Properties

Label 4012.2.b.b.237.24
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 237.24
Character \(\chi\) \(=\) 4012.237
Dual form 4012.2.b.b.237.23

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.123544i q^{3} +0.454269i q^{5} +3.56076i q^{7} +2.98474 q^{9} +O(q^{10})\) \(q+0.123544i q^{3} +0.454269i q^{5} +3.56076i q^{7} +2.98474 q^{9} +2.55914i q^{11} +1.05907 q^{13} -0.0561222 q^{15} +(2.55663 + 3.23476i) q^{17} -5.45088 q^{19} -0.439910 q^{21} +5.82839i q^{23} +4.79364 q^{25} +0.739378i q^{27} +10.2975i q^{29} -0.560481i q^{31} -0.316166 q^{33} -1.61754 q^{35} -10.4994i q^{37} +0.130842i q^{39} -6.36520i q^{41} -9.85998 q^{43} +1.35587i q^{45} -4.05578 q^{47} -5.67898 q^{49} +(-0.399634 + 0.315856i) q^{51} +10.3974 q^{53} -1.16254 q^{55} -0.673423i q^{57} +1.00000 q^{59} -6.58255i q^{61} +10.6279i q^{63} +0.481103i q^{65} -2.95294 q^{67} -0.720062 q^{69} +4.41564i q^{71} +7.04827i q^{73} +0.592225i q^{75} -9.11246 q^{77} -7.96911i q^{79} +8.86287 q^{81} +11.8663 q^{83} +(-1.46945 + 1.16140i) q^{85} -1.27220 q^{87} -18.3374 q^{89} +3.77109i q^{91} +0.0692441 q^{93} -2.47617i q^{95} +9.68685i q^{97} +7.63835i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46 q - 54 q^{9}+O(q^{10}) \) 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
\(3\) 0.123544i 0.0713281i 0.999364 + 0.0356641i \(0.0113546\pi\)
−0.999364 + 0.0356641i \(0.988645\pi\)
\(4\) 0 0
\(5\) 0.454269i 0.203155i 0.994828 + 0.101578i \(0.0323891\pi\)
−0.994828 + 0.101578i \(0.967611\pi\)
\(6\) 0 0
\(7\) 3.56076i 1.34584i 0.739716 + 0.672920i \(0.234960\pi\)
−0.739716 + 0.672920i \(0.765040\pi\)
\(8\) 0 0
\(9\) 2.98474 0.994912
\(10\) 0 0
\(11\) 2.55914i 0.771609i 0.922581 + 0.385805i \(0.126076\pi\)
−0.922581 + 0.385805i \(0.873924\pi\)
\(12\) 0 0
\(13\) 1.05907 0.293733 0.146867 0.989156i \(-0.453081\pi\)
0.146867 + 0.989156i \(0.453081\pi\)
\(14\) 0 0
\(15\) −0.0561222 −0.0144907
\(16\) 0 0
\(17\) 2.55663 + 3.23476i 0.620074 + 0.784543i
\(18\) 0 0
\(19\) −5.45088 −1.25052 −0.625258 0.780418i \(-0.715006\pi\)
−0.625258 + 0.780418i \(0.715006\pi\)
\(20\) 0 0
\(21\) −0.439910 −0.0959962
\(22\) 0 0
\(23\) 5.82839i 1.21530i 0.794204 + 0.607652i \(0.207888\pi\)
−0.794204 + 0.607652i \(0.792112\pi\)
\(24\) 0 0
\(25\) 4.79364 0.958728
\(26\) 0 0
\(27\) 0.739378i 0.142293i
\(28\) 0 0
\(29\) 10.2975i 1.91220i 0.293034 + 0.956102i \(0.405335\pi\)
−0.293034 + 0.956102i \(0.594665\pi\)
\(30\) 0 0
\(31\) 0.560481i 0.100665i −0.998733 0.0503327i \(-0.983972\pi\)
0.998733 0.0503327i \(-0.0160282\pi\)
\(32\) 0 0
\(33\) −0.316166 −0.0550374
\(34\) 0 0
\(35\) −1.61754 −0.273415
\(36\) 0 0
\(37\) 10.4994i 1.72610i −0.505122 0.863048i \(-0.668552\pi\)
0.505122 0.863048i \(-0.331448\pi\)
\(38\) 0 0
\(39\) 0.130842i 0.0209514i
\(40\) 0 0
\(41\) 6.36520i 0.994078i −0.867728 0.497039i \(-0.834421\pi\)
0.867728 0.497039i \(-0.165579\pi\)
\(42\) 0 0
\(43\) −9.85998 −1.50363 −0.751816 0.659372i \(-0.770822\pi\)
−0.751816 + 0.659372i \(0.770822\pi\)
\(44\) 0 0
\(45\) 1.35587i 0.202122i
\(46\) 0 0
\(47\) −4.05578 −0.591597 −0.295798 0.955250i \(-0.595586\pi\)
−0.295798 + 0.955250i \(0.595586\pi\)
\(48\) 0 0
\(49\) −5.67898 −0.811283
\(50\) 0 0
\(51\) −0.399634 + 0.315856i −0.0559600 + 0.0442287i
\(52\) 0 0
\(53\) 10.3974 1.42820 0.714099 0.700044i \(-0.246836\pi\)
0.714099 + 0.700044i \(0.246836\pi\)
\(54\) 0 0
\(55\) −1.16254 −0.156757
\(56\) 0 0
\(57\) 0.673423i 0.0891970i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 6.58255i 0.842809i −0.906873 0.421404i \(-0.861537\pi\)
0.906873 0.421404i \(-0.138463\pi\)
\(62\) 0 0
\(63\) 10.6279i 1.33899i
\(64\) 0 0
\(65\) 0.481103i 0.0596735i
\(66\) 0 0
\(67\) −2.95294 −0.360759 −0.180379 0.983597i \(-0.557733\pi\)
−0.180379 + 0.983597i \(0.557733\pi\)
\(68\) 0 0
\(69\) −0.720062 −0.0866853
\(70\) 0 0
\(71\) 4.41564i 0.524040i 0.965062 + 0.262020i \(0.0843887\pi\)
−0.965062 + 0.262020i \(0.915611\pi\)
\(72\) 0 0
\(73\) 7.04827i 0.824938i 0.910972 + 0.412469i \(0.135333\pi\)
−0.910972 + 0.412469i \(0.864667\pi\)
\(74\) 0 0
\(75\) 0.592225i 0.0683842i
\(76\) 0 0
\(77\) −9.11246 −1.03846
\(78\) 0 0
\(79\) 7.96911i 0.896595i −0.893884 0.448297i \(-0.852031\pi\)
0.893884 0.448297i \(-0.147969\pi\)
\(80\) 0 0
\(81\) 8.86287 0.984763
