188 1.24324 0.621621 0.783318i \(-0.286474\pi\)
0.621621 + 0.783318i \(0.286474\pi\)
\(332\) 0 0
\(333\) 11.5486i 0.632859i
\(334\) 0 0
\(335\) 3.40112i 0.185823i
\(336\) 0 0
\(337\) 22.3964i 1.22001i −0.792398 0.610004i \(-0.791168\pi\)
0.792398 0.610004i \(-0.208832\pi\)
\(338\) 0 0
\(339\) −6.45144 −0.350394
\(340\) 0 0
\(341\) 10.0275 0.543019
\(342\) 0 0
\(343\) 1.71260i 0.0924717i
\(344\) 0 0
\(345\) 0.631211i 0.0339833i
\(346\) 0 0
\(347\) 17.9296i 0.962511i 0.876580 + 0.481256i \(0.159819\pi\)
−0.876580 + 0.481256i \(0.840181\pi\)
\(348\) 0 0
\(349\) −15.7956 −0.845517 −0.422758 0.906242i \(-0.638938\pi\)
−0.422758 + 0.906242i \(0.638938\pi\)
\(350\) 0 0
\(351\) 2.87481i 0.153446i
\(352\) 0 0
\(353\) −14.9174 −0.793971 −0.396986 0.917825i \(-0.629944\pi\)
−0.396986 + 0.917825i \(0.629944\pi\)
\(354\) 0 0
\(355\) −15.0780 −0.800259
\(356\) 0 0
\(357\) 0.203330 + 0.0221410i 0.0107613 + 0.00117183i
\(358\) 0 0
\(359\) 8.98954 0.474450 0.237225 0.971455i \(-0.423762\pi\)
0.237225 + 0.971455i \(0.423762\pi\)
\(360\) 0 0
\(361\) 11.2317 0.591144
\(362\) 0 0
\(363\) 3.34809i 0.175729i
\(364\) 0 0
\(365\) −17.8234 −0.932918
\(366\) 0 0
\(367\) 4.92123i 0.256886i 0.991717 + 0.128443i \(0.0409979\pi\)
−0.991717 + 0.128443i \(0.959002\pi\)
\(368\) 0 0
\(369\) 5.61939i 0.292534i
\(370\) 0 0
\(371\) 0.577593i 0.0299872i
\(372\) 0 0
\(373\) 2.22961 0.115445 0.0577225 0.998333i \(-0.481616\pi\)
0.0577225 + 0.998333i \(0.481616\pi\)
\(374\) 0 0
\(375\) 4.17244 0.215464
\(376\) 0 0
\(377\) 11.3269i 0.583364i
\(378\) 0 0
\(379\) 27.0371i 1.38880i −0.719588 0.694401i \(-0.755669\pi\)
0.719588 0.694401i \(-0.244331\pi\)
\(380\) 0 0
\(381\) 0.0256673i 0.00131498i
\(382\) 0 0
\(383\) −9.31266 −0.475855 −0.237927 0.971283i \(-0.576468\pi\)
−0.237927 + 0.971283i \(0.576468\pi\)
\(384\) 0 0
\(385\) 0.244026i 0.0124367i
\(386\) 0 0
\(387\) 21.1197 1.07358
\(388\) 0 0
\(389\) −12.7814 −0.648045 −0.324023 0.946049i \(-0.605035\pi\)
−0.324023 + 0.946049i \(0.605035\pi\)
\(390\) 0 0
\(391\) 0.577180 5.30047i 0.0291893 0.268056i
\(392\) 0 0
\(393\) −3.12814 −0.157794
\(394\) 0 0
\(395\) 1.10544 0.0556208
\(396\) 0 0
\(397\) 20.4352i 1.02561i 0.858505 + 0.512806i \(0.171394\pi\)
−0.858505 + 0.512806i \(0.828606\pi\)
\(398\) 0 0
\(399\) −0.272752 −0.0136547
\(400\) 0 0
\(401\) 22.5597i 1.12658i 0.826259 + 0.563290i \(0.190465\pi\)
−0.826259 + 0.563290i \(0.809535\pi\)
\(402\) 0 0
\(403\) 7.37379i 0.367315i
\(404\) 0 0
\(405\) 9.09778i 0.452072i
\(406\) 0 0
\(407\) −6.73437 −0.333810
\(408\) 0 0
\(409\) 16.2860 0.805291 0.402646 0.915356i \(-0.368091\pi\)
0.402646 + 0.915356i \(0.368091\pi\)
\(410\) 0 0
\(411\) 2.39983i 0.118375i
