Properties

Label 4014.2.d.a.4013.18
Level $4014$
Weight $2$
Character 4014.4013
Analytic conductor $32.052$
Analytic rank $0$
Dimension $72$
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] = [4014,2,Mod(4013,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4014.4013");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.d (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: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4013.18
Character \(\chi\) \(=\) 4014.4013
Dual form 4014.2.d.a.4013.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +0.707934 q^{5} +0.916361 q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +0.707934 q^{5} +0.916361 q^{7} -1.00000i q^{8} +0.707934i q^{10} +4.30663 q^{11} -4.29049i q^{13} +0.916361i q^{14} +1.00000 q^{16} -2.79279i q^{17} +6.09016 q^{19} -0.707934 q^{20} +4.30663i q^{22} +0.682495 q^{23} -4.49883 q^{25} +4.29049 q^{26} -0.916361 q^{28} -3.21724i q^{29} -0.315439 q^{31} +1.00000i q^{32} +2.79279 q^{34} +0.648723 q^{35} -4.22523 q^{37} +6.09016i q^{38} -0.707934i q^{40} +0.607558i q^{41} +1.08767 q^{43} -4.30663 q^{44} +0.682495i q^{46} -2.90784i q^{47} -6.16028 q^{49} -4.49883i q^{50} +4.29049i q^{52} -11.3029i q^{53} +3.04881 q^{55} -0.916361i q^{56} +3.21724 q^{58} +0.693451 q^{59} -9.12711i q^{61} -0.315439i q^{62} -1.00000 q^{64} -3.03738i q^{65} -12.9662i q^{67} +2.79279i q^{68} +0.648723i q^{70} -7.15131 q^{71} -2.17812 q^{73} -4.22523i q^{74} -6.09016 q^{76} +3.94643 q^{77} +13.1897i q^{79} +0.707934 q^{80} -0.607558 q^{82} +0.529002i q^{83} -1.97711i q^{85} +1.08767i q^{86} -4.30663i q^{88} -5.43684i q^{89} -3.93164i q^{91} -0.682495 q^{92} +2.90784 q^{94} +4.31143 q^{95} +13.4837i q^{97} -6.16028i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times\).

\(n\) \(893\) \(2233\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0.707934 0.316598 0.158299 0.987391i \(-0.449399\pi\)
0.158299 + 0.987391i \(0.449399\pi\)
\(6\) 0 0
\(7\) 0.916361 0.346352 0.173176 0.984891i \(-0.444597\pi\)
0.173176 + 0.984891i \(0.444597\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0.707934i 0.223868i
\(11\) 4.30663 1.29850 0.649249 0.760576i \(-0.275083\pi\)
0.649249 + 0.760576i \(0.275083\pi\)
\(12\) 0 0
\(13\) 4.29049i 1.18997i −0.803738 0.594984i \(-0.797159\pi\)
0.803738 0.594984i \(-0.202841\pi\)
\(14\) 0.916361i 0.244908i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.79279i 0.677351i −0.940903 0.338676i \(-0.890021\pi\)
0.940903 0.338676i \(-0.109979\pi\)
\(18\) 0 0
\(19\) 6.09016 1.39718 0.698590 0.715523i \(-0.253811\pi\)
0.698590 + 0.715523i \(0.253811\pi\)
\(20\) −0.707934 −0.158299
\(21\) 0 0
\(22\) 4.30663i 0.918177i
\(23\) 0.682495 0.142310 0.0711550 0.997465i \(-0.477332\pi\)
0.0711550 + 0.997465i \(0.477332\pi\)
\(24\) 0 0
\(25\) −4.49883 −0.899766
\(26\) 4.29049 0.841434
\(27\) 0 0
\(28\) −0.916361 −0.173176
\(29\) 3.21724i 0.597427i −0.954343 0.298714i \(-0.903443\pi\)
0.954343 0.298714i \(-0.0965575\pi\)
\(30\) 0 0
\(31\) −0.315439 −0.0566544 −0.0283272 0.999599i \(-0.509018\pi\)
−0.0283272 + 0.999599i \(0.509018\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 2.79279 0.478960
\(35\) 0.648723 0.109654
\(36\) 0 0
\(37\) −4.22523 −0.694623 −0.347311 0.937750i \(-0.612905\pi\)
−0.347311 + 0.937750i \(0.612905\pi\)
\(38\) 6.09016i 0.987955i
\(39\) 0 0
\(40\) 0.707934i 0.111934i
\(41\) 0.607558i 0.0948847i 0.998874 + 0.0474423i \(0.0151070\pi\)
−0.998874 + 0.0474423i \(0.984893\pi\)
\(42\) 0 0
\(43\) 1.08767 0.165869 0.0829343 0.996555i \(-0.473571\pi\)
0.0829343 + 0.996555i \(0.473571\pi\)
\(44\) −4.30663 −0.649249
\(45\) 0 0
\(46\) 0.682495i 0.100628i
\(47\) 2.90784i 0.424152i −0.977253 0.212076i \(-0.931978\pi\)
0.977253 0.212076i \(-0.0680224\pi\)
\(48\) 0 0
\(49\) −6.16028 −0.880040
\(50\) 4.49883i 0.636231i
\(51\) 0 0
\(52\) 4.29049i 0.594984i
\(53\) 11.3029i 1.55257i −0.630380 0.776287i \(-0.717101\pi\)
0.630380 0.776287i \(-0.282899\pi\)
\(54\) 0 0
\(55\) 3.04881 0.411101
\(56\) 0.916361i 0.122454i
\(57\) 0 0
\(58\) 3.21724 0.422445
\(59\) 0.693451 0.0902797 0.0451398 0.998981i \(-0.485627\pi\)
0.0451398 + 0.998981i \(0.485627\pi\)
\(60\) 0 0
\(61\) 9.12711i 1.16861i −0.811535 0.584303i \(-0.801368\pi\)
0.811535 0.584303i \(-0.198632\pi\)
\(62\) 0.315439i 0.0400607i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 3.03738i 0.376741i
\(66\) 0 0
\(67\) 12.9662i 1.58407i −0.610476 0.792035i \(-0.709022\pi\)
0.610476 0.792035i \(-0.290978\pi\)
\(68\) 2.79279i 0.338676i
\(69\) 0 0
\(70\) 0.648723i 0.0775373i
\(71\) −7.15131 −0.848704 −0.424352 0.905497i \(-0.639498\pi\)