\(82\) 0 0
\(83\) 11.8663 1.30249 0.651247 0.758866i \(-0.274246\pi\)
0.651247 + 0.758866i \(0.274246\pi\)
\(84\) 0 0
\(85\) −1.46945 + 1.16140i −0.159384 + 0.125971i
\(86\) 0 0
\(87\) −1.27220 −0.136394
\(88\) 0 0
\(89\) −18.3374 −1.94376 −0.971880 0.235475i \(-0.924335\pi\)
−0.971880 + 0.235475i \(0.924335\pi\)
\(90\) 0 0
\(91\) 3.77109i 0.395318i
\(92\) 0 0
\(93\) 0.0692441 0.00718028
\(94\) 0 0
\(95\) 2.47617i 0.254049i
\(96\) 0 0
\(97\) 9.68685i 0.983551i 0.870722 + 0.491775i \(0.163652\pi\)
−0.870722 + 0.491775i \(0.836348\pi\)
\(98\) 0 0
\(99\) 7.63835i 0.767683i
\(100\) 0 0
\(101\) 10.7833 1.07297 0.536487 0.843909i \(-0.319751\pi\)
0.536487 + 0.843909i \(0.319751\pi\)
\(102\) 0 0
\(103\) −2.61985 −0.258141 −0.129071 0.991635i \(-0.541199\pi\)
−0.129071 + 0.991635i \(0.541199\pi\)
\(104\) 0 0
\(105\) 0.199838i 0.0195021i
\(106\) 0 0
\(107\) 14.1782i 1.37066i −0.728234 0.685329i \(-0.759658\pi\)
0.728234 0.685329i \(-0.240342\pi\)
\(108\) 0 0
\(109\) 2.82164i 0.270264i 0.990828 + 0.135132i \(0.0431459\pi\)
−0.990828 + 0.135132i \(0.956854\pi\)
\(110\) 0 0
\(111\) 1.29714 0.123119
\(112\) 0 0
\(113\) 7.43401i 0.699333i 0.936874 + 0.349666i \(0.113705\pi\)
−0.936874 + 0.349666i \(0.886295\pi\)
\(114\) 0 0
\(115\) −2.64766 −0.246896
\(116\) 0 0
\(117\) 3.16104 0.292239
\(118\) 0 0
\(119\) −11.5182 + 9.10354i −1.05587 + 0.834520i
\(120\) 0 0
\(121\) 4.45081 0.404619
\(122\) 0 0
\(123\) 0.786382 0.0709057
\(124\) 0 0
\(125\) 4.44895i 0.397926i
\(126\) 0 0
\(127\) −19.0541 −1.69078 −0.845388 0.534153i \(-0.820631\pi\)
−0.845388 + 0.534153i \(0.820631\pi\)
\(128\) 0 0
\(129\) 1.21814i 0.107251i
\(130\) 0 0
\(131\) 17.4469i 1.52434i −0.647377 0.762170i \(-0.724134\pi\)
0.647377 0.762170i \(-0.275866\pi\)
\(132\) 0 0
\(133\) 19.4092i 1.68299i
\(134\) 0 0
\(135\) −0.335877 −0.0289077
\(136\) 0 0
\(137\) −15.2168 −1.30006 −0.650029 0.759910i \(-0.725243\pi\)
−0.650029 + 0.759910i \(0.725243\pi\)
\(138\) 0 0
\(139\) 12.3854i 1.05052i −0.850943 0.525258i \(-0.823969\pi\)
0.850943 0.525258i \(-0.176031\pi\)
\(140\) 0 0
\(141\) 0.501067i 0.0421975i
\(142\) 0 0
\(143\) 2.71031i 0.226647i
\(144\) 0 0
\(145\) −4.67786 −0.388475
\(146\) 0 0
\(147\) 0.701603i 0.0578673i
\(148\) 0 0
\(149\) 7.21156 0.590794 0.295397 0.955375i \(-0.404548\pi\)
0.295397 + 0.955375i \(0.404548\pi\)
\(150\) 0 0
\(151\) −19.9609 −1.62440 −0.812199 0.583381i \(-0.801730\pi\)
−0.812199 + 0.583381i \(0.801730\pi\)
\(152\) 0 0
\(153\) 7.63087 + 9.65489i 0.616919 + 0.780552i
\(154\) 0 0
\(155\) 0.254610 0.0204507
\(156\) 0 0
\(157\) 15.9907 1.27619 0.638096 0.769956i \(-0.279722\pi\)
0.638096 + 0.769956i \(0.279722\pi\)
\(158\) 0 0
\(159\) 1.28454i 0.101871i
\(160\) 0 0
\(161\) −20.7535 −1.63560
\(162\) 0 0
\(163\) 1.98112i 0.155173i −0.996986 0.0775867i \(-0.975279\pi\)
0.996986 0.0775867i \(-0.0247215\pi\)
\(164\) 0 0
\(165\) 0.143625i 0.0111812i
\(166\) 0 0
\(167\) 3.16345i 0.244795i 0.992481 + 0.122397i \(0.0390583\pi\)
−0.992481 + 0.122397i \(0.960942\pi\)
\(168\) 0 0
\(169\) −11.8784 −0.913721
\(170\) 0 0
\(171\) −16.2694 −1.24415
\(172\) 0 0
\(173\) 17.3966i 1.32264i −0.750105 0.661319i \(-0.769997\pi\)
0.750105 0.661319i \(-0.230003\pi\)
\(174\) 0 0
\(175\) 17.0690i 1.29029i
\(176\) 0 0
\(177\) 0.123544i 0.00928613i
\(178\) 0 0
\(179\) 12.5748 0.939887 0.469943 0.882697i \(-0.344274\pi\)
0.469943 + 0.882697i \(0.344274\pi\)
\(180\) 0 0
\(181\) 22.8834i 1.70091i 0.526051 + 0.850453i \(0.323672\pi\)
−0.526051 + 0.850453i \(0.676328\pi\)
\(182\) 0 0
\(183\) 0.813233 0.0601159
\(184\) 0 0
\(185\) 4.76957 0.350666
\(186\) 0 0
\(187\) −8.27818 + 6.54277i −0.605361 + 0.478455i
\(188\) 0 0
\(189\) −2.63274 −0.191504
\(190\) 0 0
\(191\) 11.5865 0.838372 0.419186 0.907900i \(-0.362316\pi\)
0.419186 + 0.907900i \(0.362316\pi\)
\(192\) 0 0
\(193\) 12.8891i 0.927777i 0.885893 + 0.463889i \(0.153546\pi\)
−0.885893 + 0.463889i \(0.846454\pi\)
\(194\) 0 0
\(195\) −0.0594374 −0.00425640
\(196\) 0 0
\(197\) 21.4768i 1.53016i 0.643937 + 0.765078i \(0.277300\pi\)
−0.643937 + 0.765078i \(0.722700\pi\)
\(198\) 0 0
\(199\) 8.87015i 0.628788i 0.949293 + 0.314394i \(0.101801\pi\)
−0.949293 + 0.314394i \(0.898199\pi\)
\(200\) 0 0
\(201\) 0.364817i 0.0257322i
\(202\) 0 0
