Properties

Label 4028.2.c.a.3497.18
Level $4028$
Weight $2$
Character 4028.3497
Analytic conductor $32.164$
Analytic rank $0$
Dimension $82$
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] = [4028,2,Mod(3497,4028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4028.3497");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4028.c (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: \(32.1637419342\)
Analytic rank: \(0\)
Dimension: \(82\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3497.18
Character \(\chi\) \(=\) 4028.3497
Dual form 4028.2.c.a.3497.65

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.07591i q^{3} -1.80942i q^{5} +1.11331 q^{7} -1.30941 q^{9} +O(q^{10})\) \(q-2.07591i q^{3} -1.80942i q^{5} +1.11331 q^{7} -1.30941 q^{9} +3.61828 q^{11} +6.43779 q^{13} -3.75619 q^{15} -6.97195 q^{17} -1.00000i q^{19} -2.31114i q^{21} +6.32108i q^{23} +1.72601 q^{25} -3.50952i q^{27} +1.59306 q^{29} -0.346279i q^{31} -7.51123i q^{33} -2.01445i q^{35} +8.84318 q^{37} -13.3643i q^{39} -10.2048i q^{41} -3.64187 q^{43} +2.36926i q^{45} +4.75561 q^{47} -5.76053 q^{49} +14.4731i q^{51} +(6.70311 + 2.84048i) q^{53} -6.54698i q^{55} -2.07591 q^{57} +14.2914 q^{59} +7.53081i q^{61} -1.45778 q^{63} -11.6487i q^{65} -5.06783i q^{67} +13.1220 q^{69} -11.4007i q^{71} -11.1518i q^{73} -3.58304i q^{75} +4.02828 q^{77} +6.51640i q^{79} -11.2137 q^{81} +3.38380i q^{83} +12.6152i q^{85} -3.30706i q^{87} -5.49994 q^{89} +7.16728 q^{91} -0.718843 q^{93} -1.80942 q^{95} +7.13179 q^{97} -4.73780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 82 q - 8 q^{7} - 82 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 82 q - 8 q^{7} - 82 q^{9} + 4 q^{13} + 4 q^{15} - 4 q^{17} - 58 q^{25} - 16 q^{29} - 12 q^{37} - 32 q^{43} + 8 q^{47} + 98 q^{49} + 6 q^{53} - 4 q^{57} + 4 q^{59} + 8 q^{63} + 28 q^{69} - 8 q^{77} + 154 q^{81} - 20 q^{89} + 48 q^{91} - 56 q^{93} - 44 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4028\mathbb{Z}\right)^\times\).

\(n\) \(2015\) \(2281\) \(2757\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 2.07591i 1.19853i −0.800552 0.599264i \(-0.795460\pi\)
0.800552 0.599264i \(-0.204540\pi\)
\(4\) 0 0
\(5\) 1.80942i 0.809196i −0.914495 0.404598i \(-0.867411\pi\)
0.914495 0.404598i \(-0.132589\pi\)
\(6\) 0 0
\(7\) 1.11331 0.420793 0.210396 0.977616i \(-0.432525\pi\)
0.210396 + 0.977616i \(0.432525\pi\)
\(8\) 0 0
\(9\) −1.30941 −0.436469
\(10\) 0 0
\(11\) 3.61828 1.09095 0.545477 0.838126i \(-0.316349\pi\)
0.545477 + 0.838126i \(0.316349\pi\)
\(12\) 0 0
\(13\) 6.43779 1.78552 0.892761 0.450530i \(-0.148765\pi\)
0.892761 + 0.450530i \(0.148765\pi\)
\(14\) 0 0
\(15\) −3.75619 −0.969844
\(16\) 0 0
\(17\) −6.97195 −1.69095 −0.845473 0.534019i \(-0.820681\pi\)
−0.845473 + 0.534019i \(0.820681\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 2.31114i 0.504332i
\(22\) 0 0
\(23\) 6.32108i 1.31804i 0.752127 + 0.659018i \(0.229028\pi\)
−0.752127 + 0.659018i \(0.770972\pi\)
\(24\) 0 0
\(25\) 1.72601 0.345202
\(26\) 0 0
\(27\) 3.50952i 0.675408i
\(28\) 0 0
\(29\) 1.59306 0.295824 0.147912 0.989000i \(-0.452745\pi\)
0.147912 + 0.989000i \(0.452745\pi\)
\(30\) 0 0
\(31\) 0.346279i 0.0621935i −0.999516 0.0310967i \(-0.990100\pi\)
0.999516 0.0310967i \(-0.00989999\pi\)
\(32\) 0 0
\(33\) 7.51123i 1.30754i
\(34\) 0 0
\(35\) 2.01445i 0.340504i
\(36\) 0 0
\(37\) 8.84318 1.45381 0.726905 0.686738i \(-0.240958\pi\)
0.726905 + 0.686738i \(0.240958\pi\)
\(38\) 0 0
\(39\) 13.3643i 2.14000i
\(40\) 0 0
\(41\) 10.2048i 1.59372i −0.604165 0.796859i \(-0.706493\pi\)
0.604165 0.796859i \(-0.293507\pi\)
\(42\) 0 0
\(43\) −3.64187 −0.555380 −0.277690 0.960671i \(-0.589569\pi\)
−0.277690 + 0.960671i \(0.589569\pi\)
\(44\) 0 0
\(45\) 2.36926i 0.353189i
\(46\) 0 0
\(47\) 4.75561 0.693676 0.346838 0.937925i \(-0.387255\pi\)
0.346838 + 0.937925i \(0.387255\pi\)
\(48\) 0 0
\(49\) −5.76053 −0.822934
\(50\) 0 0
\(51\) 14.4731i 2.02664i
\(52\) 0 0
\(53\) 6.70311 + 2.84048i 0.920743 + 0.390169i
\(54\) 0 0
\(55\) 6.54698i 0.882795i
\(56\) 0 0
\(57\) −2.07591 −0.274961
\(58\) 0 0
\(59\) 14.2914 1.86059 0.930293 0.366818i \(-0.119553\pi\)
0.930293 + 0.366818i \(0.119553\pi\)