−0.424352 + 0.905497i \(0.639498\pi\)
\(72\) 0 0
\(73\) −2.17812 −0.254930 −0.127465 0.991843i \(-0.540684\pi\)
−0.127465 + 0.991843i \(0.540684\pi\)
\(74\) 4.22523i 0.491173i
\(75\) 0 0
\(76\) −6.09016 −0.698590
\(77\) 3.94643 0.449737
\(78\) 0 0
\(79\) 13.1897i 1.48396i 0.670424 + 0.741978i \(0.266112\pi\)
−0.670424 + 0.741978i \(0.733888\pi\)
\(80\) 0.707934 0.0791494
\(81\) 0 0
\(82\) −0.607558 −0.0670936
\(83\) 0.529002i 0.0580655i 0.999578 + 0.0290328i \(0.00924272\pi\)
−0.999578 + 0.0290328i \(0.990757\pi\)
\(84\) 0 0
\(85\) 1.97711i 0.214448i
\(86\) 1.08767i 0.117287i
\(87\) 0 0
\(88\) 4.30663i 0.459088i
\(89\) 5.43684i 0.576304i −0.957585 0.288152i \(-0.906959\pi\)
0.957585 0.288152i \(-0.0930408\pi\)
\(90\) 0 0
\(91\) 3.93164i 0.412148i
\(92\) −0.682495 −0.0711550
\(93\) 0 0
\(94\) 2.90784 0.299921
\(95\) 4.31143 0.442344
\(96\) 0 0
\(97\) 13.4837i 1.36906i 0.728982 + 0.684532i \(0.239994\pi\)
−0.728982 + 0.684532i \(0.760006\pi\)
\(98\) 6.16028i 0.622282i
\(99\) 0 0
\(100\) 4.49883 0.449883
\(101\) 15.9010i 1.58221i −0.611680 0.791106i \(-0.709506\pi\)
0.611680 0.791106i \(-0.290494\pi\)
\(102\) 0 0
\(103\) 9.59780i 0.945699i 0.881143 + 0.472850i \(0.156775\pi\)
−0.881143 + 0.472850i \(0.843225\pi\)
\(104\) −4.29049 −0.420717
\(105\) 0 0
\(106\) 11.3029 1.09784
\(107\) 13.2748 1.28333 0.641663 0.766987i \(-0.278245\pi\)
0.641663 + 0.766987i \(0.278245\pi\)
\(108\) 0 0
\(109\) 14.2315 1.36313 0.681567 0.731755i \(-0.261299\pi\)
0.681567 + 0.731755i \(0.261299\pi\)
\(110\) 3.04881i 0.290693i
\(111\) 0 0
\(112\) 0.916361 0.0865880
\(113\) −19.8205 −1.86455 −0.932277 0.361745i \(-0.882181\pi\)
−0.932277 + 0.361745i \(0.882181\pi\)
\(114\) 0 0
\(115\) 0.483161 0.0450550
\(116\) 3.21724i 0.298714i
\(117\) 0 0
\(118\) 0.693451i 0.0638374i
\(119\) 2.55921i 0.234602i
\(120\) 0 0
\(121\) 7.54707 0.686097
\(122\) 9.12711 0.826330
\(123\) 0 0
\(124\) 0.315439 0.0283272
\(125\) −6.72454 −0.601461
\(126\) 0 0
\(127\) −0.123444 −0.0109539 −0.00547696 0.999985i \(-0.501743\pi\)
−0.00547696 + 0.999985i \(0.501743\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 3.03738 0.266396
\(131\) 5.18225i 0.452776i −0.974037 0.226388i \(-0.927308\pi\)
0.974037 0.226388i \(-0.0726917\pi\)
\(132\) 0 0
\(133\) 5.58079 0.483916
\(134\) 12.9662 1.12011
\(135\) 0 0
\(136\) −2.79279 −0.239480
\(137\) −9.58358 −0.818780 −0.409390 0.912359i \(-0.634258\pi\)
−0.409390 + 0.912359i \(0.634258\pi\)
\(138\) 0 0
\(139\) −15.2856 −1.29651 −0.648256 0.761423i \(-0.724501\pi\)
−0.648256 + 0.761423i \(0.724501\pi\)
\(140\) −0.648723 −0.0548271
\(141\) 0 0
\(142\) 7.15131i 0.600125i
\(143\) 18.4775i 1.54517i
\(144\) 0 0
\(145\) 2.27760i 0.189144i
\(146\) 2.17812i 0.180263i
\(147\) 0 0
\(148\) 4.22523 0.347311
\(149\) 21.8677 1.79147 0.895734 0.444590i \(-0.146651\pi\)
0.895734 + 0.444590i \(0.146651\pi\)
\(150\) 0 0
\(151\) 12.3956i 1.00874i 0.863487 + 0.504370i \(0.168275\pi\)
−0.863487 + 0.504370i \(0.831725\pi\)
\(152\) 6.09016i 0.493977i
\(153\) 0 0
\(154\) 3.94643i 0.318012i
\(155\) −0.223310 −0.0179367
\(156\) 0 0
\(157\) 8.86184i 0.707252i 0.935387 + 0.353626i \(0.115051\pi\)
−0.935387 + 0.353626i \(0.884949\pi\)
\(158\) −13.1897 −1.04932
\(159\) 0 0
\(160\) 0.707934i 0.0559671i
\(161\) 0.625412 0.0492893
\(162\) 0 0
\(163\) 11.6089i 0.909278i 0.890676 + 0.454639i \(0.150232\pi\)
−0.890676 + 0.454639i \(0.849768\pi\)
\(164\) 0.607558i 0.0474423i
\(165\) 0 0
\(166\) −0.529002 −0.0410585
\(167\) 20.0127 1.54863 0.774316 0.632799i \(-0.218094\pi\)
0.774316 + 0.632799i \(0.218094\pi\)
\(168\) 0 0
\(169\) −5.40829 −0.416022
\(170\) 1.97711 0.151638
\(171\) 0 0
\(172\) −1.08767 −0.0829343
\(173\) 6.40756 0.487158 0.243579 0.969881i \(-0.421678\pi\)
0.243579 + 0.969881i \(0.421678\pi\)
\(174\) 0 0
\(175\) −4.12255 −0.311636
\(176\) 4.30663 0.324625
\(177\) 0 0
\(178\) 5.43684 0.407508
\(179\) 26.6168i 1.98943i −0.102677 0.994715i \(-0.532741\pi\)
0.102677 0.994715i \(-0.467259\pi\)
\(180\) 0 0
\(181\) −6.55228 −0.487027 −0.243513 0.969898i \(-0.578300\pi\)
−0.243513 + 0.969898i \(0.578300\pi\)
\(182\) 3.93164 0.291432
\(183\) 0 0
\(184\) 0.682495i 0.0503142i
\(185\) −2.99118 −0.219916
\(186\) 0 0
\(187\) 12.0275i 0.879539i
\(188\) 2.90784i 0.212076i
\(189\) 0 0
\(190\) 4.31143i 0.312784i
\(191\) 18.7825 1.35906 0.679528 0.733650i \(-0.262185\pi\)
0.679528 + 0.733650i \(0.262185\pi\)
\(192\) 0 0
\(193\) 3.23686i 0.232994i 0.993191 + 0.116497i \(0.0371666\pi\)
−0.993191 + 0.116497i \(0.962833\pi\)
\(194\) −13.4837 −0.968075
\(195\) 0 0
\(196\) 6.16028 0.440020