\(203\) −36.6670 −2.57352
\(204\) 0 0
\(205\) 2.89152 0.201952
\(206\) 0 0
\(207\) 17.3962i 1.20912i
\(208\) 0 0
\(209\) 13.9495i 0.964910i
\(210\) 0 0
\(211\) 23.9563i 1.64922i 0.565701 + 0.824610i \(0.308606\pi\)
−0.565701 + 0.824610i \(0.691394\pi\)
\(212\) 0 0
\(213\) −0.545526 −0.0373788
\(214\) 0 0
\(215\) 4.47909i 0.305471i
\(216\) 0 0
\(217\) 1.99574 0.135479
\(218\) 0 0
\(219\) −0.870771 −0.0588413
\(220\) 0 0
\(221\) 2.70765 + 3.42583i 0.182136 + 0.230446i
\(222\) 0 0
\(223\) 16.6350 1.11396 0.556981 0.830525i \(-0.311960\pi\)
0.556981 + 0.830525i \(0.311960\pi\)
\(224\) 0 0
\(225\) 14.3078 0.953850
\(226\) 0 0
\(227\) 10.8491i 0.720079i 0.932937 + 0.360039i \(0.117237\pi\)
−0.932937 + 0.360039i \(0.882763\pi\)
\(228\) 0 0
\(229\) −8.90060 −0.588168 −0.294084 0.955780i \(-0.595015\pi\)
−0.294084 + 0.955780i \(0.595015\pi\)
\(230\) 0 0
\(231\) 1.12579i 0.0740715i
\(232\) 0 0
\(233\) 4.73114i 0.309947i −0.987919 0.154974i \(-0.950471\pi\)
0.987919 0.154974i \(-0.0495292\pi\)
\(234\) 0 0
\(235\) 1.84242i 0.120186i
\(236\) 0 0
\(237\) 0.984535 0.0639524
\(238\) 0 0
\(239\) −0.214811 −0.0138950 −0.00694748 0.999976i \(-0.502211\pi\)
−0.00694748 + 0.999976i \(0.502211\pi\)
\(240\) 0 0
\(241\) 17.5712i 1.13186i 0.824454 + 0.565929i \(0.191482\pi\)
−0.824454 + 0.565929i \(0.808518\pi\)
\(242\) 0 0
\(243\) 3.31309i 0.212535i
\(244\) 0 0
\(245\) 2.57979i 0.164817i
\(246\) 0 0
\(247\) −5.77286 −0.367318
\(248\) 0 0
\(249\) 1.46601i 0.0929044i
\(250\) 0 0
\(251\) −22.3649 −1.41166 −0.705830 0.708382i \(-0.749426\pi\)
−0.705830 + 0.708382i \(0.749426\pi\)
\(252\) 0 0
\(253\) −14.9157 −0.937739
\(254\) 0 0
\(255\) −0.143484 0.181542i −0.00898530 0.0113686i
\(256\) 0 0
\(257\) 6.34853 0.396010 0.198005 0.980201i \(-0.436554\pi\)
0.198005 + 0.980201i \(0.436554\pi\)
\(258\) 0 0
\(259\) 37.3859 2.32305
\(260\) 0 0
\(261\) 30.7354i 1.90248i
\(262\) 0 0
\(263\) 1.36608 0.0842362 0.0421181 0.999113i \(-0.486589\pi\)
0.0421181 + 0.999113i \(0.486589\pi\)
\(264\) 0 0
\(265\) 4.72324i 0.290146i
\(266\) 0 0
\(267\) 2.26547i 0.138645i
\(268\) 0 0
\(269\) 30.0617i 1.83289i −0.400159 0.916446i \(-0.631045\pi\)
0.400159 0.916446i \(-0.368955\pi\)
\(270\) 0 0
\(271\) 26.7633 1.62575 0.812876 0.582436i \(-0.197900\pi\)
0.812876 + 0.582436i \(0.197900\pi\)
\(272\) 0 0
\(273\) −0.465895 −0.0281973
\(274\) 0 0
\(275\) 12.2676i 0.739763i
\(276\) 0 0
\(277\) 3.74897i 0.225254i −0.993637 0.112627i \(-0.964073\pi\)
0.993637 0.112627i \(-0.0359265\pi\)
\(278\) 0 0
\(279\) 1.67289i 0.100153i
\(280\) 0 0
\(281\) −23.3772 −1.39457 −0.697283 0.716796i \(-0.745608\pi\)
−0.697283 + 0.716796i \(0.745608\pi\)
\(282\) 0 0
\(283\) 13.9107i 0.826906i −0.910525 0.413453i \(-0.864323\pi\)
0.910525 0.413453i \(-0.135677\pi\)
\(284\) 0 0
\(285\) 0.305915 0.0181209
\(286\) 0 0
\(287\) 22.6649 1.33787
\(288\) 0 0
\(289\) −3.92728 + 16.5401i −0.231017 + 0.972950i
\(290\) 0 0
\(291\) −1.19675 −0.0701548
\(292\) 0 0
\(293\) −14.5996 −0.852916 −0.426458 0.904507i \(-0.640239\pi\)
−0.426458 + 0.904507i \(0.640239\pi\)
\(294\) 0 0
\(295\) 0.454269i 0.0264486i
\(296\) 0 0
\(297\) −1.89217 −0.109795
\(298\) 0 0
\(299\) 6.17267i 0.356975i
\(300\) 0 0
\(301\) 35.1090i 2.02365i
\(302\) 0 0
\(303\) 1.33221i 0.0765332i
\(304\) 0 0
\(305\) 2.99025 0.171221
\(306\) 0 0
\(307\) 14.6278 0.834854 0.417427 0.908710i \(-0.362932\pi\)
0.417427 + 0.908710i \(0.362932\pi\)
\(308\) 0 0
\(309\) 0.323666i 0.0184127i
\(310\) 0 0
\(311\) 23.1670i 1.31368i −0.754030 0.656840i \(-0.771892\pi\)
0.754030 0.656840i \(-0.228108\pi\)
\(312\) 0 0
\(313\) 30.7854i 1.74009i 0.492968 + 0.870047i \(0.335912\pi\)
−0.492968 + 0.870047i \(0.664088\pi\)
\(314\) 0 0
\(315\) −4.82794 −0.272024
\(316\) 0 0
\(317\) 4.28361i 0.240591i 0.992738 + 0.120296i \(0.0383843\pi\)
−0.992738 + 0.120296i \(0.961616\pi\)
\(318\) 0 0
\(319\) −26.3528 −1.47547
\(320\) 0 0
\(321\) 1.75163 0.0977664
\(322\) 0 0
\(323\) −13.9359 17.6322i −0.775413 0.981085i
\(324\) 0 0
\(325\) 5.07680 0.281610
\(326\) 0 0
\(327\) −0.348597 −0.0192774
\(328\) 0 0
\(329\) 14.4417i 0.796194i
\(330\) 0 0
\(331\) 4.61534 0.253682 0.126841 0.991923i \(-0.459516\pi\)
0.126841 + 0.991923i \(0.459516\pi\)
\(332\) 0 0
\(333\) 31.3380i 1.71731i
\(334\) 0 0