\(60\) 0 0
\(61\) 7.53081i 0.964221i 0.876110 + 0.482111i \(0.160130\pi\)
−0.876110 + 0.482111i \(0.839870\pi\)
\(62\) 0 0
\(63\) −1.45778 −0.183663
\(64\) 0 0
\(65\) 11.6487i 1.44484i
\(66\) 0 0
\(67\) 5.06783i 0.619134i −0.950878 0.309567i \(-0.899816\pi\)
0.950878 0.309567i \(-0.100184\pi\)
\(68\) 0 0
\(69\) 13.1220 1.57970
\(70\) 0 0
\(71\) 11.4007i 1.35301i −0.736437 0.676506i \(-0.763493\pi\)
0.736437 0.676506i \(-0.236507\pi\)
\(72\) 0 0
\(73\) 11.1518i 1.30521i −0.757696 0.652607i \(-0.773675\pi\)
0.757696 0.652607i \(-0.226325\pi\)
\(74\) 0 0
\(75\) 3.58304i 0.413734i
\(76\) 0 0
\(77\) 4.02828 0.459065
\(78\) 0 0
\(79\) 6.51640i 0.733152i 0.930388 + 0.366576i \(0.119470\pi\)
−0.930388 + 0.366576i \(0.880530\pi\)
\(80\) 0 0
\(81\) −11.2137 −1.24596
\(82\) 0 0
\(83\) 3.38380i 0.371421i 0.982605 + 0.185710i \(0.0594586\pi\)
−0.982605 + 0.185710i \(0.940541\pi\)
\(84\) 0 0
\(85\) 12.6152i 1.36831i
\(86\) 0 0
\(87\) 3.30706i 0.354554i
\(88\) 0 0
\(89\) −5.49994 −0.582992 −0.291496 0.956572i \(-0.594153\pi\)
−0.291496 + 0.956572i \(0.594153\pi\)
\(90\) 0 0
\(91\) 7.16728 0.751335
\(92\) 0 0
\(93\) −0.718843 −0.0745406
\(94\) 0 0
\(95\) −1.80942 −0.185642
\(96\) 0 0
\(97\) 7.13179 0.724124 0.362062 0.932154i \(-0.382073\pi\)
0.362062 + 0.932154i \(0.382073\pi\)
\(98\) 0 0
\(99\) −4.73780 −0.476167
\(100\) 0 0
\(101\) 3.07135i 0.305610i 0.988256 + 0.152805i \(0.0488307\pi\)
−0.988256 + 0.152805i \(0.951169\pi\)
\(102\) 0 0
\(103\) 5.51664i 0.543571i 0.962358 + 0.271785i \(0.0876140\pi\)
−0.962358 + 0.271785i \(0.912386\pi\)
\(104\) 0 0
\(105\) −4.18181 −0.408103
\(106\) 0 0
\(107\) −1.59900 −0.154581 −0.0772907 0.997009i \(-0.524627\pi\)
−0.0772907 + 0.997009i \(0.524627\pi\)
\(108\) 0 0
\(109\) 18.2931i 1.75217i 0.482161 + 0.876083i \(0.339852\pi\)
−0.482161 + 0.876083i \(0.660148\pi\)
\(110\) 0 0
\(111\) 18.3577i 1.74243i
\(112\) 0 0
\(113\) −15.0175 −1.41273 −0.706363 0.707849i \(-0.749666\pi\)
−0.706363 + 0.707849i \(0.749666\pi\)
\(114\) 0 0
\(115\) 11.4375 1.06655
\(116\) 0 0
\(117\) −8.42969 −0.779325
\(118\) 0 0
\(119\) −7.76196 −0.711537
\(120\) 0 0
\(121\) 2.09196 0.190179
\(122\) 0 0
\(123\) −21.1842 −1.91011
\(124\) 0 0
\(125\) 12.1702i 1.08853i
\(126\) 0 0
\(127\) 18.8012i 1.66834i 0.551508 + 0.834170i \(0.314053\pi\)
−0.551508 + 0.834170i \(0.685947\pi\)
\(128\) 0 0
\(129\) 7.56020i 0.665638i
\(130\) 0 0
\(131\) 2.52680 0.220768 0.110384 0.993889i \(-0.464792\pi\)
0.110384 + 0.993889i \(0.464792\pi\)
\(132\) 0 0
\(133\) 1.11331i 0.0965365i
\(134\) 0 0
\(135\) −6.35019 −0.546537
\(136\) 0 0
\(137\) 0.301874i 0.0257908i −0.999917 0.0128954i \(-0.995895\pi\)
0.999917 0.0128954i \(-0.00410485\pi\)
\(138\) 0 0
\(139\) 10.5999i 0.899070i −0.893263 0.449535i \(-0.851590\pi\)
0.893263 0.449535i \(-0.148410\pi\)
\(140\) 0 0
\(141\) 9.87221i 0.831390i
\(142\) 0 0
\(143\) 23.2937 1.94792
\(144\) 0 0
\(145\) 2.88252i 0.239380i
\(146\) 0 0
\(147\) 11.9584i 0.986309i
\(148\) 0 0
\(149\) −11.7004 −0.958536 −0.479268 0.877669i \(-0.659098\pi\)
−0.479268 + 0.877669i \(0.659098\pi\)
\(150\) 0 0
\(151\) 0.200264i 0.0162973i 0.999967 + 0.00814863i \(0.00259382\pi\)
−0.999967 + 0.00814863i \(0.997406\pi\)
\(152\) 0 0
\(153\) 9.12911 0.738045
\(154\) 0 0
\(155\) −0.626562 −0.0503267
\(156\) 0 0
\(157\) 6.37658i 0.508906i −0.967085 0.254453i \(-0.918105\pi\)
0.967085 0.254453i \(-0.0818955\pi\)
\(158\) 0 0
\(159\) 5.89657 13.9151i 0.467629 1.10354i
\(160\) 0 0
\(161\) 7.03734i 0.554620i
\(162\) 0 0
\(163\) −3.91950 −0.306999 −0.153499 0.988149i \(-0.549054\pi\)
−0.153499 + 0.988149i \(0.549054\pi\)
\(164\) 0 0
\(165\) −13.5910 −1.05805
\(166\) 0 0
\(167\) 17.5709i 1.35968i −0.733362 0.679838i \(-0.762050\pi\)
0.733362 0.679838i \(-0.237950\pi\)
\(168\) 0 0
\(169\) 28.4452 2.18809
\(170\) 0 0
\(171\) 1.30941i 0.100133i
\(172\) 0 0
\(173\) 18.1021i 1.37628i −0.725580 0.688138i \(-0.758429\pi\)
0.725580 0.688138i \(-0.241571\pi\)
\(174\) 0 0
\(175\) 1.92159 0.145258
\(176\) 0 0
\(177\) 29.6677i 2.22996i
\(178\) 0 0
\(179\) 12.4269i 0.928833i −0.885617 0.464416i \(-0.846264\pi\)
0.885617 0.464416i \(-0.153736\pi\)
\(180\) 0 0
\(181\) 7.03256i 0.522726i 0.965241 + 0.261363i \(0.0841720\pi\)