\(197\) 9.64258i 0.687005i 0.939152 + 0.343503i \(0.111613\pi\)
−0.939152 + 0.343503i \(0.888387\pi\)
\(198\) 0 0
\(199\) 26.4561 1.87542 0.937711 0.347417i \(-0.112941\pi\)
0.937711 + 0.347417i \(0.112941\pi\)
\(200\) 4.49883i 0.318115i
\(201\) 0 0
\(202\) 15.9010 1.11879
\(203\) 2.94816i 0.206920i
\(204\) 0 0
\(205\) 0.430111i 0.0300403i
\(206\) −9.59780 −0.668710
\(207\) 0 0
\(208\) 4.29049i 0.297492i
\(209\) 26.2281 1.81423
\(210\) 0 0
\(211\) 11.8071 0.812832 0.406416 0.913688i \(-0.366778\pi\)
0.406416 + 0.913688i \(0.366778\pi\)
\(212\) 11.3029i 0.776287i
\(213\) 0 0
\(214\) 13.2748i 0.907449i
\(215\) 0.770001 0.0525136
\(216\) 0 0
\(217\) −0.289056 −0.0196224
\(218\) 14.2315i 0.963882i
\(219\) 0 0
\(220\) −3.04881 −0.205551
\(221\) −11.9824 −0.806026
\(222\) 0 0
\(223\) −14.0849 + 4.96135i −0.943196 + 0.332237i
\(224\) 0.916361i 0.0612270i
\(225\) 0 0
\(226\) 19.8205i 1.31844i
\(227\) 5.13389i 0.340748i −0.985379 0.170374i \(-0.945502\pi\)
0.985379 0.170374i \(-0.0544976\pi\)
\(228\) 0 0
\(229\) 3.67624i 0.242932i −0.992596 0.121466i \(-0.961240\pi\)
0.992596 0.121466i \(-0.0387596\pi\)
\(230\) 0.483161i 0.0318587i
\(231\) 0 0
\(232\) −3.21724 −0.211222
\(233\) −2.85827 −0.187252 −0.0936259 0.995607i \(-0.529846\pi\)
−0.0936259 + 0.995607i \(0.529846\pi\)
\(234\) 0 0
\(235\) 2.05856i 0.134285i
\(236\) −0.693451 −0.0451398
\(237\) 0 0
\(238\) 2.55921 0.165889
\(239\) 22.3971i 1.44875i 0.689407 + 0.724374i \(0.257871\pi\)
−0.689407 + 0.724374i \(0.742129\pi\)
\(240\) 0 0
\(241\) 27.8693 1.79522 0.897609 0.440792i \(-0.145302\pi\)
0.897609 + 0.440792i \(0.145302\pi\)
\(242\) 7.54707i 0.485144i
\(243\) 0 0
\(244\) 9.12711i 0.584303i
\(245\) −4.36107 −0.278619
\(246\) 0 0
\(247\) 26.1298i 1.66260i
\(248\) 0.315439i 0.0200304i
\(249\) 0 0
\(250\) 6.72454i 0.425297i
\(251\) 21.7690i 1.37405i −0.726636 0.687023i \(-0.758917\pi\)
0.726636 0.687023i \(-0.241083\pi\)
\(252\) 0 0
\(253\) 2.93925 0.184789
\(254\) 0.123444i 0.00774559i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 29.0480i 1.81196i −0.423319 0.905981i \(-0.639135\pi\)
0.423319 0.905981i \(-0.360865\pi\)
\(258\) 0 0
\(259\) −3.87183 −0.240584
\(260\) 3.03738i 0.188370i
\(261\) 0 0
\(262\) 5.18225 0.320161
\(263\) −24.8682 −1.53344 −0.766720 0.641981i \(-0.778113\pi\)
−0.766720 + 0.641981i \(0.778113\pi\)
\(264\) 0 0
\(265\) 8.00171i 0.491541i
\(266\) 5.58079i 0.342180i
\(267\) 0 0
\(268\) 12.9662i 0.792035i
\(269\) −9.76578 −0.595430 −0.297715 0.954655i \(-0.596225\pi\)
−0.297715 + 0.954655i \(0.596225\pi\)
\(270\) 0 0
\(271\) 7.60049i 0.461697i −0.972990 0.230848i \(-0.925850\pi\)
0.972990 0.230848i \(-0.0741502\pi\)
\(272\) 2.79279i 0.169338i
\(273\) 0 0
\(274\) 9.58358i 0.578965i
\(275\) −19.3748 −1.16834
\(276\) 0 0
\(277\) 20.1990i 1.21364i 0.794839 + 0.606820i \(0.207555\pi\)
−0.794839 + 0.606820i \(0.792445\pi\)
\(278\) 15.2856i 0.916772i
\(279\) 0 0
\(280\) 0.648723i 0.0387686i
\(281\) 6.43316i 0.383770i −0.981417 0.191885i \(-0.938540\pi\)
0.981417 0.191885i \(-0.0614601\pi\)
\(282\) 0 0
\(283\) 10.5196 0.625323 0.312661 0.949865i \(-0.398780\pi\)
0.312661 + 0.949865i \(0.398780\pi\)
\(284\) 7.15131 0.424352
\(285\) 0 0
\(286\) 18.4775 1.09260
\(287\) 0.556743i 0.0328635i
\(288\) 0 0
\(289\) 9.20032 0.541195
\(290\) 2.27760 0.133745
\(291\) 0 0
\(292\) 2.17812 0.127465
\(293\) 20.0197 1.16956 0.584781 0.811191i \(-0.301180\pi\)
0.584781 + 0.811191i \(0.301180\pi\)
\(294\) 0 0
\(295\) 0.490918 0.0285823
\(296\) 4.22523i 0.245586i
\(297\) 0 0
\(298\) 21.8677i 1.26676i
\(299\) 2.92823i 0.169344i
\(300\) 0 0
\(301\) 0.996702 0.0574489
\(302\) −12.3956 −0.713287
\(303\) 0 0
\(304\) 6.09016 0.349295
\(305\) 6.46139i 0.369978i
\(306\) 0 0
\(307\) 21.1982i 1.20985i 0.796284 + 0.604923i \(0.206796\pi\)
−0.796284 + 0.604923i \(0.793204\pi\)
\(308\) −3.94643 −0.224869
\(309\) 0 0
\(310\) 0.223310i 0.0126831i
\(311\) 27.2673 1.54619 0.773094 0.634291i \(-0.218708\pi\)
0.773094 + 0.634291i \(0.218708\pi\)
\(312\) 0 0
\(313\) 6.28652i 0.355335i 0.984091 + 0.177668i \(0.0568552\pi\)
−0.984091 + 0.177668i \(0.943145\pi\)
\(314\) −8.86184 −0.500103
\(315\) 0 0
\(316\) 13.1897i 0.741978i
\(317\) 16.6895i 0.937378i 0.883363 + 0.468689i \(0.155273\pi\)
−0.883363 + 0.468689i \(0.844727\pi\)
\(318\) 0 0
\(319\) 13.8555i 0.775758i
\(320\) −0.707934 −0.0395747
\(321\) 0 0
\(322\) 0.625412i 0.0348528i
\(323\) 17.0085i 0.946381i
\(324\) 0 0
\(325\) 19.3022i 1.07069i
\(326\) −11.6089 −0.642957
\(327\) 0 0
\(328\) 0.607558 0.0335468
\(329\) 2.66463i 0.146906i
\(330\) 0 0