\(335\) 1.34143i 0.0732901i
\(336\) 0 0
\(337\) 4.81381i 0.262225i 0.991368 + 0.131112i \(0.0418549\pi\)
−0.991368 + 0.131112i \(0.958145\pi\)
\(338\) 0 0
\(339\) −0.918426 −0.0498821
\(340\) 0 0
\(341\) 1.43435 0.0776744
\(342\) 0 0
\(343\) 4.70383i 0.253983i
\(344\) 0 0
\(345\) 0.327102i 0.0176106i
\(346\) 0 0
\(347\) 28.7989i 1.54600i 0.634404 + 0.773002i \(0.281246\pi\)
−0.634404 + 0.773002i \(0.718754\pi\)
\(348\) 0 0
\(349\) 1.14168 0.0611125 0.0305563 0.999533i \(-0.490272\pi\)
0.0305563 + 0.999533i \(0.490272\pi\)
\(350\) 0 0
\(351\) 0.783053i 0.0417963i
\(352\) 0 0
\(353\) −6.61788 −0.352234 −0.176117 0.984369i \(-0.556354\pi\)
−0.176117 + 0.984369i \(0.556354\pi\)
\(354\) 0 0
\(355\) −2.00589 −0.106462
\(356\) 0 0
\(357\) −1.12469 1.42300i −0.0595247 0.0753131i
\(358\) 0 0
\(359\) 22.4815 1.18653 0.593263 0.805009i \(-0.297839\pi\)
0.593263 + 0.805009i \(0.297839\pi\)
\(360\) 0 0
\(361\) 10.7121 0.563792
\(362\) 0 0
\(363\) 0.549871i 0.0288607i
\(364\) 0 0
\(365\) −3.20182 −0.167591
\(366\) 0 0
\(367\) 0.731676i 0.0381932i 0.999818 + 0.0190966i \(0.00607900\pi\)
−0.999818 + 0.0190966i \(0.993921\pi\)
\(368\) 0 0
\(369\) 18.9985i 0.989020i
\(370\) 0 0
\(371\) 37.0228i 1.92213i
\(372\) 0 0
\(373\) 18.2319 0.944013 0.472006 0.881595i \(-0.343530\pi\)
0.472006 + 0.881595i \(0.343530\pi\)
\(374\) 0 0
\(375\) −0.549641 −0.0283833
\(376\) 0 0
\(377\) 10.9058i 0.561678i
\(378\) 0 0
\(379\) 17.1721i 0.882071i 0.897490 + 0.441036i \(0.145389\pi\)
−0.897490 + 0.441036i \(0.854611\pi\)
\(380\) 0 0
\(381\) 2.35401i 0.120600i
\(382\) 0 0
\(383\) −8.19399 −0.418693 −0.209347 0.977841i \(-0.567134\pi\)
−0.209347 + 0.977841i \(0.567134\pi\)
\(384\) 0 0
\(385\) 4.13951i 0.210969i
\(386\) 0 0
\(387\) −29.4294 −1.49598
\(388\) 0 0
\(389\) −5.87389 −0.297818 −0.148909 0.988851i \(-0.547576\pi\)
−0.148909 + 0.988851i \(0.547576\pi\)
\(390\) 0 0
\(391\) −18.8534 + 14.9010i −0.953458 + 0.753578i
\(392\) 0 0
\(393\) 2.15546 0.108728
\(394\) 0 0
\(395\) 3.62012 0.182148
\(396\) 0 0
\(397\) 14.9831i 0.751982i −0.926623 0.375991i \(-0.877302\pi\)
0.926623 0.375991i \(-0.122698\pi\)
\(398\) 0 0
\(399\) 2.39789 0.120045
\(400\) 0 0
\(401\) 11.5382i 0.576188i −0.957602 0.288094i \(-0.906978\pi\)
0.957602 0.288094i \(-0.0930216\pi\)
\(402\) 0 0
\(403\) 0.593589i 0.0295688i
\(404\) 0 0
\(405\) 4.02613i 0.200060i
\(406\) 0 0
\(407\) 26.8695 1.33187
\(408\) 0 0
\(409\) −15.2899 −0.756039 −0.378020 0.925798i \(-0.623395\pi\)
−0.378020 + 0.925798i \(0.623395\pi\)
\(410\) 0 0
\(411\) 1.87994i 0.0927306i
\(412\) 0 0
\(413\) 3.56076i 0.175213i
\(414\) 0 0
\(415\) 5.39049i 0.264609i
\(416\) 0 0
\(417\) 1.53014 0.0749313
\(418\) 0 0
\(419\) 18.7295i 0.914997i −0.889210 0.457499i \(-0.848745\pi\)
0.889210 0.457499i \(-0.151255\pi\)
\(420\) 0 0
\(421\) 37.8151 1.84300 0.921498 0.388384i \(-0.126967\pi\)
0.921498 + 0.388384i \(0.126967\pi\)
\(422\) 0 0
\(423\) −12.1054 −0.588587
\(424\) 0 0
\(425\) 12.2556 + 15.5062i 0.594482 + 0.752164i
\(426\) 0 0
\(427\) 23.4388 1.13428
\(428\) 0 0
\(429\) −0.334842 −0.0161663
\(430\) 0 0
\(431\) 1.06536i 0.0513164i 0.999671 + 0.0256582i \(0.00816815\pi\)
−0.999671 + 0.0256582i \(0.991832\pi\)
\(432\) 0 0
\(433\) −0.599855 −0.0288272 −0.0144136 0.999896i \(-0.504588\pi\)
−0.0144136 + 0.999896i \(0.504588\pi\)
\(434\) 0 0
\(435\) 0.577921i 0.0277092i
\(436\) 0 0
\(437\) 31.7698i 1.51976i
\(438\) 0 0
\(439\) 5.56664i 0.265681i 0.991137 + 0.132841i \(0.0424098\pi\)
−0.991137 + 0.132841i \(0.957590\pi\)
\(440\) 0 0
\(441\) −16.9503 −0.807155
\(442\) 0 0
\(443\) 33.0999 1.57262 0.786311 0.617831i \(-0.211988\pi\)
0.786311 + 0.617831i \(0.211988\pi\)
\(444\) 0 0
\(445\) 8.33012i 0.394886i
\(446\) 0 0
\(447\) 0.890944i 0.0421402i
\(448\) 0 0
\(449\) 6.85503i 0.323509i 0.986831 + 0.161754i \(0.0517152\pi\)
−0.986831 + 0.161754i \(0.948285\pi\)
\(450\) 0 0
\(451\) 16.2894 0.767040
\(452\) 0 0
\(453\) 2.46605i 0.115865i
\(454\) 0 0
\(455\) −1.71309 −0.0803109
\(456\) 0 0
\(457\) 4.38315 0.205035 0.102518 0.994731i \(-0.467310\pi\)
0.102518 + 0.994731i \(0.467310\pi\)
\(458\) 0 0
\(459\) −2.39171 + 1.89032i −0.111635 + 0.0882324i
\(460\) 0 0
\(461\) 32.2461 1.50185 0.750925 0.660388i \(-0.229608\pi\)
0.750925 + 0.660388i \(0.229608\pi\)