−0.965241 + 0.261363i \(0.915828\pi\)
\(182\) 0 0
\(183\) 15.6333 1.15565
\(184\) 0 0
\(185\) 16.0010i 1.17642i
\(186\) 0 0
\(187\) −25.2265 −1.84474
\(188\) 0 0
\(189\) 3.90720i 0.284207i
\(190\) 0 0
\(191\) 8.63409i 0.624740i −0.949960 0.312370i \(-0.898877\pi\)
0.949960 0.312370i \(-0.101123\pi\)
\(192\) 0 0
\(193\) 15.0817i 1.08561i 0.839860 + 0.542804i \(0.182637\pi\)
−0.839860 + 0.542804i \(0.817363\pi\)
\(194\) 0 0
\(195\) −24.1816 −1.73168
\(196\) 0 0
\(197\) −7.24576 −0.516239 −0.258120 0.966113i \(-0.583103\pi\)
−0.258120 + 0.966113i \(0.583103\pi\)
\(198\) 0 0
\(199\) 1.02486 0.0726504 0.0363252 0.999340i \(-0.488435\pi\)
0.0363252 + 0.999340i \(0.488435\pi\)
\(200\) 0 0
\(201\) −10.5204 −0.742050
\(202\) 0 0
\(203\) 1.77358 0.124481
\(204\) 0 0
\(205\) −18.4647 −1.28963
\(206\) 0 0
\(207\) 8.27686i 0.575282i
\(208\) 0 0
\(209\) 3.61828i 0.250282i
\(210\) 0 0
\(211\) 24.3791 1.67832 0.839162 0.543882i \(-0.183046\pi\)
0.839162 + 0.543882i \(0.183046\pi\)
\(212\) 0 0
\(213\) −23.6668 −1.62162
\(214\) 0 0
\(215\) 6.58966i 0.449411i
\(216\) 0 0
\(217\) 0.385516i 0.0261706i
\(218\) 0 0
\(219\) −23.1501 −1.56434
\(220\) 0 0
\(221\) −44.8839 −3.01922
\(222\) 0 0
\(223\) −12.4371 −0.832847 −0.416424 0.909171i \(-0.636717\pi\)
−0.416424 + 0.909171i \(0.636717\pi\)
\(224\) 0 0
\(225\) −2.26005 −0.150670
\(226\) 0 0
\(227\) −18.7540 −1.24475 −0.622373 0.782721i \(-0.713831\pi\)
−0.622373 + 0.782721i \(0.713831\pi\)
\(228\) 0 0
\(229\) −5.96459 −0.394151 −0.197076 0.980388i \(-0.563144\pi\)
−0.197076 + 0.980388i \(0.563144\pi\)
\(230\) 0 0
\(231\) 8.36235i 0.550202i
\(232\) 0 0
\(233\) 14.5654i 0.954210i −0.878846 0.477105i \(-0.841686\pi\)
0.878846 0.477105i \(-0.158314\pi\)
\(234\) 0 0
\(235\) 8.60488i 0.561320i
\(236\) 0 0
\(237\) 13.5275 0.878703
\(238\) 0 0
\(239\) 25.2877i 1.63573i 0.575412 + 0.817864i \(0.304842\pi\)
−0.575412 + 0.817864i \(0.695158\pi\)
\(240\) 0 0
\(241\) −24.3921 −1.57123 −0.785616 0.618715i \(-0.787654\pi\)
−0.785616 + 0.618715i \(0.787654\pi\)
\(242\) 0 0
\(243\) 12.7500i 0.817914i
\(244\) 0 0
\(245\) 10.4232i 0.665915i
\(246\) 0 0
\(247\) 6.43779i 0.409627i
\(248\) 0 0
\(249\) 7.02448 0.445158
\(250\) 0 0
\(251\) 1.51779i 0.0958023i −0.998852 0.0479012i \(-0.984747\pi\)
0.998852 0.0479012i \(-0.0152533\pi\)
\(252\) 0 0
\(253\) 22.8715i 1.43792i
\(254\) 0 0
\(255\) 26.1880 1.63995
\(256\) 0 0
\(257\) 13.9365i 0.869337i 0.900590 + 0.434669i \(0.143134\pi\)
−0.900590 + 0.434669i \(0.856866\pi\)
\(258\) 0 0
\(259\) 9.84523 0.611753
\(260\) 0 0
\(261\) −2.08597 −0.129118
\(262\) 0 0
\(263\) 22.1130i 1.36355i 0.731562 + 0.681775i \(0.238791\pi\)
−0.731562 + 0.681775i \(0.761209\pi\)
\(264\) 0 0
\(265\) 5.13961 12.1287i 0.315724 0.745062i
\(266\) 0 0
\(267\) 11.4174i 0.698733i
\(268\) 0 0
\(269\) −10.6401 −0.648737 −0.324368 0.945931i \(-0.605152\pi\)
−0.324368 + 0.945931i \(0.605152\pi\)
\(270\) 0 0
\(271\) −27.9844 −1.69993 −0.849967 0.526836i \(-0.823378\pi\)
−0.849967 + 0.526836i \(0.823378\pi\)
\(272\) 0 0
\(273\) 14.8786i 0.900495i
\(274\) 0 0
\(275\) 6.24518 0.376599
\(276\) 0 0
\(277\) 24.1032i 1.44822i −0.689683 0.724112i \(-0.742250\pi\)
0.689683 0.724112i \(-0.257750\pi\)
\(278\) 0 0
\(279\) 0.453419i 0.0271455i
\(280\) 0 0
\(281\) −21.5683 −1.28666 −0.643330 0.765589i \(-0.722448\pi\)
−0.643330 + 0.765589i \(0.722448\pi\)
\(282\) 0 0
\(283\) 8.64733i 0.514030i 0.966407 + 0.257015i \(0.0827390\pi\)
−0.966407 + 0.257015i \(0.917261\pi\)
\(284\) 0 0
\(285\) 3.75619i 0.222497i
\(286\) 0 0
\(287\) 11.3611i 0.670625i
\(288\) 0 0
\(289\) 31.6080 1.85930
\(290\) 0 0
\(291\) 14.8050i 0.867883i
\(292\) 0 0
\(293\) 9.75977 0.570172 0.285086 0.958502i \(-0.407978\pi\)
0.285086 + 0.958502i \(0.407978\pi\)
\(294\) 0 0
\(295\) 25.8592i 1.50558i
\(296\) 0 0
\(297\) 12.6984i 0.736838i
\(298\) 0 0
\(299\) 40.6938i 2.35338i
\(300\) 0 0
\(301\) −4.05454 −0.233700
\(302\) 0 0
\(303\) 6.37584 0.366283
\(304\) 0 0
\(305\) 13.6264 0.780244
\(306\) 0 0
\(307\) −0.210125 −0.0119925 −0.00599623 0.999982i \(-0.501909\pi\)
−0.00599623 + 0.999982i \(0.501909\pi\)
\(308\) 0 0
\(309\) 11.4520 0.651484
\(310\) 0 0
\(311\) 25.2791 1.43345 0.716724 0.697357i \(-0.245641\pi\)