\(412\) 0 0
\(413\) 0.122460i 0.00602585i
\(414\) 0 0
\(415\) 15.3361i 0.752818i
\(416\) 0 0
\(417\) −6.67748 −0.326998
\(418\) 0 0
\(419\) 3.56522i 0.174172i 0.996201 + 0.0870862i \(0.0277556\pi\)
−0.996201 + 0.0870862i \(0.972244\pi\)
\(420\) 0 0
\(421\) −36.8154 −1.79427 −0.897136 0.441755i \(-0.854356\pi\)
−0.897136 + 0.441755i \(0.854356\pi\)
\(422\) 0 0
\(423\) −28.7434 −1.39755
\(424\) 0 0
\(425\) −14.5428 1.58360i −0.705431 0.0768160i
\(426\) 0 0
\(427\) 0.473866 0.0229320
\(428\) 0 0
\(429\) −0.814630 −0.0393307
\(430\) 0 0
\(431\) 7.27173i 0.350267i −0.984545 0.175134i \(-0.943964\pi\)
0.984545 0.175134i \(-0.0560357\pi\)
\(432\) 0 0
\(433\) −13.9835 −0.672004 −0.336002 0.941861i \(-0.609075\pi\)
−0.336002 + 0.941861i \(0.609075\pi\)
\(434\) 0 0
\(435\) 4.54650i 0.217988i
\(436\) 0 0
\(437\) 7.11019i 0.340127i
\(438\) 0 0
\(439\) 20.9580i 1.00027i 0.865947 + 0.500136i \(0.166717\pi\)
−0.865947 + 0.500136i \(0.833283\pi\)
\(440\) 0 0
\(441\) 19.8088 0.943278
\(442\) 0 0
\(443\) −10.8826 −0.517046 −0.258523 0.966005i \(-0.583236\pi\)
−0.258523 + 0.966005i \(0.583236\pi\)
\(444\) 0 0
\(445\) 2.70046i 0.128014i
\(446\) 0 0
\(447\) 0.161137i 0.00762151i
\(448\) 0 0
\(449\) 21.3267i 1.00647i 0.864150 + 0.503235i \(0.167857\pi\)
−0.864150 + 0.503235i \(0.832143\pi\)
\(450\) 0 0
\(451\) −3.27685 −0.154301
\(452\) 0 0
\(453\) 7.79045i 0.366027i
\(454\) 0 0
\(455\) −0.179446 −0.00841257
\(456\) 0 0
\(457\) 3.67720 0.172012 0.0860061 0.996295i \(-0.472590\pi\)
0.0860061 + 0.996295i \(0.472590\pi\)
\(458\) 0 0
\(459\) 1.05515 9.68982i 0.0492500 0.452282i
\(460\) 0 0
\(461\) 1.43905 0.0670233 0.0335116 0.999438i \(-0.489331\pi\)
0.0335116 + 0.999438i \(0.489331\pi\)
\(462\) 0 0
\(463\) −3.24480 −0.150799 −0.0753994 0.997153i \(-0.524023\pi\)
−0.0753994 + 0.997153i \(0.524023\pi\)
\(464\) 0 0
\(465\) 2.95976i 0.137256i
\(466\) 0 0
\(467\) −23.9357 −1.10761 −0.553805 0.832646i \(-0.686825\pi\)
−0.553805 + 0.832646i \(0.686825\pi\)
\(468\) 0 0
\(469\) 0.345647i 0.0159605i
\(470\) 0 0
\(471\) 2.94340i 0.135625i
\(472\) 0 0
\(473\) 12.3156i 0.566272i
\(474\) 0 0
\(475\) 19.5081 0.895095
\(476\) 0 0
\(477\) 13.3758 0.612438
\(478\) 0 0
\(479\) 2.84658i 0.130063i −0.997883 0.0650317i \(-0.979285\pi\)
0.997883 0.0650317i \(-0.0207149\pi\)
\(480\) 0 0
\(481\) 4.95217i 0.225799i
\(482\) 0 0
\(483\) 0.0641484i 0.00291885i
\(484\) 0 0
\(485\) 9.55211 0.433739
\(486\) 0 0
\(487\) 21.1362i 0.957773i 0.877877 + 0.478887i \(0.158959\pi\)
−0.877877 + 0.478887i \(0.841041\pi\)
\(488\) 0 0
\(489\) 4.20287 0.190061
\(490\) 0 0
\(491\) −7.61991 −0.343882 −0.171941 0.985107i \(-0.555004\pi\)
−0.171941 + 0.985107i \(0.555004\pi\)