\(462\) 0 0
\(463\) −20.5847 −0.956651 −0.478325 0.878183i \(-0.658756\pi\)
−0.478325 + 0.878183i \(0.658756\pi\)
\(464\) 0 0
\(465\) 0.0314555i 0.00145871i
\(466\) 0 0
\(467\) 10.4467 0.483416 0.241708 0.970349i \(-0.422292\pi\)
0.241708 + 0.970349i \(0.422292\pi\)
\(468\) 0 0
\(469\) 10.5147i 0.485523i
\(470\) 0 0
\(471\) 1.97555i 0.0910284i
\(472\) 0 0
\(473\) 25.2330i 1.16022i
\(474\) 0 0
\(475\) −26.1295 −1.19891
\(476\) 0 0
\(477\) 31.0336 1.42093
\(478\) 0 0
\(479\) 35.1744i 1.60716i 0.595196 + 0.803581i \(0.297074\pi\)
−0.595196 + 0.803581i \(0.702926\pi\)
\(480\) 0 0
\(481\) 11.1196i 0.507012i
\(482\) 0 0
\(483\) 2.56396i 0.116664i
\(484\) 0 0
\(485\) −4.40044 −0.199814
\(486\) 0 0
\(487\) 16.9137i 0.766434i 0.923658 + 0.383217i \(0.125184\pi\)
−0.923658 + 0.383217i \(0.874816\pi\)
\(488\) 0 0
\(489\) 0.244756 0.0110682
\(490\) 0 0
\(491\) −33.5236 −1.51290 −0.756449 0.654052i \(-0.773068\pi\)
−0.756449 + 0.654052i \(0.773068\pi\)
\(492\) 0 0
\(493\) −33.3100 + 26.3270i −1.50021 + 1.18571i
\(494\) 0 0
\(495\) −3.46987 −0.155959
\(496\) 0 0
\(497\) −15.7230 −0.705274
\(498\) 0 0
\(499\) 5.60167i 0.250765i 0.992108 + 0.125383i \(0.0400159\pi\)
−0.992108 + 0.125383i \(0.959984\pi\)
\(500\) 0 0
\(501\) −0.390825 −0.0174608
\(502\) 0 0
\(503\) 33.9719i 1.51473i −0.652990 0.757367i \(-0.726486\pi\)
0.652990 0.757367i \(-0.273514\pi\)
\(504\) 0 0
\(505\) 4.89850i 0.217981i
\(506\) 0 0
\(507\) 1.46750i 0.0651740i
\(508\) 0 0
\(509\) −39.6811 −1.75883 −0.879417 0.476052i \(-0.842067\pi\)
−0.879417 + 0.476052i \(0.842067\pi\)
\(510\) 0 0
\(511\) −25.0972 −1.11023
\(512\) 0 0
\(513\) 4.03026i 0.177940i
\(514\) 0 0
\(515\) 1.19012i 0.0524428i
\(516\) 0 0
\(517\) 10.3793i 0.456482i
\(518\) 0 0
\(519\) 2.14924 0.0943413
\(520\) 0 0
\(521\) 39.9053i 1.74828i −0.485671 0.874142i \(-0.661425\pi\)
0.485671 0.874142i \(-0.338575\pi\)
\(522\) 0 0
\(523\) −4.43348 −0.193863 −0.0969314 0.995291i \(-0.530903\pi\)
−0.0969314 + 0.995291i \(0.530903\pi\)
\(524\) 0 0
\(525\) −2.10877 −0.0920342
\(526\) 0 0
\(527\) 1.81302 1.43294i 0.0789764 0.0624200i
\(528\) 0 0
\(529\) −10.9701 −0.476962
\(530\) 0 0
\(531\) 2.98474 0.129527
\(532\) 0 0
\(533\) 6.74120i 0.291994i
\(534\) 0 0
\(535\) 6.44072 0.278457
\(536\) 0 0
\(537\) 1.55354i 0.0670404i
\(538\) 0 0
\(539\) 14.5333i 0.625993i
\(540\) 0 0
\(541\) 22.5883i 0.971149i −0.874195 0.485574i \(-0.838611\pi\)
0.874195 0.485574i \(-0.161389\pi\)
\(542\) 0 0
\(543\) −2.82710 −0.121322
\(544\) 0 0
\(545\) −1.28179 −0.0549057
\(546\) 0 0
\(547\) 19.1095i 0.817062i 0.912744 + 0.408531i \(0.133959\pi\)
−0.912744 + 0.408531i \(0.866041\pi\)
\(548\) 0 0
\(549\) 19.6472i 0.838521i
\(550\) 0 0
\(551\) 56.1306i 2.39124i
\(552\) 0 0
\(553\) 28.3761 1.20667
\(554\) 0 0
\(555\) 0.589252i 0.0250123i
\(556\) 0 0
\(557\) −6.81109 −0.288595 −0.144298 0.989534i \(-0.546092\pi\)
−0.144298 + 0.989534i \(0.546092\pi\)
\(558\) 0 0
\(559\) −10.4424 −0.441667
\(560\) 0 0
\(561\) −0.808319 1.02272i −0.0341273 0.0431792i
\(562\) 0 0
\(563\) −15.6517 −0.659640 −0.329820 0.944044i \(-0.606988\pi\)
−0.329820 + 0.944044i \(0.606988\pi\)
\(564\) 0 0
\(565\) −3.37704 −0.142073
\(566\) 0 0
\(567\) 31.5585i 1.32533i
\(568\) 0 0
\(569\) −34.9961 −1.46711 −0.733557 0.679628i \(-0.762141\pi\)
−0.733557 + 0.679628i \(0.762141\pi\)
\(570\) 0 0
\(571\) 24.3105i 1.01736i −0.860955 0.508682i \(-0.830133\pi\)
0.860955 0.508682i \(-0.169867\pi\)
\(572\) 0 0
\(573\) 1.43145i 0.0597995i
\(574\) 0 0
\(575\) 27.9392i 1.16515i
\(576\) 0 0
\(577\) 32.1976 1.34040 0.670202 0.742178i \(-0.266207\pi\)
0.670202 + 0.742178i \(0.266207\pi\)
\(578\) 0 0
\(579\) −1.59237 −0.0661766
\(580\) 0 0
\(581\) 42.2529i 1.75295i
\(582\) 0 0
\(583\) 26.6085i 1.10201i
\(584\) 0 0
\(585\) 1.43597i 0.0593699i
\(586\) 0 0
\(587\) 31.7249 1.30943 0.654713 0.755878i \(-0.272789\pi\)
0.654713 + 0.755878i \(0.272789\pi\)
\(588\) 0 0
\(589\) 3.05512i 0.125884i
\(590\) 0 0
\(591\) −2.65332 −0.109143
\(592\) 0 0
\(593\) 30.6442 1.25841 0.629204 0.777240i \(-0.283381\pi\)
0.629204 + 0.777240i \(0.283381\pi\)
\(594\) 0 0
\(595\) −4.13546 5.23235i −0.169537 0.214506i
\(596\) 0 0
\(597\) −1.09585 −0.0448503
\(598\) 0 0
\(599\) 40.4039 1.65086 0.825429 0.564506i \(-0.190933\pi\)