0.716724 + 0.697357i \(0.245641\pi\)
\(312\) 0 0
\(313\) 16.9644i 0.958883i −0.877574 0.479441i \(-0.840839\pi\)
0.877574 0.479441i \(-0.159161\pi\)
\(314\) 0 0
\(315\) 2.63773i 0.148619i
\(316\) 0 0
\(317\) −27.5277 −1.54611 −0.773054 0.634340i \(-0.781272\pi\)
−0.773054 + 0.634340i \(0.781272\pi\)
\(318\) 0 0
\(319\) 5.76415 0.322731
\(320\) 0 0
\(321\) 3.31939i 0.185270i
\(322\) 0 0
\(323\) 6.97195i 0.387929i
\(324\) 0 0
\(325\) 11.1117 0.616365
\(326\) 0 0
\(327\) 37.9749 2.10002
\(328\) 0 0
\(329\) 5.29448 0.291894
\(330\) 0 0
\(331\) 13.1765 0.724246 0.362123 0.932130i \(-0.382052\pi\)
0.362123 + 0.932130i \(0.382052\pi\)
\(332\) 0 0
\(333\) −11.5793 −0.634543
\(334\) 0 0
\(335\) −9.16983 −0.501001
\(336\) 0 0
\(337\) 20.8326i 1.13482i 0.823435 + 0.567411i \(0.192055\pi\)
−0.823435 + 0.567411i \(0.807945\pi\)
\(338\) 0 0
\(339\) 31.1750i 1.69319i
\(340\) 0 0
\(341\) 1.25293i 0.0678502i
\(342\) 0 0
\(343\) −14.2065 −0.767077
\(344\) 0 0
\(345\) 23.7432i 1.27829i
\(346\) 0 0
\(347\) −21.7844 −1.16945 −0.584723 0.811233i \(-0.698797\pi\)
−0.584723 + 0.811233i \(0.698797\pi\)
\(348\) 0 0
\(349\) 4.10963i 0.219983i −0.993933 0.109992i \(-0.964918\pi\)
0.993933 0.109992i \(-0.0350824\pi\)
\(350\) 0 0
\(351\) 22.5936i 1.20596i
\(352\) 0 0
\(353\) 28.9921i 1.54309i 0.636172 + 0.771547i \(0.280517\pi\)
−0.636172 + 0.771547i \(0.719483\pi\)
\(354\) 0 0
\(355\) −20.6286 −1.09485
\(356\) 0 0
\(357\) 16.1131i 0.852797i
\(358\) 0 0
\(359\) 16.7496i 0.884009i 0.897013 + 0.442005i \(0.145733\pi\)
−0.897013 + 0.442005i \(0.854267\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 4.34273i 0.227934i
\(364\) 0 0
\(365\) −20.1782 −1.05617
\(366\) 0 0
\(367\) −4.07482 −0.212704 −0.106352 0.994329i \(-0.533917\pi\)
−0.106352 + 0.994329i \(0.533917\pi\)
\(368\) 0 0
\(369\) 13.3622i 0.695608i
\(370\) 0 0
\(371\) 7.46266 + 3.16234i 0.387442 + 0.164180i
\(372\) 0 0
\(373\) 2.39014i 0.123757i −0.998084 0.0618783i \(-0.980291\pi\)
0.998084 0.0618783i \(-0.0197091\pi\)
\(374\) 0 0
\(375\) −25.2642 −1.30464
\(376\) 0 0
\(377\) 10.2558 0.528201
\(378\) 0 0
\(379\) 30.6785i 1.57585i 0.615773 + 0.787924i \(0.288844\pi\)
−0.615773 + 0.787924i \(0.711156\pi\)
\(380\) 0 0
\(381\) 39.0297 1.99955
\(382\) 0 0
\(383\) 23.9732i 1.22497i 0.790481 + 0.612486i \(0.209830\pi\)
−0.790481 + 0.612486i \(0.790170\pi\)
\(384\) 0 0
\(385\) 7.28884i 0.371474i
\(386\) 0 0
\(387\) 4.76869 0.242406
\(388\) 0 0
\(389\) 2.00795i 0.101807i −0.998704 0.0509035i \(-0.983790\pi\)
0.998704 0.0509035i \(-0.0162101\pi\)
\(390\) 0 0
\(391\) 44.0702i 2.22873i
\(392\) 0 0
\(393\) 5.24542i 0.264596i
\(394\) 0 0
\(395\) 11.7909 0.593264
\(396\) 0 0
\(397\) 33.6928i 1.69100i 0.533979 + 0.845498i \(0.320696\pi\)
−0.533979 + 0.845498i \(0.679304\pi\)
\(398\) 0 0
\(399\) −2.31114 −0.115702
\(400\) 0 0
\(401\) 32.6847i 1.63220i −0.577912 0.816099i \(-0.696132\pi\)
0.577912 0.816099i \(-0.303868\pi\)
\(402\) 0 0
\(403\) 2.22927i 0.111048i
\(404\) 0 0
\(405\) 20.2902i 1.00823i
\(406\) 0 0
\(407\) 31.9971 1.58604
\(408\) 0 0
\(409\) −36.7668 −1.81800 −0.909000 0.416797i \(-0.863153\pi\)
−0.909000 + 0.416797i \(0.863153\pi\)
\(410\) 0 0
\(411\) −0.626663 −0.0309110
\(412\) 0 0
\(413\) 15.9108 0.782921
\(414\) 0 0
\(415\) 6.12272 0.300552
\(416\) 0 0
\(417\) −22.0044 −1.07756
\(418\) 0 0
\(419\) 1.73470i 0.0847458i −0.999102 0.0423729i \(-0.986508\pi\)
0.999102 0.0423729i \(-0.0134918\pi\)
\(420\) 0 0
\(421\) 16.1548i 0.787335i −0.919253 0.393667i \(-0.871206\pi\)
0.919253 0.393667i \(-0.128794\pi\)
\(422\) 0 0
\(423\) −6.22702 −0.302768
\(424\) 0 0
\(425\) −12.0336 −0.583717
\(426\) 0 0
\(427\) 8.38415i 0.405737i
\(428\) 0 0
\(429\) 48.3557i 2.33464i
\(430\) 0 0
\(431\) −0.470516 −0.0226640 −0.0113320 0.999936i \(-0.503607\pi\)
−0.0113320 + 0.999936i \(0.503607\pi\)
\(432\) 0 0
\(433\) 2.49971 0.120128 0.0600641 0.998195i \(-0.480869\pi\)
0.0600641 + 0.998195i \(0.480869\pi\)
\(434\) 0 0
\(435\) −5.98385 −0.286904
\(436\) 0 0
\(437\) 6.32108 0.302378
\(438\) 0 0
\(439\) 24.9522 1.19090 0.595452 0.803391i \(-0.296973\pi\)
0.595452 + 0.803391i \(0.296973\pi\)
\(440\) 0 0
\(441\) 7.54288 0.359185
\(442\) 0 0