\(331\) 7.40295i 0.406903i −0.979085 0.203452i \(-0.934784\pi\)
0.979085 0.203452i \(-0.0652160\pi\)
\(332\) 0.529002i 0.0290328i
\(333\) 0 0
\(334\) 20.0127i 1.09505i
\(335\) 9.17919i 0.501513i
\(336\) 0 0
\(337\) 3.32095i 0.180904i −0.995901 0.0904519i \(-0.971169\pi\)
0.995901 0.0904519i \(-0.0288311\pi\)
\(338\) 5.40829i 0.294172i
\(339\) 0 0
\(340\) 1.97711i 0.107224i
\(341\) −1.35848 −0.0735657
\(342\) 0 0
\(343\) −12.0596 −0.651156
\(344\) 1.08767i 0.0586434i
\(345\) 0 0
\(346\) 6.40756i 0.344473i
\(347\) 2.50543i 0.134498i −0.997736 0.0672492i \(-0.978578\pi\)
0.997736 0.0672492i \(-0.0214223\pi\)
\(348\) 0 0
\(349\) 26.2768 1.40656 0.703282 0.710911i \(-0.251717\pi\)
0.703282 + 0.710911i \(0.251717\pi\)
\(350\) 4.12255i 0.220360i
\(351\) 0 0
\(352\) 4.30663i 0.229544i
\(353\) 15.0382i 0.800402i 0.916427 + 0.400201i \(0.131060\pi\)
−0.916427 + 0.400201i \(0.868940\pi\)
\(354\) 0 0
\(355\) −5.06266 −0.268698
\(356\) 5.43684i 0.288152i
\(357\) 0 0
\(358\) 26.6168 1.40674
\(359\) 8.25711i 0.435794i −0.975972 0.217897i \(-0.930080\pi\)
0.975972 0.217897i \(-0.0699196\pi\)
\(360\) 0 0
\(361\) 18.0901 0.952110
\(362\) 6.55228i 0.344380i
\(363\) 0 0
\(364\) 3.93164i 0.206074i
\(365\) −1.54197 −0.0807102
\(366\) 0 0
\(367\) 17.5620 0.916726 0.458363 0.888765i \(-0.348436\pi\)
0.458363 + 0.888765i \(0.348436\pi\)
\(368\) 0.682495 0.0355775
\(369\) 0 0
\(370\) 2.99118i 0.155504i
\(371\) 10.3575i 0.537737i
\(372\) 0 0
\(373\) 7.72130i 0.399794i −0.979817 0.199897i \(-0.935939\pi\)
0.979817 0.199897i \(-0.0640607\pi\)
\(374\) 12.0275 0.621928
\(375\) 0 0
\(376\) −2.90784 −0.149960
\(377\) −13.8035 −0.710919
\(378\) 0 0
\(379\) −16.9842 −0.872419 −0.436210 0.899845i \(-0.643679\pi\)
−0.436210 + 0.899845i \(0.643679\pi\)
\(380\) −4.31143 −0.221172
\(381\) 0 0
\(382\) 18.7825i 0.960998i
\(383\) −9.74547 −0.497970 −0.248985 0.968507i \(-0.580097\pi\)
−0.248985 + 0.968507i \(0.580097\pi\)
\(384\) 0 0
\(385\) 2.79381 0.142386
\(386\) −3.23686 −0.164752
\(387\) 0 0
\(388\) 13.4837i 0.684532i
\(389\) 24.1878i 1.22637i 0.789940 + 0.613184i \(0.210112\pi\)
−0.789940 + 0.613184i \(0.789888\pi\)
\(390\) 0 0
\(391\) 1.90606i 0.0963938i
\(392\) 6.16028i 0.311141i
\(393\) 0 0
\(394\) −9.64258 −0.485786
\(395\) 9.33743i 0.469817i
\(396\) 0 0
\(397\) 6.72413i 0.337475i 0.985661 + 0.168737i \(0.0539689\pi\)
−0.985661 + 0.168737i \(0.946031\pi\)
\(398\) 26.4561i 1.32612i
\(399\) 0 0
\(400\) −4.49883 −0.224941
\(401\) 19.7180i 0.984669i 0.870406 + 0.492334i \(0.163856\pi\)
−0.870406 + 0.492334i \(0.836144\pi\)
\(402\) 0 0
\(403\) 1.35339i 0.0674169i
\(404\) 15.9010i 0.791106i
\(405\) 0 0
\(406\) 2.94816 0.146315
\(407\) −18.1965 −0.901967
\(408\) 0 0
\(409\) 9.07773i 0.448865i 0.974490 + 0.224432i \(0.0720528\pi\)
−0.974490 + 0.224432i \(0.927947\pi\)
\(410\) −0.430111 −0.0212417
\(411\) 0 0
\(412\) 9.59780i 0.472850i
\(413\) 0.635452 0.0312686
\(414\) 0 0
\(415\) 0.374499i 0.0183834i
\(416\) 4.29049 0.210358
\(417\) 0 0
\(418\) 26.2281i 1.28286i
\(419\) 16.4153i 0.801941i 0.916091 + 0.400971i \(0.131327\pi\)
−0.916091 + 0.400971i \(0.868673\pi\)
\(420\) 0 0
\(421\) 37.0141i 1.80396i −0.431783 0.901978i \(-0.642115\pi\)
0.431783 0.901978i \(-0.357885\pi\)
\(422\) 11.8071i 0.574759i
\(423\) 0 0
\(424\) −11.3029 −0.548918
\(425\) 12.5643i 0.609458i
\(426\) 0 0
\(427\) 8.36373i 0.404749i
\(428\) −13.2748 −0.641663
\(429\) 0 0
\(430\) 0.770001i 0.0371327i
\(431\) 27.6342 1.33109 0.665545 0.746357i \(-0.268199\pi\)
0.665545 + 0.746357i \(0.268199\pi\)
\(432\) 0 0
\(433\) −17.2451 −0.828747 −0.414374 0.910107i \(-0.635999\pi\)
−0.414374 + 0.910107i \(0.635999\pi\)
\(434\) 0.289056i 0.0138751i
\(435\) 0 0
\(436\) −14.2315 −0.681567
\(437\) 4.15650 0.198832
\(438\) 0 0
\(439\) 31.9404i 1.52443i 0.647323 + 0.762216i \(0.275889\pi\)
−0.647323 + 0.762216i \(0.724111\pi\)
\(440\) 3.04881i 0.145346i
\(441\) 0 0
\(442\) 11.9824i 0.569946i
\(443\) 5.99899i 0.285021i −0.989793 0.142510i \(-0.954483\pi\)
0.989793 0.142510i \(-0.0455174\pi\)
\(444\) 0 0
\(445\) 3.84892i 0.182456i
\(446\) −4.96135 14.0849i −0.234927 0.666940i
\(447\) 0 0
\(448\) −0.916361 −0.0432940
\(449\) 13.1828 0.622134 0.311067 0.950388i \(-0.399314\pi\)
0.311067 + 0.950388i \(0.399314\pi\)
\(450\) 0 0
\(451\) 2.61653i 0.123208i
\(452\) 19.8205 0.932277
\(453\) 0 0
\(454\) 5.13389 0.240945
\(455\) 2.78334i 0.130485i
\(456\) 0 0
\(457\) 13.0199i 0.609046i 0.952505 + 0.304523i \(0.0984971\pi\)
−0.952505 + 0.304523i \(0.901503\pi\)
\(458\) 3.67624 0.171779
\(459\) 0 0
\(460\) −0.483161 −0.0225275
\(461\) 2.85620i 0.133026i 0.997786 + 0.0665132i \(0.0211875\pi\)