0.825429 + 0.564506i \(0.190933\pi\)
\(600\) 0 0
\(601\) 7.46970i 0.304695i −0.988327 0.152348i \(-0.951317\pi\)
0.988327 0.152348i \(-0.0486833\pi\)
\(602\) 0 0
\(603\) −8.81374 −0.358923
\(604\) 0 0
\(605\) 2.02187i 0.0822006i
\(606\) 0 0
\(607\) 21.4720i 0.871521i −0.900063 0.435761i \(-0.856479\pi\)
0.900063 0.435761i \(-0.143521\pi\)
\(608\) 0 0
\(609\) 4.52999i 0.183564i
\(610\) 0 0
\(611\) −4.29536 −0.173772
\(612\) 0 0
\(613\) 33.6928 1.36084 0.680421 0.732822i \(-0.261797\pi\)
0.680421 + 0.732822i \(0.261797\pi\)
\(614\) 0 0
\(615\) 0.357229i 0.0144049i
\(616\) 0 0
\(617\) 17.4231i 0.701428i −0.936483 0.350714i \(-0.885939\pi\)
0.936483 0.350714i \(-0.114061\pi\)
\(618\) 0 0
\(619\) 35.5804i 1.43010i 0.699075 + 0.715049i \(0.253595\pi\)
−0.699075 + 0.715049i \(0.746405\pi\)
\(620\) 0 0
\(621\) −4.30938 −0.172930
\(622\) 0 0
\(623\) 65.2950i 2.61599i
\(624\) 0 0
\(625\) 21.9472 0.877887
\(626\) 0 0
\(627\) 1.72338 0.0688252
\(628\) 0 0
\(629\) 33.9631 26.8432i 1.35420 1.07031i
\(630\) 0 0
\(631\) 13.4260 0.534479 0.267240 0.963630i \(-0.413889\pi\)
0.267240 + 0.963630i \(0.413889\pi\)
\(632\) 0 0
\(633\) −2.95966 −0.117636
\(634\) 0 0
\(635\) 8.65568i 0.343490i
\(636\) 0 0
\(637\) −6.01444 −0.238301
\(638\) 0 0
\(639\) 13.1795i 0.521374i
\(640\) 0 0
\(641\) 1.17967i 0.0465941i 0.999729 + 0.0232971i \(0.00741635\pi\)
−0.999729 + 0.0232971i \(0.992584\pi\)
\(642\) 0 0
\(643\) 42.4219i 1.67296i −0.547999 0.836479i \(-0.684610\pi\)
0.547999 0.836479i \(-0.315390\pi\)
\(644\) 0 0
\(645\) 0.553364 0.0217887
\(646\) 0 0
\(647\) 43.4812 1.70942 0.854711 0.519105i \(-0.173734\pi\)
0.854711 + 0.519105i \(0.173734\pi\)
\(648\) 0 0
\(649\) 2.55914i 0.100455i
\(650\) 0 0
\(651\) 0.246561i 0.00966350i
\(652\) 0 0
\(653\) 6.75483i 0.264337i −0.991227 0.132168i \(-0.957806\pi\)
0.991227 0.132168i \(-0.0421940\pi\)
\(654\) 0 0
\(655\) 7.92558 0.309678
\(656\) 0 0
\(657\) 21.0372i 0.820741i
\(658\) 0 0
\(659\) 17.1645 0.668635 0.334317 0.942461i \(-0.391494\pi\)
0.334317 + 0.942461i \(0.391494\pi\)
\(660\) 0 0
\(661\) −40.4195 −1.57214 −0.786068 0.618140i \(-0.787886\pi\)
−0.786068 + 0.618140i \(0.787886\pi\)
\(662\) 0 0
\(663\) −0.423241 + 0.334514i −0.0164373 + 0.0129914i
\(664\) 0 0
\(665\) 8.81702 0.341910
\(666\) 0 0
\(667\) −60.0180 −2.32391
\(668\) 0 0
\(669\) 2.05515i 0.0794568i
\(670\) 0 0
\(671\) 16.8456 0.650319
\(672\) 0 0
\(673\) 18.7981i 0.724612i −0.932059 0.362306i \(-0.881990\pi\)
0.932059 0.362306i \(-0.118010\pi\)
\(674\) 0 0
\(675\) 3.54431i 0.136421i
\(676\) 0 0
\(677\) 31.3773i 1.20593i −0.797769 0.602963i \(-0.793987\pi\)
0.797769 0.602963i \(-0.206013\pi\)
\(678\) 0 0
\(679\) −34.4925 −1.32370
\(680\) 0 0
\(681\) −1.34034 −0.0513619
\(682\) 0 0
\(683\) 13.9333i 0.533141i −0.963815 0.266571i \(-0.914109\pi\)
0.963815 0.266571i \(-0.0858905\pi\)
\(684\) 0 0
\(685\) 6.91252i 0.264114i
\(686\) 0 0
\(687\) 1.09962i 0.0419529i
\(688\) 0 0
\(689\) 11.0116 0.419509
\(690\) 0 0
\(691\) 27.0868i 1.03043i 0.857061 + 0.515215i \(0.172288\pi\)
−0.857061 + 0.515215i \(0.827712\pi\)
\(692\) 0 0
\(693\) −27.1983 −1.03318
\(694\) 0 0
\(695\) 5.62631 0.213418
\(696\) 0 0
\(697\) 20.5899 16.2735i 0.779897 0.616402i
\(698\) 0 0
\(699\) 0.584503 0.0221079
\(700\) 0 0
\(701\) 7.57170 0.285979 0.142990 0.989724i \(-0.454328\pi\)
0.142990 + 0.989724i \(0.454328\pi\)
\(702\) 0 0
\(703\) 57.2311i 2.15851i
\(704\) 0 0
\(705\) 0.227620 0.00857265
\(706\) 0 0
\(707\) 38.3965i 1.44405i
\(708\) 0 0
\(709\) 2.50772i 0.0941793i 0.998891 + 0.0470896i \(0.0149946\pi\)
−0.998891 + 0.0470896i \(0.985005\pi\)
\(710\) 0 0
\(711\) 23.7857i 0.892033i
\(712\) 0 0
\(713\) 3.26670 0.122339
\(714\) 0 0
\(715\) −1.23121 −0.0460446
\(716\) 0 0
\(717\) 0.0265386i 0.000991102i
\(718\) 0 0
\(719\) 51.2654i 1.91188i 0.293570 + 0.955938i \(0.405157\pi\)
−0.293570 + 0.955938i \(0.594843\pi\)
\(720\) 0 0
\(721\) 9.32864i 0.347417i
\(722\) 0 0
\(723\) −2.17081 −0.0807333
\(724\) 0 0
\(725\) 49.3627i 1.83328i
\(726\) 0 0
\(727\) 35.3839 1.31232 0.656159 0.754623i \(-0.272180\pi\)
0.656159 + 0.754623i \(0.272180\pi\)
\(728\) 0 0
\(729\) 26.1793 0.969603
\(730\) 0 0
\(731\) −25.2083 31.8946i −0.932364 1.17967i
\(732\) 0 0
\(733\) −46.5913 −1.72089 −0.860444 0.509546i \(-0.829814\pi\)