\(443\) 9.11705i 0.433164i 0.976264 + 0.216582i \(0.0694909\pi\)
−0.976264 + 0.216582i \(0.930509\pi\)
\(444\) 0 0
\(445\) 9.95169i 0.471755i
\(446\) 0 0
\(447\) 24.2890i 1.14883i
\(448\) 0 0
\(449\) −18.5551 −0.875669 −0.437834 0.899056i \(-0.644254\pi\)
−0.437834 + 0.899056i \(0.644254\pi\)
\(450\) 0 0
\(451\) 36.9237i 1.73867i
\(452\) 0 0
\(453\) 0.415731 0.0195327
\(454\) 0 0
\(455\) 12.9686i 0.607977i
\(456\) 0 0
\(457\) 10.5736i 0.494612i −0.968937 0.247306i \(-0.920455\pi\)
0.968937 0.247306i \(-0.0795454\pi\)
\(458\) 0 0
\(459\) 24.4682i 1.14208i
\(460\) 0 0
\(461\) 1.83054 0.0852568 0.0426284 0.999091i \(-0.486427\pi\)
0.0426284 + 0.999091i \(0.486427\pi\)
\(462\) 0 0
\(463\) 11.2235i 0.521599i −0.965393 0.260799i \(-0.916014\pi\)
0.965393 0.260799i \(-0.0839861\pi\)
\(464\) 0 0
\(465\) 1.30069i 0.0603180i
\(466\) 0 0
\(467\) 2.55857 0.118397 0.0591983 0.998246i \(-0.481146\pi\)
0.0591983 + 0.998246i \(0.481146\pi\)
\(468\) 0 0
\(469\) 5.64208i 0.260527i
\(470\) 0 0
\(471\) −13.2372 −0.609938
\(472\) 0 0
\(473\) −13.1773 −0.605894
\(474\) 0 0
\(475\) 1.72601i 0.0791947i
\(476\) 0 0
\(477\) −8.77710 3.71934i −0.401876 0.170297i
\(478\) 0 0
\(479\) 11.1251i 0.508318i 0.967162 + 0.254159i \(0.0817987\pi\)
−0.967162 + 0.254159i \(0.918201\pi\)
\(480\) 0 0
\(481\) 56.9306 2.59581
\(482\) 0 0
\(483\) 14.6089 0.664727
\(484\) 0 0
\(485\) 12.9044i 0.585958i
\(486\) 0 0
\(487\) 23.5147 1.06555 0.532777 0.846256i \(-0.321149\pi\)
0.532777 + 0.846256i \(0.321149\pi\)
\(488\) 0 0
\(489\) 8.13652i 0.367946i
\(490\) 0 0
\(491\) 20.9397i 0.944995i −0.881332 0.472498i \(-0.843353\pi\)
0.881332 0.472498i \(-0.156647\pi\)
\(492\) 0 0
\(493\) −11.1067 −0.500223
\(494\) 0 0
\(495\) 8.57266i 0.385312i
\(496\) 0 0
\(497\) 12.6925i 0.569338i
\(498\) 0 0
\(499\) 18.0171i 0.806557i 0.915077 + 0.403279i \(0.132129\pi\)
−0.915077 + 0.403279i \(0.867871\pi\)
\(500\) 0 0
\(501\) −36.4756 −1.62961
\(502\) 0 0
\(503\) 8.50820i 0.379362i −0.981846 0.189681i \(-0.939255\pi\)
0.981846 0.189681i \(-0.0607453\pi\)
\(504\) 0 0
\(505\) 5.55735 0.247299
\(506\) 0 0
\(507\) 59.0496i 2.62249i
\(508\) 0 0
\(509\) 20.4527i 0.906549i 0.891371 + 0.453275i \(0.149744\pi\)
−0.891371 + 0.453275i \(0.850256\pi\)
\(510\) 0 0
\(511\) 12.4154i 0.549225i
\(512\) 0 0
\(513\) −3.50952 −0.154949
\(514\) 0 0
\(515\) 9.98190 0.439855
\(516\) 0 0
\(517\) 17.2071 0.756768
\(518\) 0 0
\(519\) −37.5783 −1.64950
\(520\) 0 0
\(521\) −27.1086 −1.18765 −0.593824 0.804595i \(-0.702382\pi\)
−0.593824 + 0.804595i \(0.702382\pi\)
\(522\) 0 0
\(523\) 36.3889 1.59117 0.795587 0.605839i \(-0.207162\pi\)
0.795587 + 0.605839i \(0.207162\pi\)
\(524\) 0 0
\(525\) 3.98904i 0.174096i
\(526\) 0 0
\(527\) 2.41424i 0.105166i
\(528\) 0 0
\(529\) −16.9561 −0.737220
\(530\) 0 0
\(531\) −18.7133 −0.812088
\(532\) 0 0
\(533\) 65.6962i 2.84562i
\(534\) 0 0
\(535\) 2.89326i 0.125087i
\(536\) 0 0
\(537\) −25.7972 −1.11323
\(538\) 0 0
\(539\) −20.8432 −0.897782
\(540\) 0 0
\(541\) 25.3899 1.09160 0.545798 0.837917i \(-0.316226\pi\)
0.545798 + 0.837917i \(0.316226\pi\)
\(542\) 0 0
\(543\) 14.5990 0.626502
\(544\) 0 0
\(545\) 33.0999 1.41785
\(546\) 0 0
\(547\) 36.6208 1.56579 0.782896 0.622153i \(-0.213742\pi\)
0.782896 + 0.622153i \(0.213742\pi\)
\(548\) 0 0
\(549\) 9.86089i 0.420852i
\(550\) 0 0
\(551\) 1.59306i 0.0678668i
\(552\) 0 0
\(553\) 7.25479i 0.308505i
\(554\) 0 0
\(555\) −33.2167 −1.40997
\(556\) 0 0
\(557\) 33.9146i 1.43701i −0.695523 0.718504i \(-0.744827\pi\)
0.695523 0.718504i \(-0.255173\pi\)
\(558\) 0 0
\(559\) −23.4456 −0.991643
\(560\) 0 0
\(561\) 52.3679i 2.21097i
\(562\) 0 0
\(563\) 1.23922i 0.0522268i −0.999659 0.0261134i \(-0.991687\pi\)
0.999659 0.0261134i \(-0.00831309\pi\)
\(564\) 0 0
\(565\) 27.1729i 1.14317i
\(566\) 0 0
\(567\) −12.4843 −0.524292
\(568\) 0 0
\(569\) 17.4079i 0.729779i −0.931051 0.364889i \(-0.881107\pi\)
0.931051 0.364889i \(-0.118893\pi\)
\(570\) 0 0
\(571\) 2.41087i 0.100892i 0.998727 + 0.0504458i \(0.0160642\pi\)
−0.998727 + 0.0504458i \(0.983936\pi\)
\(572\) 0 0
\(573\) −17.9236 −0.748769
\(574\) 0 0
\(575\) 10.9102i 0.454988i
\(576\) 0 0
\(577\) 26.3394 1.09652 0.548261 0.836307i \(-0.315290\pi\)
0.548261 + 0.836307i \(0.315290\pi\)