−0.997786 + 0.0665132i \(0.978813\pi\)
\(462\) 0 0
\(463\) −20.8442 −0.968712 −0.484356 0.874871i \(-0.660946\pi\)
−0.484356 + 0.874871i \(0.660946\pi\)
\(464\) 3.21724i 0.149357i
\(465\) 0 0
\(466\) 2.85827i 0.132407i
\(467\) 11.5277 0.533440 0.266720 0.963774i \(-0.414060\pi\)
0.266720 + 0.963774i \(0.414060\pi\)
\(468\) 0 0
\(469\) 11.8817i 0.548646i
\(470\) 2.05856 0.0949541
\(471\) 0 0
\(472\) 0.693451i 0.0319187i
\(473\) 4.68421 0.215380
\(474\) 0 0
\(475\) −27.3986 −1.25713
\(476\) 2.55921i 0.117301i
\(477\) 0 0
\(478\) −22.3971 −1.02442
\(479\) 1.79824i 0.0821636i 0.999156 + 0.0410818i \(0.0130804\pi\)
−0.999156 + 0.0410818i \(0.986920\pi\)
\(480\) 0 0
\(481\) 18.1283i 0.826579i
\(482\) 27.8693i 1.26941i
\(483\) 0 0
\(484\) −7.54707 −0.343049
\(485\) 9.54559i 0.433443i
\(486\) 0 0
\(487\) −33.1233 −1.50096 −0.750479 0.660894i \(-0.770177\pi\)
−0.750479 + 0.660894i \(0.770177\pi\)
\(488\) −9.12711 −0.413165
\(489\) 0 0
\(490\) 4.36107i 0.197013i
\(491\) −11.0600 −0.499132 −0.249566 0.968358i \(-0.580288\pi\)
−0.249566 + 0.968358i \(0.580288\pi\)
\(492\) 0 0
\(493\) −8.98509 −0.404668
\(494\) 26.1298 1.17563
\(495\) 0 0
\(496\) −0.315439 −0.0141636
\(497\) −6.55318 −0.293951
\(498\) 0 0
\(499\) −23.1514 −1.03640 −0.518200 0.855260i \(-0.673398\pi\)
−0.518200 + 0.855260i \(0.673398\pi\)
\(500\) 6.72454 0.300731
\(501\) 0 0
\(502\) 21.7690 0.971597
\(503\) 2.50969 0.111902 0.0559509 0.998434i \(-0.482181\pi\)
0.0559509 + 0.998434i \(0.482181\pi\)
\(504\) 0 0
\(505\) 11.2569i 0.500924i
\(506\) 2.93925i 0.130666i
\(507\) 0 0
\(508\) 0.123444 0.00547696
\(509\) 0.432553i 0.0191726i 0.999954 + 0.00958628i \(0.00305145\pi\)
−0.999954 + 0.00958628i \(0.996949\pi\)
\(510\) 0 0
\(511\) −1.99595 −0.0882955
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 29.0480 1.28125
\(515\) 6.79461i 0.299406i
\(516\) 0 0
\(517\) 12.5230i 0.550760i
\(518\) 3.87183i 0.170119i
\(519\) 0 0
\(520\) −3.03738 −0.133198
\(521\) 30.2220 1.32405 0.662026 0.749481i \(-0.269697\pi\)
0.662026 + 0.749481i \(0.269697\pi\)
\(522\) 0 0
\(523\) 29.5062i 1.29022i −0.764091 0.645108i \(-0.776812\pi\)
0.764091 0.645108i \(-0.223188\pi\)
\(524\) 5.18225i 0.226388i
\(525\) 0 0
\(526\) 24.8682i 1.08431i
\(527\) 0.880954i 0.0383750i
\(528\) 0 0
\(529\) −22.5342 −0.979748
\(530\) 8.00171 0.347572
\(531\) 0 0
\(532\) −5.58079 −0.241958
\(533\) 2.60672 0.112910
\(534\) 0 0
\(535\) 9.39770 0.406298
\(536\) −12.9662 −0.560053
\(537\) 0 0
\(538\) 9.76578i 0.421033i
\(539\) −26.5301 −1.14273
\(540\) 0 0
\(541\) 28.3366i 1.21828i −0.793061 0.609142i \(-0.791514\pi\)
0.793061 0.609142i \(-0.208486\pi\)
\(542\) 7.60049 0.326469
\(543\) 0 0
\(544\) 2.79279 0.119740
\(545\) 10.0750 0.431565
\(546\) 0 0
\(547\) 31.7429 1.35723 0.678615 0.734494i \(-0.262580\pi\)
0.678615 + 0.734494i \(0.262580\pi\)
\(548\) 9.58358 0.409390
\(549\) 0 0
\(550\) 19.3748i 0.826144i
\(551\) 19.5935i 0.834713i
\(552\) 0 0
\(553\) 12.0865i 0.513971i
\(554\) −20.1990 −0.858173
\(555\) 0 0
\(556\) 15.2856 0.648256
\(557\) −19.9132 −0.843747 −0.421874 0.906655i \(-0.638627\pi\)
−0.421874 + 0.906655i \(0.638627\pi\)
\(558\) 0 0
\(559\) 4.66665i 0.197378i
\(560\) 0.648723 0.0274136
\(561\) 0 0
\(562\) 6.43316 0.271366
\(563\) −21.6663 −0.913124 −0.456562 0.889692i \(-0.650919\pi\)
−0.456562 + 0.889692i \(0.650919\pi\)
\(564\) 0 0
\(565\) −14.0316 −0.590313
\(566\) 10.5196i 0.442170i
\(567\) 0 0
\(568\) 7.15131i 0.300062i
\(569\) −12.4174 −0.520564 −0.260282 0.965533i \(-0.583816\pi\)
−0.260282 + 0.965533i \(0.583816\pi\)
\(570\) 0 0
\(571\) 1.56438i 0.0654672i −0.999464 0.0327336i \(-0.989579\pi\)
0.999464 0.0327336i \(-0.0104213\pi\)
\(572\) 18.4775i 0.772585i
\(573\) 0 0
\(574\) −0.556743 −0.0232380
\(575\) −3.07043 −0.128046
\(576\) 0 0
\(577\) −20.8990 −0.870037 −0.435019 0.900421i \(-0.643258\pi\)
−0.435019 + 0.900421i \(0.643258\pi\)
\(578\) 9.20032i 0.382683i
\(579\) 0 0
\(580\) 2.27760i 0.0945720i
\(581\) 0.484757i 0.0201111i
\(582\) 0 0
\(583\) 48.6774i 2.01601i
\(584\) 2.17812i 0.0901314i
\(585\) 0 0
\(586\) 20.0197i 0.827005i
\(587\) 1.75943 0.0726195 0.0363097 0.999341i \(-0.488440\pi\)
0.0363097 + 0.999341i \(0.488440\pi\)
\(588\) 0 0
\(589\) −1.92107 −0.0791564
\(590\) 0.490918i 0.0202108i
\(591\) 0 0
\(592\) −4.22523 −0.173656
\(593\) 10.5072 0.431479 0.215739 0.976451i \(-0.430784\pi\)
0.215739 + 0.976451i \(0.430784\pi\)
\(594\) 0 0
\(595\) 1.81175i 0.0742744i
\(596\) −21.8677 −0.895734
\(597\) 0 0
\(598\) 2.92823 0.119744
\(599\) 19.5031i 0.796877i 0.917195 + 0.398438i \(0.130448\pi\)