\(492\) 0 0
\(493\) 4.15732 38.1783i 0.187236 1.71947i
\(494\) 0 0
\(495\) −5.65112 −0.253999
\(496\) 0 0
\(497\) 1.53234 0.0687349
\(498\) 0 0
\(499\) 12.5751i 0.562938i −0.959570 0.281469i \(-0.909178\pi\)
0.959570 0.281469i \(-0.0908217\pi\)
\(500\) 0 0
\(501\) −5.74520 −0.256676
\(502\) 0 0
\(503\) 26.3701i 1.17579i 0.808939 + 0.587893i \(0.200042\pi\)
−0.808939 + 0.587893i \(0.799958\pi\)
\(504\) 0 0
\(505\) 5.37893i 0.239359i
\(506\) 0 0
\(507\) 4.66701i 0.207269i
\(508\) 0 0
\(509\) −5.57527 −0.247119 −0.123560 0.992337i \(-0.539431\pi\)
−0.123560 + 0.992337i \(0.539431\pi\)
\(510\) 0 0
\(511\) 1.81134 0.0801291
\(512\) 0 0
\(513\) 12.9982i 0.573884i
\(514\) 0 0
\(515\) 10.4896i 0.462228i
\(516\) 0 0
\(517\) 16.7612i 0.737157i
\(518\) 0 0
\(519\) −2.71066 −0.118985
\(520\) 0 0
\(521\) 8.87437i 0.388793i −0.980923 0.194397i \(-0.937725\pi\)
0.980923 0.194397i \(-0.0622749\pi\)
\(522\) 0 0
\(523\) 4.70502 0.205736 0.102868 0.994695i \(-0.467198\pi\)
0.102868 + 0.994695i \(0.467198\pi\)
\(524\) 0 0
\(525\) −0.176003 −0.00768140
\(526\) 0 0
\(527\) 2.70641 24.8540i 0.117893 1.08266i
\(528\) 0 0
\(529\) 21.3278 0.927294
\(530\) 0 0
\(531\) 2.83591 0.123068
\(532\) 0 0
\(533\) 2.40966i 0.104374i
\(534\) 0 0
\(535\) −0.739086 −0.0319535
\(536\) 0 0
\(537\) 4.79599i 0.206962i
\(538\) 0 0
\(539\) 11.5512i 0.497545i
\(540\) 0 0
\(541\) 35.0127i 1.50531i 0.658413 + 0.752657i \(0.271228\pi\)
−0.658413 + 0.752657i \(0.728772\pi\)
\(542\) 0 0
\(543\) 6.33669 0.271934
\(544\) 0 0
\(545\) −11.3867 −0.487751
\(546\) 0 0
\(547\) 33.3916i 1.42772i 0.700287 + 0.713862i \(0.253056\pi\)
−0.700287 + 0.713862i \(0.746944\pi\)
\(548\) 0 0
\(549\) 10.9737i 0.468347i
\(550\) 0 0
\(551\) 51.2134i 2.18176i
\(552\) 0 0
\(553\) −0.112343 −0.00477732
\(554\) 0 0
\(555\) 1.98775i 0.0843753i
\(556\) 0 0
\(557\) −15.5185 −0.657540 −0.328770 0.944410i \(-0.606634\pi\)
−0.328770 + 0.944410i \(0.606634\pi\)
\(558\) 0 0
\(559\) −9.05638 −0.383044
\(560\) 0 0
\(561\) −2.74579 0.298995i −0.115927 0.0126236i
\(562\) 0 0
\(563\) 9.09863 0.383462 0.191731 0.981448i \(-0.438590\pi\)
0.191731 + 0.981448i \(0.438590\pi\)
\(564\) 0 0
\(565\) 19.1910 0.807370
\(566\) 0 0
\(567\) 0.924584i 0.0388289i
\(568\) 0 0
\(569\) −19.1267 −0.801834 −0.400917 0.916114i \(-0.631308\pi\)
−0.400917 + 0.916114i \(0.631308\pi\)
\(570\) 0 0
\(571\) 3.78922i 0.158574i −0.996852 0.0792869i \(-0.974736\pi\)
0.996852 0.0792869i \(-0.0252643\pi\)
\(572\) 0 0
\(573\) 3.34998i 0.139947i
\(574\) 0 0
\(575\) 4.58811i 0.191337i
\(576\) 0 0
\(577\) −24.9530 −1.03881 −0.519403 0.854529i \(-0.673846\pi\)
−0.519403 + 0.854529i \(0.673846\pi\)
\(578\) 0 0
\(579\) 6.78282 0.281885