\(578\) 0 0
\(579\) 31.3083 1.30113
\(580\) 0 0
\(581\) 3.76723i 0.156291i
\(582\) 0 0
\(583\) 24.2537 + 10.2776i 1.00449 + 0.425656i
\(584\) 0 0
\(585\) 15.2528i 0.630627i
\(586\) 0 0
\(587\) 3.00249 0.123926 0.0619630 0.998078i \(-0.480264\pi\)
0.0619630 + 0.998078i \(0.480264\pi\)
\(588\) 0 0
\(589\) −0.346279 −0.0142682
\(590\) 0 0
\(591\) 15.0416i 0.618727i
\(592\) 0 0
\(593\) 19.8221 0.813996 0.406998 0.913429i \(-0.366576\pi\)
0.406998 + 0.913429i \(0.366576\pi\)
\(594\) 0 0
\(595\) 14.0446i 0.575773i
\(596\) 0 0
\(597\) 2.12752i 0.0870735i
\(598\) 0 0
\(599\) 8.53257 0.348632 0.174316 0.984690i \(-0.444229\pi\)
0.174316 + 0.984690i \(0.444229\pi\)
\(600\) 0 0
\(601\) 21.0541i 0.858815i −0.903111 0.429408i \(-0.858722\pi\)
0.903111 0.429408i \(-0.141278\pi\)
\(602\) 0 0
\(603\) 6.63585i 0.270233i
\(604\) 0 0
\(605\) 3.78524i 0.153892i
\(606\) 0 0
\(607\) 4.28330 0.173854 0.0869269 0.996215i \(-0.472295\pi\)
0.0869269 + 0.996215i \(0.472295\pi\)
\(608\) 0 0
\(609\) 3.68179i 0.149194i
\(610\) 0 0
\(611\) 30.6156 1.23857
\(612\) 0 0
\(613\) 19.3708i 0.782378i −0.920310 0.391189i \(-0.872064\pi\)
0.920310 0.391189i \(-0.127936\pi\)
\(614\) 0 0
\(615\) 38.3311i 1.54566i
\(616\) 0 0
\(617\) 15.6768i 0.631124i 0.948905 + 0.315562i \(0.102193\pi\)
−0.948905 + 0.315562i \(0.897807\pi\)
\(618\) 0 0
\(619\) −2.02148 −0.0812500 −0.0406250 0.999174i \(-0.512935\pi\)
−0.0406250 + 0.999174i \(0.512935\pi\)
\(620\) 0 0
\(621\) 22.1840 0.890212
\(622\) 0 0
\(623\) −6.12315 −0.245319
\(624\) 0 0
\(625\) −13.3909 −0.535634
\(626\) 0 0
\(627\) −7.51123 −0.299970
\(628\) 0 0
\(629\) −61.6542 −2.45831
\(630\) 0 0
\(631\) 24.2114i 0.963842i 0.876215 + 0.481921i \(0.160061\pi\)
−0.876215 + 0.481921i \(0.839939\pi\)
\(632\) 0 0
\(633\) 50.6087i 2.01152i
\(634\) 0 0
\(635\) 34.0193 1.35001
\(636\) 0 0
\(637\) −37.0851 −1.46937
\(638\) 0 0
\(639\) 14.9281i 0.590548i
\(640\) 0 0
\(641\) 25.8246i 1.02001i −0.860172 0.510004i \(-0.829644\pi\)
0.860172 0.510004i \(-0.170356\pi\)
\(642\) 0 0
\(643\) 38.9182 1.53478 0.767392 0.641178i \(-0.221554\pi\)
0.767392 + 0.641178i \(0.221554\pi\)
\(644\) 0 0
\(645\) 13.6796 0.538632
\(646\) 0 0
\(647\) 23.1605 0.910534 0.455267 0.890355i \(-0.349544\pi\)
0.455267 + 0.890355i \(0.349544\pi\)
\(648\) 0 0
\(649\) 51.7104 2.02981
\(650\) 0 0
\(651\) −0.800298 −0.0313661
\(652\) 0 0
\(653\) 13.7838 0.539400 0.269700 0.962944i \(-0.413075\pi\)
0.269700 + 0.962944i \(0.413075\pi\)
\(654\) 0 0
\(655\) 4.57204i 0.178645i
\(656\) 0 0
\(657\) 14.6022i 0.569685i
\(658\) 0 0
\(659\) 33.0183i 1.28621i 0.765777 + 0.643106i \(0.222355\pi\)
−0.765777 + 0.643106i \(0.777645\pi\)
\(660\) 0 0
\(661\) −6.25752 −0.243389 −0.121695 0.992568i \(-0.538833\pi\)
−0.121695 + 0.992568i \(0.538833\pi\)
\(662\) 0 0
\(663\) 93.1751i 3.61862i
\(664\) 0 0
\(665\) −2.01445 −0.0781169
\(666\) 0 0
\(667\) 10.0699i 0.389907i
\(668\) 0 0
\(669\) 25.8182i 0.998191i
\(670\) 0 0
\(671\) 27.2486i 1.05192i
\(672\) 0 0
\(673\) −18.1994 −0.701536 −0.350768 0.936462i \(-0.614079\pi\)
−0.350768 + 0.936462i \(0.614079\pi\)
\(674\) 0 0
\(675\) 6.05746i 0.233152i
\(676\) 0 0
\(677\) 9.18112i 0.352859i 0.984313 + 0.176429i \(0.0564548\pi\)
−0.984313 + 0.176429i \(0.943545\pi\)
\(678\) 0 0
\(679\) 7.93992 0.304706
\(680\) 0 0
\(681\) 38.9316i 1.49186i
\(682\) 0 0
\(683\) 33.1304 1.26770 0.633849 0.773457i \(-0.281474\pi\)
0.633849 + 0.773457i \(0.281474\pi\)
\(684\) 0 0
\(685\) −0.546215 −0.0208698
\(686\) 0 0
\(687\) 12.3820i 0.472401i
\(688\) 0 0
\(689\) 43.1532 + 18.2864i 1.64401 + 0.696656i
\(690\) 0 0
\(691\) 12.4262i 0.472714i 0.971666 + 0.236357i \(0.0759536\pi\)
−0.971666 + 0.236357i \(0.924046\pi\)
\(692\) 0 0
\(693\) −5.27465 −0.200368
\(694\) 0 0
\(695\) −19.1796 −0.727524
\(696\) 0 0
\(697\) 71.1471i 2.69489i
\(698\) 0 0
\(699\) −30.2364 −1.14365
\(700\) 0 0
\(701\) 6.92468i 0.261542i −0.991413 0.130771i \(-0.958255\pi\)
0.991413 0.130771i \(-0.0417452\pi\)
\(702\) 0 0
\(703\) 8.84318i 0.333527i
\(704\) 0 0
\(705\) −17.8630 −0.672758
\(706\) 0 0
\(707\) 3.41937i 0.128599i
\(708\) 0 0
\(709\) 7.46483i 0.280347i −0.990127 0.140174i \(-0.955234\pi\)
0.990127 0.140174i \(-0.0447661\pi\)
\(710\) 0 0