−0.917195 + 0.398438i \(0.869552\pi\)
\(600\) 0 0
\(601\) 27.6894i 1.12947i 0.825271 + 0.564736i \(0.191022\pi\)
−0.825271 + 0.564736i \(0.808978\pi\)
\(602\) 0.996702i 0.0406225i
\(603\) 0 0
\(604\) 12.3956i 0.504370i
\(605\) 5.34283 0.217217
\(606\) 0 0
\(607\) 8.64182i 0.350761i 0.984501 + 0.175380i \(0.0561155\pi\)
−0.984501 + 0.175380i \(0.943884\pi\)
\(608\) 6.09016i 0.246989i
\(609\) 0 0
\(610\) 6.46139 0.261614
\(611\) −12.4760 −0.504727
\(612\) 0 0
\(613\) 6.66647i 0.269256i 0.990896 + 0.134628i \(0.0429840\pi\)
−0.990896 + 0.134628i \(0.957016\pi\)
\(614\) −21.1982 −0.855490
\(615\) 0 0
\(616\) 3.94643i 0.159006i
\(617\) 28.1483i 1.13321i 0.823990 + 0.566604i \(0.191743\pi\)
−0.823990 + 0.566604i \(0.808257\pi\)
\(618\) 0 0
\(619\) 28.6495i 1.15152i −0.817619 0.575760i \(-0.804706\pi\)
0.817619 0.575760i \(-0.195294\pi\)
\(620\) 0.223310 0.00896833
\(621\) 0 0
\(622\) 27.2673i 1.09332i
\(623\) 4.98211i 0.199604i
\(624\) 0 0
\(625\) 17.7336 0.709345
\(626\) −6.28652 −0.251260
\(627\) 0 0
\(628\) 8.86184i 0.353626i
\(629\) 11.8002i 0.470504i
\(630\) 0 0
\(631\) 32.3640i 1.28839i 0.764861 + 0.644195i \(0.222808\pi\)
−0.764861 + 0.644195i \(0.777192\pi\)
\(632\) 13.1897 0.524658
\(633\) 0 0
\(634\) −16.6895 −0.662826
\(635\) −0.0873904 −0.00346798
\(636\) 0 0
\(637\) 26.4306i 1.04722i
\(638\) 13.8555 0.548544
\(639\) 0 0
\(640\) 0.707934i 0.0279835i
\(641\) 41.7534 1.64916 0.824581 0.565744i \(-0.191411\pi\)
0.824581 + 0.565744i \(0.191411\pi\)
\(642\) 0 0
\(643\) −35.0838 −1.38357 −0.691785 0.722104i \(-0.743175\pi\)
−0.691785 + 0.722104i \(0.743175\pi\)
\(644\) −0.625412 −0.0246447
\(645\) 0 0
\(646\) 17.0085 0.669192
\(647\) 13.9549i 0.548622i −0.961641 0.274311i \(-0.911550\pi\)
0.961641 0.274311i \(-0.0884499\pi\)
\(648\) 0 0
\(649\) 2.98644 0.117228
\(650\) −19.3022 −0.757094
\(651\) 0 0
\(652\) 11.6089i 0.454639i
\(653\) −6.75889 −0.264496 −0.132248 0.991217i \(-0.542220\pi\)
−0.132248 + 0.991217i \(0.542220\pi\)
\(654\) 0 0
\(655\) 3.66869i 0.143348i
\(656\) 0.607558i 0.0237212i
\(657\) 0 0
\(658\) 2.66463 0.103878
\(659\) 18.0530i 0.703246i −0.936142 0.351623i \(-0.885630\pi\)
0.936142 0.351623i \(-0.114370\pi\)
\(660\) 0 0
\(661\) 36.8348i 1.43271i 0.697738 + 0.716353i \(0.254190\pi\)
−0.697738 + 0.716353i \(0.745810\pi\)
\(662\) 7.40295 0.287724
\(663\) 0 0
\(664\) 0.529002 0.0205293
\(665\) 3.95083 0.153207
\(666\) 0 0
\(667\) 2.19575i 0.0850198i
\(668\) −20.0127 −0.774316
\(669\) 0 0
\(670\) 9.17919 0.354623
\(671\) 39.3071i 1.51743i
\(672\) 0 0
\(673\) 7.13676 0.275102 0.137551 0.990495i \(-0.456077\pi\)
0.137551 + 0.990495i \(0.456077\pi\)
\(674\) 3.32095 0.127918
\(675\) 0 0
\(676\) 5.40829 0.208011
\(677\) 19.2900i 0.741375i 0.928758 + 0.370688i \(0.120878\pi\)
−0.928758 + 0.370688i \(0.879122\pi\)
\(678\) 0 0
\(679\) 12.3560i 0.474178i
\(680\) −1.97711 −0.0758188
\(681\) 0 0
\(682\) 1.35848i 0.0520188i
\(683\) 20.6801i 0.791300i −0.918401 0.395650i \(-0.870519\pi\)
0.918401 0.395650i \(-0.129481\pi\)
\(684\) 0 0
\(685\) −6.78454 −0.259224
\(686\) 12.0596i 0.460437i
\(687\) 0 0
\(688\) 1.08767 0.0414672
\(689\) −48.4950 −1.84751
\(690\) 0 0
\(691\) 40.9355i 1.55726i −0.627484 0.778629i \(-0.715915\pi\)
0.627484 0.778629i \(-0.284085\pi\)
\(692\) −6.40756 −0.243579
\(693\) 0 0
\(694\) 2.50543 0.0951047
\(695\) −10.8212 −0.410473
\(696\) 0 0
\(697\) 1.69678 0.0642703
\(698\) 26.2768i 0.994592i
\(699\) 0 0
\(700\) 4.12255 0.155818
\(701\) 25.5815i 0.966198i 0.875566 + 0.483099i \(0.160489\pi\)
−0.875566 + 0.483099i \(0.839511\pi\)
\(702\) 0 0
\(703\) −25.7323 −0.970513
\(704\) −4.30663 −0.162312
\(705\) 0 0
\(706\) −15.0382 −0.565969
\(707\) 14.5711i 0.548002i
\(708\) 0 0
\(709\) 38.1785i 1.43383i −0.697163 0.716913i \(-0.745555\pi\)
0.697163 0.716913i \(-0.254445\pi\)
\(710\) 5.06266i 0.189998i
\(711\) 0 0
\(712\) −5.43684 −0.203754
\(713\) −0.215285 −0.00806249
\(714\) 0 0
\(715\) 13.0809i 0.489197i
\(716\) 26.6168i 0.994715i
\(717\) 0 0
\(718\) 8.25711 0.308153
\(719\) 35.1730i 1.31173i 0.754877 + 0.655866i \(0.227696\pi\)
−0.754877 + 0.655866i \(0.772304\pi\)
\(720\) 0 0
\(721\) 8.79505i 0.327545i
\(722\) 18.0901i 0.673243i
\(723\) 0 0
\(724\) 6.55228 0.243513
\(725\) 14.4738i 0.537544i
\(726\) 0 0
\(727\) −15.4730 −0.573861 −0.286930 0.957951i \(-0.592635\pi\)
−0.286930 + 0.957951i \(0.592635\pi\)
\(728\) −3.93164 −0.145716
\(729\) 0 0
\(730\) 1.54197i 0.0570708i
\(731\) 3.03764i 0.112351i
\(732\) 0 0
\(733\) 19.8853 0.734478 0.367239 0.930127i \(-0.380303\pi\)
0.367239 + 0.930127i \(0.380303\pi\)
\(734\) 17.5620i 0.648223i