\(580\) 0 0
\(581\) 1.55857i 0.0646602i
\(582\) 0 0
\(583\) 7.79989i 0.323039i
\(584\) 0 0
\(585\) 4.15559i 0.171813i
\(586\) 0 0
\(587\) 5.06343 0.208990 0.104495 0.994525i \(-0.466677\pi\)
0.104495 + 0.994525i \(0.466677\pi\)
\(588\) 0 0
\(589\) 33.3399i 1.37375i
\(590\) 0 0
\(591\) −3.42785 −0.141003
\(592\) 0 0
\(593\) 11.2442 0.461746 0.230873 0.972984i \(-0.425842\pi\)
0.230873 + 0.972984i \(0.425842\pi\)
\(594\) 0 0
\(595\) −0.604840 0.0658624i −0.0247960 0.00270010i
\(596\) 0 0
\(597\) 11.0083 0.450539
\(598\) 0 0
\(599\) −14.4981 −0.592378 −0.296189 0.955129i \(-0.595716\pi\)
−0.296189 + 0.955129i \(0.595716\pi\)
\(600\) 0 0
\(601\) 12.1474i 0.495503i −0.968824 0.247751i \(-0.920308\pi\)
0.968824 0.247751i \(-0.0796916\pi\)
\(602\) 0 0
\(603\) 8.00446 0.325967
\(604\) 0 0
\(605\) 9.95951i 0.404912i
\(606\) 0 0
\(607\) 45.1298i 1.83176i −0.401451 0.915881i \(-0.631494\pi\)
0.401451 0.915881i \(-0.368506\pi\)
\(608\) 0 0
\(609\) 0.462049i 0.0187232i
\(610\) 0 0
\(611\) 12.3255 0.498635
\(612\) 0 0
\(613\) −42.8452 −1.73050 −0.865250 0.501340i \(-0.832841\pi\)
−0.865250 + 0.501340i \(0.832841\pi\)
\(614\) 0 0
\(615\) 0.967211i 0.0390017i
\(616\) 0 0
\(617\) 18.9274i 0.761988i −0.924577 0.380994i \(-0.875582\pi\)
0.924577 0.380994i \(-0.124418\pi\)
\(618\) 0 0
\(619\) 18.8350i 0.757041i 0.925593 + 0.378520i \(0.123567\pi\)
−0.925593 + 0.378520i \(0.876433\pi\)
\(620\) 0 0
\(621\) −3.05704 −0.122675
\(622\) 0 0
\(623\) 0.274441i 0.0109952i
\(624\) 0 0
\(625\) 5.32837 0.213135
\(626\) 0 0
\(627\) 3.68327 0.147096
\(628\) 0 0
\(629\) −1.81760 + 16.6917i −0.0724725 + 0.665543i
\(630\) 0 0
\(631\) 13.5185 0.538164 0.269082 0.963117i \(-0.413280\pi\)
0.269082 + 0.963117i \(0.413280\pi\)
\(632\) 0 0
\(633\) 3.61113 0.143530
\(634\) 0 0
\(635\) 0.0763521i 0.00302994i
\(636\) 0 0
\(637\) −8.49425 −0.336554
\(638\) 0 0
\(639\) 35.4858i 1.40380i
\(640\) 0 0
\(641\) 20.1887i 0.797407i −0.917080 0.398704i \(-0.869460\pi\)
0.917080 0.398704i \(-0.130540\pi\)
\(642\) 0 0
\(643\) 13.4682i 0.531135i 0.964092 + 0.265567i \(0.0855592\pi\)
−0.964092 + 0.265567i \(0.914441\pi\)
\(644\) 0 0
\(645\) 3.63514 0.143133
\(646\) 0 0
\(647\) 6.54797 0.257427 0.128714 0.991682i \(-0.458915\pi\)
0.128714 + 0.991682i \(0.458915\pi\)
\(648\) 0 0
\(649\) 1.65371i 0.0649139i
\(650\) 0 0
\(651\) 0.300793i 0.0117890i
\(652\) 0 0
\(653\) 27.3249i 1.06931i −0.845071 0.534654i \(-0.820442\pi\)
0.845071 0.534654i \(-0.179558\pi\)
\(654\) 0 0
\(655\) 9.30522 0.363585
\(656\) 0 0
\(657\) 41.9469i 1.63650i
\(658\) 0 0
\(659\) −47.4256 −1.84744 −0.923719 0.383070i \(-0.874867\pi\)
−0.923719 + 0.383070i \(0.874867\pi\)
\(660\) 0 0