\(711\) 8.53261i 0.319998i
\(712\) 0 0
\(713\) 2.18885 0.0819733
\(714\) 0 0
\(715\) 42.1481i 1.57625i
\(716\) 0 0
\(717\) 52.4951 1.96047
\(718\) 0 0
\(719\) 18.5673i 0.692444i −0.938153 0.346222i \(-0.887464\pi\)
0.938153 0.346222i \(-0.112536\pi\)
\(720\) 0 0
\(721\) 6.14174i 0.228730i
\(722\) 0 0
\(723\) 50.6358i 1.88316i
\(724\) 0 0
\(725\) 2.74964 0.102119
\(726\) 0 0
\(727\) 43.5615 1.61561 0.807803 0.589453i \(-0.200657\pi\)
0.807803 + 0.589453i \(0.200657\pi\)
\(728\) 0 0
\(729\) −7.17311 −0.265671
\(730\) 0 0
\(731\) 25.3909 0.939117
\(732\) 0 0
\(733\) −35.2011 −1.30018 −0.650092 0.759856i \(-0.725270\pi\)
−0.650092 + 0.759856i \(0.725270\pi\)
\(734\) 0 0
\(735\) 21.6377 0.798117
\(736\) 0 0
\(737\) 18.3369i 0.675447i
\(738\) 0 0
\(739\) 35.7783i 1.31613i 0.752963 + 0.658063i \(0.228624\pi\)
−0.752963 + 0.658063i \(0.771376\pi\)
\(740\) 0 0
\(741\) −13.3643 −0.490949
\(742\) 0 0
\(743\) −8.21910 −0.301530 −0.150765 0.988570i \(-0.548174\pi\)
−0.150765 + 0.988570i \(0.548174\pi\)
\(744\) 0 0
\(745\) 21.1709i 0.775643i
\(746\) 0 0
\(747\) 4.43077i 0.162114i
\(748\) 0 0
\(749\) −1.78019 −0.0650467
\(750\) 0 0
\(751\) 5.71438 0.208521 0.104260 0.994550i \(-0.466753\pi\)
0.104260 + 0.994550i \(0.466753\pi\)
\(752\) 0 0
\(753\) −3.15081 −0.114822
\(754\) 0 0
\(755\) 0.362362 0.0131877
\(756\) 0 0
\(757\) −9.56822 −0.347763 −0.173881 0.984767i \(-0.555631\pi\)
−0.173881 + 0.984767i \(0.555631\pi\)
\(758\) 0 0
\(759\) 47.4791 1.72338
\(760\) 0 0
\(761\) 4.28182i 0.155216i −0.996984 0.0776079i \(-0.975272\pi\)
0.996984 0.0776079i \(-0.0247282\pi\)
\(762\) 0 0
\(763\) 20.3660i 0.737298i
\(764\) 0 0
\(765\) 16.5184i 0.597223i
\(766\) 0 0
\(767\) 92.0053 3.32212
\(768\) 0 0
\(769\) 30.1138i 1.08593i −0.839754 0.542966i \(-0.817301\pi\)
0.839754 0.542966i \(-0.182699\pi\)
\(770\) 0 0
\(771\) 28.9310 1.04192
\(772\) 0 0
\(773\) 25.6605i 0.922943i −0.887155 0.461471i \(-0.847322\pi\)
0.887155 0.461471i \(-0.152678\pi\)
\(774\) 0 0
\(775\) 0.597680i 0.0214693i
\(776\) 0 0
\(777\) 20.4378i 0.733203i
\(778\) 0 0
\(779\) −10.2048 −0.365624
\(780\) 0 0
\(781\) 41.2509i 1.47607i
\(782\) 0 0
\(783\) 5.59089i 0.199802i
\(784\) 0 0
\(785\) −11.5379 −0.411805
\(786\) 0 0
\(787\) 13.4167i 0.478255i 0.970988 + 0.239127i \(0.0768614\pi\)
−0.970988 + 0.239127i \(0.923139\pi\)
\(788\) 0 0
\(789\) 45.9047 1.63425
\(790\) 0 0
\(791\) −16.7192 −0.594465
\(792\) 0 0
\(793\) 48.4818i 1.72164i
\(794\) 0 0
\(795\) −25.1782 10.6694i −0.892977 0.378403i
\(796\) 0 0
\(797\) 0.166602i 0.00590134i 0.999996 + 0.00295067i \(0.000939229\pi\)
−0.999996 + 0.00295067i \(0.999061\pi\)
\(798\) 0 0
\(799\) −33.1558 −1.17297
\(800\) 0 0
\(801\) 7.20165 0.254458
\(802\) 0 0
\(803\) 40.3502i 1.42393i
\(804\) 0 0
\(805\) 12.7335 0.448796
\(806\) 0 0
\(807\) 22.0878i 0.777529i
\(808\) 0 0
\(809\) 19.6632i 0.691322i 0.938360 + 0.345661i \(0.112345\pi\)
−0.938360 + 0.345661i \(0.887655\pi\)
\(810\) 0 0
\(811\) −3.10444 −0.109012 −0.0545058 0.998513i \(-0.517358\pi\)
−0.0545058 + 0.998513i \(0.517358\pi\)
\(812\) 0 0
\(813\) 58.0932i 2.03742i
\(814\) 0 0
\(815\) 7.09200i 0.248422i
\(816\) 0 0
\(817\) 3.64187i 0.127413i
\(818\) 0 0
\(819\) −9.38488 −0.327934
\(820\) 0 0
\(821\) 32.5283i 1.13525i 0.823288 + 0.567623i \(0.192137\pi\)
−0.823288 + 0.567623i \(0.807863\pi\)
\(822\) 0 0
\(823\) −2.04964 −0.0714460 −0.0357230 0.999362i \(-0.511373\pi\)
−0.0357230 + 0.999362i \(0.511373\pi\)
\(824\) 0 0
\(825\) 12.9644i 0.451364i
\(826\) 0 0
\(827\) 43.0643i 1.49749i −0.662856 0.748747i \(-0.730656\pi\)
0.662856 0.748747i \(-0.269344\pi\)
\(828\) 0 0
\(829\) 31.1360i 1.08140i 0.841217 + 0.540698i \(0.181840\pi\)
−0.841217 + 0.540698i \(0.818160\pi\)
\(830\) 0 0
\(831\) −50.0362 −1.73574
\(832\) 0 0
\(833\) 40.1621 1.39154
\(834\) 0 0
\(835\) −31.7931 −1.10024
\(836\) 0 0
\(837\) −1.21527 −0.0420060
\(838\) 0 0
\(839\) −46.5149 −1.60587 −0.802936 0.596065i \(-0.796730\pi\)
−0.802936 + 0.596065i \(0.796730\pi\)
\(840\) 0 0
\(841\) −26.4621 −0.912488
\(842\) 0 0
\(843\) 44.7740i 1.54210i
\(844\) 0 0
\(845\) 51.4692i 1.77059i
\(846\) 0 0
\(847\) 2.32901 0.0800257
\(848\) 0 0
\(849\) 17.9511 0.616079
\(850\) 0 0
\(851\) 55.8985i 1.91618i