\(735\) 0 0
\(736\) 0.682495i 0.0251571i
\(737\) 55.8405i 2.05691i
\(738\) 0 0
\(739\) 4.58935i 0.168822i 0.996431 + 0.0844109i \(0.0269008\pi\)
−0.996431 + 0.0844109i \(0.973099\pi\)
\(740\) 2.99118 0.109958
\(741\) 0 0
\(742\) 10.3575 0.380238
\(743\) 15.1401i 0.555435i 0.960663 + 0.277718i \(0.0895780\pi\)
−0.960663 + 0.277718i \(0.910422\pi\)
\(744\) 0 0
\(745\) 15.4809 0.567175
\(746\) 7.72130 0.282697
\(747\) 0 0
\(748\) 12.0275i 0.439770i
\(749\) 12.1645 0.444483
\(750\) 0 0
\(751\) −12.6214 −0.460561 −0.230280 0.973124i \(-0.573964\pi\)
−0.230280 + 0.973124i \(0.573964\pi\)
\(752\) 2.90784i 0.106038i
\(753\) 0 0
\(754\) 13.8035i 0.502695i
\(755\) 8.77528i 0.319365i
\(756\) 0 0
\(757\) 10.4896i 0.381252i 0.981663 + 0.190626i \(0.0610518\pi\)
−0.981663 + 0.190626i \(0.938948\pi\)
\(758\) 16.9842i 0.616893i
\(759\) 0 0
\(760\) 4.31143i 0.156392i
\(761\) −39.8663 −1.44515 −0.722576 0.691291i \(-0.757042\pi\)
−0.722576 + 0.691291i \(0.757042\pi\)
\(762\) 0 0
\(763\) 13.0412 0.472125
\(764\) −18.7825 −0.679528
\(765\) 0 0
\(766\) 9.74547i 0.352118i
\(767\) 2.97525i 0.107430i
\(768\) 0 0
\(769\) −46.2287 −1.66705 −0.833525 0.552482i \(-0.813681\pi\)
−0.833525 + 0.552482i \(0.813681\pi\)
\(770\) 2.79381i 0.100682i
\(771\) 0 0
\(772\) 3.23686i 0.116497i
\(773\) 24.9358 0.896878 0.448439 0.893813i \(-0.351980\pi\)
0.448439 + 0.893813i \(0.351980\pi\)
\(774\) 0 0
\(775\) 1.41910 0.0509757
\(776\) 13.4837 0.484037
\(777\) 0 0
\(778\) −24.1878 −0.867173
\(779\) 3.70013i 0.132571i
\(780\) 0 0
\(781\) −30.7981 −1.10204
\(782\) 1.90606 0.0681607
\(783\) 0 0
\(784\) −6.16028 −0.220010
\(785\) 6.27360i 0.223914i
\(786\) 0 0
\(787\) 36.6326i 1.30581i 0.757439 + 0.652906i \(0.226450\pi\)
−0.757439 + 0.652906i \(0.773550\pi\)
\(788\) 9.64258i 0.343503i
\(789\) 0 0
\(790\) −9.33743 −0.332211
\(791\) −18.1627 −0.645792
\(792\) 0 0
\(793\) −39.1598 −1.39060
\(794\) −6.72413 −0.238631
\(795\) 0 0
\(796\) −26.4561 −0.937711
\(797\) 28.5419i 1.01101i 0.862825 + 0.505503i \(0.168693\pi\)
−0.862825 + 0.505503i \(0.831307\pi\)
\(798\) 0 0
\(799\) −8.12098 −0.287300
\(800\) 4.49883i 0.159058i
\(801\) 0 0
\(802\) −19.7180 −0.696266
\(803\) −9.38037 −0.331026
\(804\) 0 0
\(805\) 0.442750 0.0156049
\(806\) −1.35339 −0.0476710
\(807\) 0 0
\(808\) −15.9010 −0.559396
\(809\) −10.2364 −0.359893 −0.179946 0.983676i \(-0.557592\pi\)
−0.179946 + 0.983676i \(0.557592\pi\)
\(810\) 0 0
\(811\) 16.4382i 0.577222i 0.957446 + 0.288611i \(0.0931934\pi\)
−0.957446 + 0.288611i \(0.906807\pi\)
\(812\) 2.94816i 0.103460i
\(813\) 0 0
\(814\) 18.1965i 0.637787i
\(815\) 8.21833i 0.287875i
\(816\) 0 0
\(817\) 6.62411 0.231748
\(818\) −9.07773 −0.317395
\(819\) 0 0
\(820\) 0.430111i 0.0150201i
\(821\) 18.1557i 0.633640i −0.948486 0.316820i \(-0.897385\pi\)
0.948486 0.316820i \(-0.102615\pi\)
\(822\) 0 0
\(823\) 37.0902i 1.29288i 0.762963 + 0.646442i \(0.223744\pi\)
−0.762963 + 0.646442i \(0.776256\pi\)
\(824\) 9.59780 0.334355
\(825\) 0 0
\(826\) 0.635452i 0.0221102i
\(827\) 27.4977 0.956188 0.478094 0.878309i \(-0.341328\pi\)
0.478094 + 0.878309i \(0.341328\pi\)
\(828\) 0 0
\(829\) 17.4292i 0.605339i −0.953096 0.302670i \(-0.902122\pi\)
0.953096 0.302670i \(-0.0978779\pi\)
\(830\) −0.374499 −0.0129990
\(831\) 0 0
\(832\) 4.29049i 0.148746i
\(833\) 17.2044i 0.596096i
\(834\) 0 0
\(835\) 14.1677 0.490293
\(836\) −26.2281 −0.907117
\(837\) 0 0
\(838\) −16.4153 −0.567058
\(839\) −23.2768 −0.803605 −0.401802 0.915726i \(-0.631616\pi\)
−0.401802 + 0.915726i \(0.631616\pi\)
\(840\) 0 0
\(841\) 18.6493 0.643081
\(842\) 37.0141 1.27559
\(843\) 0 0
\(844\) −11.8071 −0.406416
\(845\) −3.82871 −0.131712
\(846\) 0 0
\(847\) 6.91584 0.237631
\(848\) 11.3029i 0.388143i
\(849\) 0 0
\(850\) −12.5643 −0.430952
\(851\) −2.88369 −0.0988518
\(852\) 0 0
\(853\) 0.628999i 0.0215365i 0.999942 + 0.0107683i \(0.00342771\pi\)
−0.999942 + 0.0107683i \(0.996572\pi\)
\(854\) 8.36373 0.286201
\(855\) 0 0
\(856\) 13.2748i 0.453724i
\(857\) 43.0666i 1.47113i −0.677455 0.735564i \(-0.736917\pi\)
0.677455 0.735564i \(-0.263083\pi\)
\(858\) 0 0
\(859\) 13.4684i 0.459537i −0.973245 0.229769i \(-0.926203\pi\)
0.973245 0.229769i \(-0.0737969\pi\)
\(860\) −0.770001 −0.0262568
\(861\) 0 0
\(862\) 27.6342i 0.941223i
\(863\) 51.8700 1.76568 0.882838 0.469677i \(-0.155630\pi\)
0.882838 + 0.469677i \(0.155630\pi\)
\(864\) 0 0
\(865\) 4.53613 0.154233
\(866\) 17.2451i 0.586013i
\(867\) 0 0
\(868\) 0.289056 0.00981119
\(869\) 56.8031i 1.92691i
\(870\) 0 0
\(871\) −55.6312 −1.88499
\(872\) 14.2315i 0.481941i
\(873\) 0 0