\(661\) −21.1368 −0.822127 −0.411064 0.911607i \(-0.634843\pi\)
−0.411064 + 0.911607i \(0.634843\pi\)
\(662\) 0 0
\(663\) −0.219868 + 2.01914i −0.00853897 + 0.0784167i
\(664\) 0 0
\(665\) 0.811349 0.0314627
\(666\) 0 0
\(667\) −12.0449 −0.466379
\(668\) 0 0
\(669\) 4.82863i 0.186686i
\(670\) 0 0
\(671\) −6.39914 −0.247036
\(672\) 0 0
\(673\) 16.7911i 0.647250i −0.946185 0.323625i \(-0.895098\pi\)
0.946185 0.323625i \(-0.104902\pi\)
\(674\) 0 0
\(675\) 8.38755i 0.322837i
\(676\) 0 0
\(677\) 36.3838i 1.39834i −0.714953 0.699172i \(-0.753552\pi\)
0.714953 0.699172i \(-0.246448\pi\)
\(678\) 0 0
\(679\) −0.970756 −0.0372542
\(680\) 0 0
\(681\) 5.35706 0.205283
\(682\) 0 0
\(683\) 11.1766i 0.427662i −0.976871 0.213831i \(-0.931406\pi\)
0.976871 0.213831i \(-0.0685941\pi\)
\(684\) 0 0
\(685\) 7.13873i 0.272757i
\(686\) 0 0
\(687\) 3.43306i 0.130980i
\(688\) 0 0
\(689\) −5.73571 −0.218513
\(690\) 0 0
\(691\) 43.1149i 1.64017i −0.572244 0.820083i \(-0.693927\pi\)
0.572244 0.820083i \(-0.306073\pi\)
\(692\) 0 0
\(693\) 0.574309 0.0218162
\(694\) 0 0
\(695\) 19.8634 0.753460
\(696\) 0 0
\(697\) −0.884419 + 8.12197i −0.0334998 + 0.307641i
\(698\) 0 0
\(699\) 1.77102 0.0669860
\(700\) 0 0
\(701\) 9.08849 0.343267 0.171634 0.985161i \(-0.445095\pi\)
0.171634 + 0.985161i \(0.445095\pi\)
\(702\) 0 0
\(703\) 22.3907i 0.844483i
\(704\) 0 0
\(705\) −4.94732 −0.186327
\(706\) 0 0
\(707\) 0.546647i 0.0205588i
\(708\) 0 0
\(709\) 27.8580i 1.04623i 0.852262 + 0.523115i \(0.175230\pi\)
−0.852262 + 0.523115i \(0.824770\pi\)
\(710\) 0 0
\(711\) 2.60163i 0.0975688i
\(712\) 0 0
\(713\) −7.84119 −0.293655
\(714\) 0 0
\(715\) 2.42326 0.0906249
\(716\) 0 0
\(717\) 12.1447i 0.453552i
\(718\) 0 0
\(719\) 35.4791i 1.32315i 0.749880 + 0.661573i \(0.230111\pi\)
−0.749880 + 0.661573i \(0.769889\pi\)
\(720\) 0 0
\(721\) 1.06603i 0.0397012i
\(722\) 0 0
\(723\) 8.30993 0.309050
\(724\) 0 0
\(725\) 33.0473i 1.22735i
\(726\) 0 0
\(727\) −15.9770 −0.592554 −0.296277 0.955102i \(-0.595745\pi\)
−0.296277 + 0.955102i \(0.595745\pi\)
\(728\) 0 0
\(729\) 18.5386 0.686613
\(730\) 0 0
\(731\) −30.5254 3.32398i −1.12902 0.122942i
\(732\) 0 0
\(733\) 6.93413 0.256118 0.128059 0.991767i \(-0.459125\pi\)
0.128059 + 0.991767i \(0.459125\pi\)
\(734\) 0 0
\(735\) 3.40951 0.125762
\(736\) 0 0
\(737\) 4.66766i 0.171936i
\(738\) 0 0
\(739\) −0.389535 −0.0143293 −0.00716464 0.999974i \(-0.502281\pi\)
−0.00716464 + 0.999974i \(0.502281\pi\)
\(740\) 0 0
\(741\) 2.70852i 0.0995000i
\(742\) 0 0
\(743\) 9.32540i 0.342116i −0.985261 0.171058i \(-0.945281\pi\)
0.985261 0.171058i \(-0.0547185\pi\)
\(744\) 0 0
\(745\) 0.479331i 0.0175613i
\(746\) 0 0
\(747\) 36.0931 1.32058
\(748\) 0 0