−0.860444 + 0.509546i \(0.829814\pi\)
\(734\) 0 0
\(735\) 0.318717 0.0117561
\(736\) 0 0
\(737\) 7.55697i 0.278365i
\(738\) 0 0
\(739\) −50.1732 −1.84565 −0.922826 0.385216i \(-0.874127\pi\)
−0.922826 + 0.385216i \(0.874127\pi\)
\(740\) 0 0
\(741\) 0.713202i 0.0262001i
\(742\) 0 0
\(743\) 21.6397i 0.793885i −0.917844 0.396942i \(-0.870071\pi\)
0.917844 0.396942i \(-0.129929\pi\)
\(744\) 0 0
\(745\) 3.27599i 0.120023i
\(746\) 0 0
\(747\) 35.4177 1.29587
\(748\) 0 0
\(749\) 50.4851 1.84469
\(750\) 0 0
\(751\) 30.4336i 1.11054i −0.831671 0.555268i \(-0.812616\pi\)
0.831671 0.555268i \(-0.187384\pi\)
\(752\) 0 0
\(753\) 2.76305i 0.100691i
\(754\) 0 0
\(755\) 9.06764i 0.330005i
\(756\) 0 0
\(757\) 34.3562 1.24870 0.624349 0.781145i \(-0.285364\pi\)
0.624349 + 0.781145i \(0.285364\pi\)
\(758\) 0 0
\(759\) 1.84274i 0.0668872i
\(760\) 0 0
\(761\) 25.6300 0.929089 0.464544 0.885550i \(-0.346218\pi\)
0.464544 + 0.885550i \(0.346218\pi\)
\(762\) 0 0
\(763\) −10.0472 −0.363732
\(764\) 0 0
\(765\) −4.38592 + 3.46647i −0.158573 + 0.125331i
\(766\) 0 0
\(767\) 1.05907 0.0382408
\(768\) 0 0
\(769\) −20.9528 −0.755579 −0.377789 0.925892i \(-0.623316\pi\)
−0.377789 + 0.925892i \(0.623316\pi\)
\(770\) 0 0
\(771\) 0.784322i 0.0282467i
\(772\) 0 0
\(773\) 31.1791 1.12143 0.560717 0.828008i \(-0.310526\pi\)
0.560717 + 0.828008i \(0.310526\pi\)
\(774\) 0 0
\(775\) 2.68675i 0.0965108i
\(776\) 0 0
\(777\) 4.61880i 0.165699i
\(778\) 0 0
\(779\) 34.6959i 1.24311i
\(780\) 0 0
\(781\) −11.3002 −0.404354
\(782\) 0 0
\(783\) −7.61377 −0.272094
\(784\) 0 0
\(785\) 7.26407i 0.259266i
\(786\) 0 0
\(787\) 4.50061i 0.160429i −0.996778 0.0802147i \(-0.974439\pi\)
0.996778 0.0802147i \(-0.0255606\pi\)
\(788\) 0 0
\(789\) 0.168771i 0.00600841i
\(790\) 0 0
\(791\) −26.4707 −0.941189
\(792\) 0 0
\(793\) 6.97138i 0.247561i
\(794\) 0 0
\(795\) −0.583528 −0.0206956
\(796\) 0 0
\(797\) 24.4829 0.867229 0.433614 0.901098i \(-0.357238\pi\)
0.433614 + 0.901098i \(0.357238\pi\)
\(798\) 0 0
\(799\) −10.3691 13.1195i −0.366834 0.464133i
\(800\) 0 0
\(801\) −54.7323 −1.93387
\(802\) 0 0
\(803\) −18.0375 −0.636530
\(804\) 0 0
\(805\) 9.42767i 0.332282i
\(806\) 0 0
\(807\) 3.71394 0.130737
\(808\) 0 0
\(809\) 3.84717i 0.135259i −0.997710 0.0676296i \(-0.978456\pi\)
0.997710 0.0676296i \(-0.0215436\pi\)
\(810\) 0 0
\(811\) 17.2547i 0.605896i −0.953007 0.302948i \(-0.902029\pi\)
0.953007 0.302948i \(-0.0979709\pi\)
\(812\) 0 0
\(813\) 3.30644i 0.115962i
\(814\) 0 0
\(815\) 0.899963 0.0315243
\(816\) 0 0
\(817\) 53.7455 1.88032
\(818\) 0 0
\(819\) 11.2557i 0.393306i
\(820\) 0 0
\(821\) 10.2125i 0.356418i −0.983993 0.178209i \(-0.942970\pi\)
0.983993 0.178209i \(-0.0570304\pi\)
\(822\) 0 0
\(823\) 0.129049i 0.00449837i −0.999997 0.00224918i \(-0.999284\pi\)
0.999997 0.00224918i \(-0.000715938\pi\)
\(824\) 0 0
\(825\) −1.51559 −0.0527659
\(826\) 0 0
\(827\) 3.86396i 0.134363i −0.997741 0.0671816i \(-0.978599\pi\)
0.997741 0.0671816i \(-0.0214007\pi\)
\(828\) 0 0
\(829\) −20.8365 −0.723683 −0.361841 0.932240i \(-0.617852\pi\)
−0.361841 + 0.932240i \(0.617852\pi\)
\(830\) 0 0
\(831\) 0.463163 0.0160669
\(832\) 0 0
\(833\) −14.5191 18.3701i −0.503055 0.636487i
\(834\) 0 0
\(835\) −1.43706 −0.0497314
\(836\) 0 0
\(837\) 0.414408 0.0143240
\(838\) 0 0
\(839\) 16.1733i 0.558365i −0.960238 0.279182i \(-0.909937\pi\)
0.960238 0.279182i \(-0.0900634\pi\)
\(840\) 0 0
\(841\) −77.0392 −2.65653
\(842\) 0 0
\(843\) 2.88811i 0.0994717i
\(844\) 0 0
\(845\) 5.39598i 0.185627i
\(846\) 0 0
\(847\) 15.8483i 0.544553i
\(848\) 0 0
\(849\) 1.71858 0.0589817
\(850\) 0 0
\(851\) 61.1948 2.09773
\(852\) 0 0
\(853\) 2.41135i 0.0825629i 0.999148 + 0.0412815i \(0.0131440\pi\)
−0.999148 + 0.0412815i \(0.986856\pi\)
\(854\) 0 0
\(855\) 7.39071i 0.252757i
\(856\) 0 0
\(857\) 24.1069i 0.823475i 0.911303 + 0.411737i \(0.135078\pi\)
−0.911303 + 0.411737i \(0.864922\pi\)
\(858\) 0 0
\(859\) 32.4368 1.10673 0.553365 0.832939i \(-0.313344\pi\)
0.553365 + 0.832939i \(0.313344\pi\)
\(860\) 0 0
\(861\) 2.80011i 0.0954277i
\(862\) 0 0
\(863\) −4.27350 −0.145472 −0.0727359 0.997351i \(-0.523173\pi\)
−0.0727359 + 0.997351i \(0.523173\pi\)
\(864\) 0 0
\(865\) 7.90274 0.268701
\(866\) 0 0