\(874\) 4.15650i 0.140596i
\(875\) −6.16211 −0.208317
\(876\) 0 0
\(877\) 25.2707i 0.853332i −0.904409 0.426666i \(-0.859688\pi\)
0.904409 0.426666i \(-0.140312\pi\)
\(878\) −31.9404 −1.07794
\(879\) 0 0
\(880\) 3.04881 0.102775
\(881\) 0.988956i 0.0333188i 0.999861 + 0.0166594i \(0.00530309\pi\)
−0.999861 + 0.0166594i \(0.994697\pi\)
\(882\) 0 0
\(883\) 40.7149i 1.37016i −0.728466 0.685082i \(-0.759766\pi\)
0.728466 0.685082i \(-0.240234\pi\)
\(884\) 11.9824 0.403013
\(885\) 0 0
\(886\) 5.99899 0.201540
\(887\) 6.62595i 0.222478i 0.993794 + 0.111239i \(0.0354819\pi\)
−0.993794 + 0.111239i \(0.964518\pi\)
\(888\) 0 0
\(889\) −0.113120 −0.00379391
\(890\) 3.84892 0.129016
\(891\) 0 0
\(892\) 14.0849 4.96135i 0.471598 0.166118i
\(893\) 17.7092i 0.592616i
\(894\) 0 0
\(895\) 18.8429i 0.629849i
\(896\) 0.916361i 0.0306135i
\(897\) 0 0
\(898\) 13.1828i 0.439915i
\(899\) 1.01484i 0.0338469i
\(900\) 0 0
\(901\) −31.5667 −1.05164
\(902\) −2.61653 −0.0871209
\(903\) 0 0
\(904\) 19.8205i 0.659219i
\(905\) −4.63858 −0.154192
\(906\) 0 0
\(907\) 21.8346 0.725006 0.362503 0.931983i \(-0.381922\pi\)
0.362503 + 0.931983i \(0.381922\pi\)
\(908\) 5.13389i 0.170374i
\(909\) 0 0
\(910\) 2.78334 0.0922668
\(911\) 34.2519i 1.13482i −0.823437 0.567408i \(-0.807946\pi\)
0.823437 0.567408i \(-0.192054\pi\)
\(912\) 0 0
\(913\) 2.27822i 0.0753980i
\(914\) −13.0199 −0.430661
\(915\) 0 0
\(916\) 3.67624i 0.121466i
\(917\) 4.74882i 0.156820i
\(918\) 0 0
\(919\) 27.6498i 0.912082i 0.889959 + 0.456041i \(0.150733\pi\)
−0.889959 + 0.456041i \(0.849267\pi\)
\(920\) 0.483161i 0.0159293i
\(921\) 0 0
\(922\) −2.85620 −0.0940639
\(923\) 30.6826i 1.00993i
\(924\) 0 0
\(925\) 19.0086 0.624998
\(926\) 20.8442i 0.684983i
\(927\) 0 0
\(928\) 3.21724 0.105611
\(929\) 42.1029i 1.38135i 0.723165 + 0.690675i \(0.242687\pi\)
−0.723165 + 0.690675i \(0.757313\pi\)
\(930\) 0 0
\(931\) −37.5171 −1.22957
\(932\) 2.85827 0.0936259
\(933\) 0 0
\(934\) 11.5277i 0.377199i
\(935\) 8.51469i 0.278460i
\(936\) 0 0
\(937\) 50.7517i 1.65799i −0.559259 0.828993i \(-0.688914\pi\)
0.559259 0.828993i \(-0.311086\pi\)
\(938\) 11.8817 0.387951
\(939\) 0 0
\(940\) 2.05856i 0.0671427i
\(941\) 36.8053i 1.19982i 0.800069 + 0.599909i \(0.204796\pi\)
−0.800069 + 0.599909i \(0.795204\pi\)
\(942\) 0 0
\(943\) 0.414655i 0.0135030i
\(944\) 0.693451 0.0225699
\(945\) 0 0
\(946\) 4.68421i 0.152297i
\(947\) 48.5280i 1.57695i −0.615068 0.788474i \(-0.710871\pi\)
0.615068 0.788474i \(-0.289129\pi\)
\(948\) 0 0
\(949\) 9.34521i 0.303358i
\(950\) 27.3986i 0.888928i
\(951\) 0 0
\(952\) −2.55921 −0.0829443
\(953\) 3.03657 0.0983642 0.0491821 0.998790i \(-0.484339\pi\)
0.0491821 + 0.998790i \(0.484339\pi\)
\(954\) 0 0
\(955\) 13.2968 0.430274
\(956\) 22.3971i 0.724374i
\(957\) 0 0
\(958\) −1.79824 −0.0580985
\(959\) −8.78202 −0.283586
\(960\) 0 0
\(961\) −30.9005 −0.996790
\(962\) −18.1283 −0.584479
\(963\) 0 0
\(964\) −27.8693 −0.897609
\(965\) 2.29148i 0.0737655i
\(966\) 0 0
\(967\) 39.2896i 1.26347i 0.775186 + 0.631734i \(0.217656\pi\)
−0.775186 + 0.631734i \(0.782344\pi\)
\(968\) 7.54707i 0.242572i
\(969\) 0 0
\(970\) −9.54559 −0.306490
\(971\) 60.2843 1.93461 0.967307 0.253608i \(-0.0816174\pi\)
0.967307 + 0.253608i \(0.0816174\pi\)
\(972\) 0 0
\(973\) −14.0072 −0.449050
\(974\) 33.1233i 1.06134i
\(975\) 0 0
\(976\) 9.12711i 0.292152i
\(977\) −30.9667 −0.990711 −0.495356 0.868690i \(-0.664962\pi\)
−0.495356 + 0.868690i \(0.664962\pi\)
\(978\) 0 0
\(979\) 23.4145i 0.748330i
\(980\) 4.36107 0.139309
\(981\) 0 0
\(982\) 11.0600i 0.352940i
\(983\) −8.84896 −0.282238 −0.141119 0.989993i \(-0.545070\pi\)
−0.141119 + 0.989993i \(0.545070\pi\)
\(984\) 0 0
\(985\) 6.82631i 0.217504i
\(986\) 8.98509i 0.286143i
\(987\) 0 0
\(988\) 26.1298i 0.831299i
\(989\) 0.742331 0.0236048
\(990\) 0 0
\(991\) 27.1979i 0.863968i 0.901881 + 0.431984i \(0.142186\pi\)
−0.901881 + 0.431984i \(0.857814\pi\)
\(992\) 0.315439i 0.0100152i
\(993\) 0 0
\(994\) 6.55318i 0.207854i
\(995\) 18.7292 0.593754
\(996\) 0 0
\(997\) 52.1564 1.65181 0.825905 0.563809i \(-0.190665\pi\)
0.825905 + 0.563809i \(0.190665\pi\)
\(998\) 23.1514i 0.732845i
\(999\) 0 0
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 4014.2.d.a.4013.18 yes 72
3.2 odd 2 inner 4014.2.d.a.4013.59 yes 72
223.222 odd 2 inner 4014.2.d.a.4013.60 yes 72
669.668 even 2 inner 4014.2.d.a.4013.17 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.d.a.4013.17 72 669.668 even 2 inner
4014.2.d.a.4013.18 yes 72 1.1 even 1 trivial
4014.2.d.a.4013.59 yes 72 3.2 odd 2 inner
4014.2.d.a.4013.60 yes 72 223.222 odd 2 inner