\(749\) 0.0751114 0.00274451
\(750\) 0 0
\(751\) 28.4223i 1.03714i −0.855034 0.518572i \(-0.826464\pi\)
0.855034 0.518572i \(-0.173536\pi\)
\(752\) 0 0
\(753\) 5.43507i 0.198065i
\(754\) 0 0
\(755\) 23.1741i 0.843392i
\(756\) 0 0
\(757\) −46.6449 −1.69534 −0.847668 0.530527i \(-0.821994\pi\)
−0.847668 + 0.530527i \(0.821994\pi\)
\(758\) 0 0
\(759\) 0.866267i 0.0314435i
\(760\) 0 0
\(761\) −14.8358 −0.537797 −0.268899 0.963168i \(-0.586660\pi\)
−0.268899 + 0.963168i \(0.586660\pi\)
\(762\) 0 0
\(763\) 1.15720 0.0418933
\(764\) 0 0
\(765\) −1.52523 + 14.0068i −0.0551449 + 0.506418i
\(766\) 0 0
\(767\) −1.21607 −0.0439097
\(768\) 0 0
\(769\) −3.96099 −0.142837 −0.0714185 0.997446i \(-0.522753\pi\)
−0.0714185 + 0.997446i \(0.522753\pi\)
\(770\) 0 0
\(771\) 6.45315i 0.232405i
\(772\) 0 0
\(773\) −45.9340 −1.65213 −0.826065 0.563575i \(-0.809426\pi\)
−0.826065 + 0.563575i \(0.809426\pi\)
\(774\) 0 0
\(775\) 21.5138i 0.772797i
\(776\) 0 0
\(777\) 0.202010i 0.00724707i
\(778\) 0 0
\(779\) 10.8950i 0.390355i
\(780\) 0 0
\(781\) −20.6929 −0.740451
\(782\) 0 0
\(783\) −22.0193 −0.786905
\(784\) 0 0
\(785\) 8.75567i 0.312503i
\(786\) 0 0
\(787\) 44.4809i 1.58557i −0.609500 0.792786i \(-0.708630\pi\)
0.609500 0.792786i \(-0.291370\pi\)
\(788\) 0 0
\(789\) 1.33807i 0.0476364i
\(790\) 0 0
\(791\) −1.95033 −0.0693457
\(792\) 0 0
\(793\) 4.70566i 0.167103i
\(794\) 0 0
\(795\) 2.30226 0.0816526
\(796\) 0 0
\(797\) −23.3443 −0.826899 −0.413449 0.910527i \(-0.635676\pi\)
−0.413449 + 0.910527i \(0.635676\pi\)
\(798\) 0 0
\(799\) 41.5442 + 4.52384i 1.46973 + 0.160042i
\(800\) 0 0
\(801\) −6.35547 −0.224559
\(802\) 0 0
\(803\) −24.4606 −0.863196
\(804\) 0 0
\(805\) 0.190821i 0.00672555i
\(806\) 0 0
\(807\) −7.29424 −0.256769
\(808\) 0 0
\(809\) 29.9827i 1.05414i 0.849823 + 0.527068i \(0.176709\pi\)
−0.849823 + 0.527068i \(0.823291\pi\)
\(810\) 0 0
\(811\) 17.5412i 0.615954i 0.951394 + 0.307977i \(0.0996520\pi\)
−0.951394 + 0.307977i \(0.900348\pi\)
\(812\) 0 0
\(813\) 3.44017i 0.120652i
\(814\) 0 0
\(815\) −12.5022 −0.437933
\(816\) 0 0
\(817\) 40.9475 1.43257
\(818\) 0 0
\(819\) 0.422322i 0.0147571i
\(820\) 0 0
\(821\) 44.7003i 1.56005i 0.625749 + 0.780025i \(0.284794\pi\)
−0.625749 + 0.780025i \(0.715206\pi\)
\(822\) 0 0
\(823\) 29.9358i 1.04350i 0.853099 + 0.521749i \(0.174720\pi\)
−0.853099 + 0.521749i \(0.825280\pi\)
\(824\) 0 0
\(825\) 2.37677 0.0827484
\(826\) 0 0
\(827\) 25.3775i 0.882462i −0.897393 0.441231i \(-0.854542\pi\)
0.897393 0.441231i \(-0.145458\pi\)
\(828\) 0 0
\(829\) −26.7433 −0.928833 −0.464416 0.885617i \(-0.653736\pi\)
−0.464416 + 0.885617i \(0.653736\pi\)
\(830\) 0 0
\(831\) −5.34273 −0.185337
\(832\) 0 0