\(867\) −2.04343 0.485192i −0.0693987 0.0164780i
\(868\) 0 0
\(869\) 20.3941 0.691821
\(870\) 0 0
\(871\) −3.12737 −0.105967
\(872\) 0 0
\(873\) 28.9127i 0.978547i
\(874\) 0 0
\(875\) −15.8416 −0.535545
\(876\) 0 0
\(877\) 20.1177i 0.679326i −0.940547 0.339663i \(-0.889687\pi\)
0.940547 0.339663i \(-0.110313\pi\)
\(878\) 0 0
\(879\) 1.80369i 0.0608369i
\(880\) 0 0
\(881\) 28.5492i 0.961846i −0.876763 0.480923i \(-0.840302\pi\)
0.876763 0.480923i \(-0.159698\pi\)
\(882\) 0 0
\(883\) 27.3231 0.919496 0.459748 0.888049i \(-0.347940\pi\)
0.459748 + 0.888049i \(0.347940\pi\)
\(884\) 0 0
\(885\) −0.0561222 −0.00188653
\(886\) 0 0
\(887\) 12.4943i 0.419518i −0.977753 0.209759i \(-0.932732\pi\)
0.977753 0.209759i \(-0.0672679\pi\)
\(888\) 0 0
\(889\) 67.8469i 2.27551i
\(890\) 0 0
\(891\) 22.6813i 0.759852i
\(892\) 0 0
\(893\) 22.1076 0.739802
\(894\) 0 0
\(895\) 5.71236i 0.190943i
\(896\) 0 0
\(897\) −0.762596 −0.0254623
\(898\) 0 0
\(899\) 5.77158 0.192493
\(900\) 0 0
\(901\) 26.5824 + 33.6332i 0.885589 + 1.12048i
\(902\) 0 0
\(903\) 4.33750 0.144343
\(904\) 0 0
\(905\) −10.3952 −0.345549
\(906\) 0 0
\(907\) 42.6576i 1.41642i −0.706000 0.708212i \(-0.749502\pi\)
0.706000 0.708212i \(-0.250498\pi\)
\(908\) 0 0
\(909\) 32.1852 1.06751
\(910\) 0 0
\(911\) 52.7572i 1.74792i 0.485994 + 0.873962i \(0.338458\pi\)
−0.485994 + 0.873962i \(0.661542\pi\)
\(912\) 0 0
\(913\) 30.3675i 1.00502i
\(914\) 0 0
\(915\) 0.369427i 0.0122129i
\(916\) 0 0
\(917\) 62.1241 2.05152
\(918\) 0 0
\(919\) −39.1272 −1.29069 −0.645344 0.763892i \(-0.723286\pi\)
−0.645344 + 0.763892i \(0.723286\pi\)
\(920\) 0 0
\(921\) 1.80718i 0.0595486i
\(922\) 0 0
\(923\) 4.67647i 0.153928i
\(924\) 0 0
\(925\) 50.3305i 1.65486i
\(926\) 0 0
\(927\) −7.81956 −0.256828
\(928\) 0 0
\(929\) 36.7996i 1.20736i −0.797228 0.603678i \(-0.793701\pi\)
0.797228 0.603678i \(-0.206299\pi\)
\(930\) 0 0
\(931\) 30.9554 1.01452
\(932\) 0 0
\(933\) 2.86214 0.0937024
\(934\) 0 0
\(935\) −2.97218 3.76053i −0.0972007 0.122982i
\(936\) 0 0
\(937\) 14.1198 0.461274 0.230637 0.973040i \(-0.425919\pi\)
0.230637 + 0.973040i \(0.425919\pi\)
\(938\) 0 0
\(939\) −3.80335 −0.124118
\(940\) 0 0
\(941\) 18.5654i 0.605215i −0.953115 0.302608i \(-0.902143\pi\)
0.953115 0.302608i \(-0.0978572\pi\)
\(942\) 0 0
\(943\) 37.0989 1.20811
\(944\) 0 0
\(945\) 1.19598i 0.0389051i
\(946\) 0 0
\(947\) 22.6296i 0.735364i 0.929952 + 0.367682i \(0.119849\pi\)
−0.929952 + 0.367682i \(0.880151\pi\)
\(948\) 0 0
\(949\) 7.46461i 0.242312i
\(950\) 0 0
\(951\) −0.529214 −0.0171609
\(952\) 0 0
\(953\) −15.2691 −0.494614 −0.247307 0.968937i \(-0.579546\pi\)
−0.247307 + 0.968937i \(0.579546\pi\)
\(954\) 0 0
\(955\) 5.26341i 0.170320i
\(956\) 0 0
\(957\) 3.25573i 0.105243i
\(958\) 0 0
\(959\) 54.1832i 1.74967i
\(960\) 0 0
\(961\) 30.6859 0.989866
\(962\) 0 0
\(963\) 42.3182i 1.36368i
\(964\) 0 0
\(965\) −5.85512 −0.188483
\(966\) 0 0
\(967\) 28.5811 0.919105 0.459552 0.888151i \(-0.348010\pi\)
0.459552 + 0.888151i \(0.348010\pi\)
\(968\) 0 0
\(969\) 2.17836 1.72169i 0.0699789 0.0553087i
\(970\) 0 0
\(971\) 52.1479 1.67351 0.836754 0.547580i \(-0.184451\pi\)
0.836754 + 0.547580i \(0.184451\pi\)
\(972\) 0 0
\(973\) 44.1014 1.41382
\(974\) 0 0
\(975\) 0.627208i 0.0200867i
\(976\) 0 0
\(977\) 4.83514 0.154690 0.0773449 0.997004i \(-0.475356\pi\)
0.0773449 + 0.997004i \(0.475356\pi\)
\(978\) 0 0
\(979\) 46.9279i 1.49982i
\(980\) 0 0
\(981\) 8.42186i 0.268889i
\(982\) 0 0
\(983\) 3.02318i 0.0964245i 0.998837 + 0.0482123i \(0.0153524\pi\)
−0.998837 + 0.0482123i \(0.984648\pi\)
\(984\) 0 0
\(985\) −9.75624 −0.310860
\(986\) 0 0
\(987\) 1.78418 0.0567910
\(988\) 0 0
\(989\) 57.4678i 1.82737i
\(990\) 0 0
\(991\) 3.84056i 0.121999i 0.998138 + 0.0609997i \(0.0194289\pi\)
−0.998138 + 0.0609997i \(0.980571\pi\)
\(992\) 0 0
\(993\) 0.570197i 0.0180947i
\(994\) 0 0
\(995\) −4.02944 −0.127742
\(996\) 0 0
\(997\) 22.1552i 0.701663i 0.936439 + 0.350832i \(0.114101\pi\)
−0.936439 + 0.350832i \(0.885899\pi\)
\(998\) 0 0
\(999\) 7.76305 0.245612
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.237.24 yes 46
17.16 even 2 inner 4012.2.b.b.237.23 46
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.b.b.237.23 46 17.16 even 2 inner
4012.2.b.b.237.24 yes 46 1.1 even 1 trivial