\(833\) −28.6307 3.11766i −0.991994 0.108020i
\(834\) 0 0
\(835\) 17.0901 0.591428
\(836\) 0 0
\(837\) −14.3345 −0.495474
\(838\) 0 0
\(839\) 40.8534i 1.41042i −0.709001 0.705208i \(-0.750854\pi\)
0.709001 0.705208i \(-0.249146\pi\)
\(840\) 0 0
\(841\) −57.7569 −1.99162
\(842\) 0 0
\(843\) 3.91298i 0.134770i
\(844\) 0 0
\(845\) 13.8829i 0.477585i
\(846\) 0 0
\(847\) 1.01216i 0.0347782i
\(848\) 0 0
\(849\) −4.64733 −0.159496
\(850\) 0 0
\(851\) 5.26607 0.180519
\(852\) 0 0
\(853\) 14.0075i 0.479607i −0.970821 0.239804i \(-0.922917\pi\)
0.970821 0.239804i \(-0.0770830\pi\)
\(854\) 0 0
\(855\) 18.7891i 0.642574i
\(856\) 0 0
\(857\) 44.9083i 1.53404i −0.641624 0.767020i \(-0.721739\pi\)
0.641624 0.767020i \(-0.278261\pi\)
\(858\) 0 0
\(859\) 46.0717 1.57195 0.785974 0.618260i \(-0.212162\pi\)
0.785974 + 0.618260i \(0.212162\pi\)
\(860\) 0 0
\(861\) 0.0982952i 0.00334989i
\(862\) 0 0
\(863\) 26.7618 0.910984 0.455492 0.890240i \(-0.349463\pi\)
0.455492 + 0.890240i \(0.349463\pi\)
\(864\) 0 0
\(865\) 8.06334 0.274162
\(866\) 0 0
\(867\) −1.48217 + 6.72499i −0.0503372 + 0.228393i
\(868\) 0 0
\(869\) 1.51710 0.0514640
\(870\) 0 0
\(871\) −3.43240 −0.116303
\(872\) 0 0
\(873\) 22.4807i 0.760855i
\(874\) 0 0
\(875\) 1.26136 0.0426419
\(876\) 0 0
\(877\) 20.0191i 0.675996i 0.941147 + 0.337998i \(0.109750\pi\)
−0.941147 + 0.337998i \(0.890250\pi\)
\(878\) 0 0
\(879\) 0.838120i 0.0282691i
\(880\) 0 0
\(881\) 24.7408i 0.833540i 0.909012 + 0.416770i \(0.136838\pi\)
−0.909012 + 0.416770i \(0.863162\pi\)
\(882\) 0 0
\(883\) 6.94873 0.233843 0.116922 0.993141i \(-0.462697\pi\)
0.116922 + 0.993141i \(0.462697\pi\)
\(884\) 0 0
\(885\) 0.488118 0.0164079
\(886\) 0 0
\(887\) 11.3154i 0.379935i −0.981790 0.189967i \(-0.939162\pi\)
0.981790 0.189967i \(-0.0608383\pi\)
\(888\) 0 0
\(889\) 0.00775947i 0.000260244i
\(890\) 0 0
\(891\) 12.4857i 0.418287i
\(892\) 0 0
\(893\) −55.7284 −1.86488
\(894\) 0 0
\(895\) 14.2665i 0.476877i
\(896\) 0 0
\(897\) 0.637016 0.0212694
\(898\) 0 0
\(899\) −56.4786 −1.88367
\(900\) 0 0
\(901\) −19.3327 2.10519i −0.644067 0.0701339i
\(902\) 0 0
\(903\) −0.369430 −0.0122939
\(904\) 0 0
\(905\) −18.8496 −0.626583
\(906\) 0 0
\(907\) 44.7670i 1.48646i 0.669035 + 0.743231i \(0.266708\pi\)
−0.669035 + 0.743231i \(0.733292\pi\)
\(908\) 0 0
\(909\) −12.6592 −0.419879
\(910\) 0 0
\(911\) 2.53273i 0.0839131i 0.999119 + 0.0419565i \(0.0133591\pi\)
−0.999119 + 0.0419565i \(0.986641\pi\)
\(912\) 0 0
\(913\) 21.0471i 0.696556i
\(914\) 0 0
\(915\) 1.88880i 0.0624419i
\(916\) 0 0
\(917\) −0.945666 −0.0312286
\(918\) 0 0
\(919\) 27.3097 0.900863 0.450431 0.892811i \(-0.351270\pi\)
0.450431 + 0.892811i \(0.351270\pi\)
\(920\) 0 0
\(921\) 7.91948i 0.260956i
\(922\) 0 0
\(923\) 15.2167i 0.500864i
\(924\) 0 0
\(925\) 14.4484i 0.475062i
\(926\) 0 0
\(927\) 24.6871 0.810830
\(928\) 0 0
\(929\) 25.0343i 0.821348i −0.911782 0.410674i \(-0.865293\pi\)
0.911782 0.410674i \(-0.134707\pi\)
\(930\) 0 0
\(931\) 38.4059 1.25870
\(932\) 0 0
\(933\) −9.34754 −0.306025
\(934\) 0 0
\(935\) 8.16784 + 0.889414i 0.267117 + 0.0290869i
\(936\) 0 0
\(937\) −23.7460 −0.775747 −0.387874 0.921713i \(-0.626790\pi\)
−0.387874 + 0.921713i \(0.626790\pi\)
\(938\) 0 0
\(939\) −5.78347 −0.188736
\(940\) 0 0
\(941\) 9.61968i 0.313592i −0.987631 0.156796i \(-0.949883\pi\)
0.987631 0.156796i \(-0.0501166\pi\)
\(942\) 0 0
\(943\) 2.56240 0.0834431
\(944\) 0 0
\(945\) 0.348840i 0.0113478i
\(946\) 0 0
\(947\) 14.5433i 0.472595i −0.971681 0.236297i \(-0.924066\pi\)
0.971681 0.236297i \(-0.0759340\pi\)
\(948\) 0 0
\(949\) 17.9873i 0.583892i
\(950\) 0 0
\(951\) 7.23443 0.234592
\(952\) 0 0
\(953\) −35.9067 −1.16313 −0.581566 0.813499i \(-0.697560\pi\)
−0.581566 + 0.813499i \(0.697560\pi\)
\(954\) 0 0
\(955\) 9.96511i 0.322463i
\(956\) 0 0
\(957\) 6.23956i 0.201696i
\(958\) 0 0
\(959\) 0.725491i 0.0234273i
\(960\) 0 0
\(961\) −5.76754 −0.186050
\(962\) 0 0
\(963\) 1.73942i 0.0560521i
\(964\) 0 0
\(965\) −20.1767 −0.649512
\(966\) 0 0
\(967\) −36.8977 −1.18655 −0.593275 0.805000i \(-0.702165\pi\)
−0.593275 + 0.805000i \(0.702165\pi\)
\(968\) 0 0
\(969\) 0.994113 9.12933i 0.0319355 0.293276i
\(970\) 0 0
\(971\) −19.8943 −0.638437 −0.319219 0.947681i \(-0.603420\pi\)
−0.319219 + 0.947681i \(0.603420\pi\)
\(972\) 0 0
\(973\) −2.01866 −0.0647154
\(974\) 0 0
\(975\) 1.74777i 0.0559735i
\(976\) 0 0
\(977\) −59.8699 −1.91541 −0.957704 0.287754i \(-0.907091\pi\)
−0.957704 + 0.287754i \(0.907091\pi\)
\(978\) 0 0
\(979\) 3.70608i 0.118447i
\(980\) 0 0
\(981\) 26.7982i 0.855601i
\(982\) 0 0
\(983\) 1.62326i 0.0517739i −0.999665 0.0258870i \(-0.991759\pi\)
0.999665 0.0258870i \(-0.00824099\pi\)
\(984\) 0 0
\(985\) 10.1967 0.324895
\(986\) 0 0
\(987\) 0.502784 0.0160038
\(988\) 0 0
\(989\) 9.63044i 0.306230i
\(990\) 0 0
\(991\) 11.9283i 0.378913i 0.981889 + 0.189457i \(0.0606727\pi\)
−0.981889 + 0.189457i \(0.939327\pi\)
\(992\) 0 0
\(993\) 9.16246i 0.290762i
\(994\) 0 0
\(995\) −32.7461 −1.03812
\(996\) 0 0
\(997\) 50.6631i 1.60452i 0.596978 + 0.802258i \(0.296368\pi\)
−0.596978 + 0.802258i \(0.703632\pi\)
\(998\) 0 0
\(999\) 9.62693 0.304583
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 4012.2.b.b.237.27 yes 46
17.16 even 2 inner 4012.2.b.b.237.20 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.b.b.237.20 46 17.16 even 2 inner
4012.2.b.b.237.27 yes 46 1.1 even 1 trivial