\(852\) 0 0
\(853\) 2.80122i 0.0959120i −0.998849 0.0479560i \(-0.984729\pi\)
0.998849 0.0479560i \(-0.0152707\pi\)
\(854\) 0 0
\(855\) 2.36926 0.0810271
\(856\) 0 0
\(857\) −46.8878 −1.60166 −0.800829 0.598893i \(-0.795608\pi\)
−0.800829 + 0.598893i \(0.795608\pi\)
\(858\) 0 0
\(859\) 20.0424 0.683837 0.341918 0.939730i \(-0.388923\pi\)
0.341918 + 0.939730i \(0.388923\pi\)
\(860\) 0 0
\(861\) −23.5846 −0.803762
\(862\) 0 0
\(863\) −0.0913133 −0.00310834 −0.00155417 0.999999i \(-0.500495\pi\)
−0.00155417 + 0.999999i \(0.500495\pi\)
\(864\) 0 0
\(865\) −32.7542 −1.11368
\(866\) 0 0
\(867\) 65.6155i 2.22842i
\(868\) 0 0
\(869\) 23.5782i 0.799834i
\(870\) 0 0
\(871\) 32.6257i 1.10548i
\(872\) 0 0
\(873\) −9.33841 −0.316057
\(874\) 0 0
\(875\) 13.5492i 0.458046i
\(876\) 0 0
\(877\) 51.0677 1.72443 0.862217 0.506539i \(-0.169075\pi\)
0.862217 + 0.506539i \(0.169075\pi\)
\(878\) 0 0
\(879\) 20.2604i 0.683367i
\(880\) 0 0
\(881\) 37.2222i 1.25405i 0.779000 + 0.627024i \(0.215727\pi\)
−0.779000 + 0.627024i \(0.784273\pi\)
\(882\) 0 0
\(883\) 53.5955i 1.80363i 0.432119 + 0.901817i \(0.357766\pi\)
−0.432119 + 0.901817i \(0.642234\pi\)
\(884\) 0 0
\(885\) −53.6813 −1.80448
\(886\) 0 0
\(887\) 31.0832i 1.04367i 0.853046 + 0.521835i \(0.174752\pi\)
−0.853046 + 0.521835i \(0.825248\pi\)
\(888\) 0 0
\(889\) 20.9316i 0.702025i
\(890\) 0 0
\(891\) −40.5742 −1.35929
\(892\) 0 0
\(893\) 4.75561i 0.159140i
\(894\) 0 0
\(895\) −22.4855 −0.751608
\(896\) 0 0
\(897\) 84.4767 2.82060
\(898\) 0 0
\(899\) 0.551644i 0.0183983i
\(900\) 0 0
\(901\) −46.7337 19.8036i −1.55693 0.659755i
\(902\) 0 0
\(903\) 8.41686i 0.280096i
\(904\) 0 0
\(905\) 12.7248 0.422988
\(906\) 0 0
\(907\) −23.5449 −0.781795 −0.390897 0.920434i \(-0.627835\pi\)
−0.390897 + 0.920434i \(0.627835\pi\)
\(908\) 0 0
\(909\) 4.02164i 0.133389i
\(910\) 0 0
\(911\) −25.1271 −0.832499 −0.416249 0.909251i \(-0.636656\pi\)
−0.416249 + 0.909251i \(0.636656\pi\)
\(912\) 0 0
\(913\) 12.2436i 0.405203i
\(914\) 0 0
\(915\) 28.2871i 0.935144i
\(916\) 0 0
\(917\) 2.81312 0.0928975
\(918\) 0 0
\(919\) 33.3629i 1.10054i 0.834987 + 0.550270i \(0.185475\pi\)
−0.834987 + 0.550270i \(0.814525\pi\)
\(920\) 0 0
\(921\) 0.436200i 0.0143733i
\(922\) 0 0
\(923\) 73.3953i 2.41583i
\(924\) 0 0
\(925\) 15.2634 0.501858
\(926\) 0 0
\(927\) 7.22352i 0.237252i
\(928\) 0 0
\(929\) 8.80316 0.288822 0.144411 0.989518i \(-0.453871\pi\)
0.144411 + 0.989518i \(0.453871\pi\)
\(930\) 0 0
\(931\) 5.76053i 0.188794i
\(932\) 0 0
\(933\) 52.4772i 1.71803i
\(934\) 0 0
\(935\) 45.6452i 1.49276i
\(936\) 0 0
\(937\) 18.0788 0.590609 0.295304 0.955403i \(-0.404579\pi\)
0.295304 + 0.955403i \(0.404579\pi\)
\(938\) 0 0
\(939\) −35.2165 −1.14925
\(940\) 0 0
\(941\) −49.9859 −1.62949 −0.814747 0.579817i \(-0.803124\pi\)
−0.814747 + 0.579817i \(0.803124\pi\)
\(942\) 0 0
\(943\) 64.5052 2.10058
\(944\) 0 0
\(945\) −7.06975 −0.229979
\(946\) 0 0
\(947\) −21.8364 −0.709588 −0.354794 0.934945i \(-0.615449\pi\)
−0.354794 + 0.934945i \(0.615449\pi\)
\(948\) 0 0
\(949\) 71.7927i 2.33049i
\(950\) 0 0
\(951\) 57.1450i 1.85305i
\(952\) 0 0
\(953\) 7.86659 0.254824 0.127412 0.991850i \(-0.459333\pi\)
0.127412 + 0.991850i \(0.459333\pi\)
\(954\) 0 0
\(955\) −15.6227 −0.505538
\(956\) 0 0
\(957\) 11.9659i 0.386801i
\(958\) 0 0
\(959\) 0.336080i 0.0108526i
\(960\) 0 0
\(961\) 30.8801 0.996132
\(962\) 0 0
\(963\) 2.09374 0.0674699
\(964\) 0 0
\(965\) 27.2892 0.878469
\(966\) 0 0
\(967\) 36.5809 1.17636 0.588181 0.808729i \(-0.299844\pi\)
0.588181 + 0.808729i \(0.299844\pi\)
\(968\) 0 0
\(969\) 14.4731 0.464944
\(970\) 0 0
\(971\) 21.7963 0.699475 0.349737 0.936848i \(-0.386271\pi\)
0.349737 + 0.936848i \(0.386271\pi\)
\(972\) 0 0
\(973\) 11.8010i 0.378322i
\(974\) 0 0
\(975\) 23.0669i 0.738731i
\(976\) 0 0
\(977\) 27.3745i 0.875788i 0.899027 + 0.437894i \(0.144276\pi\)
−0.899027 + 0.437894i \(0.855724\pi\)
\(978\) 0 0
\(979\) −19.9003 −0.636017
\(980\) 0 0
\(981\) 23.9532i 0.764765i
\(982\) 0 0
\(983\) −3.64589 −0.116286 −0.0581429 0.998308i \(-0.518518\pi\)
−0.0581429 + 0.998308i \(0.518518\pi\)
\(984\) 0 0
\(985\) 13.1106i 0.417739i
\(986\) 0 0
\(987\) 10.9909i 0.349843i
\(988\) 0 0
\(989\) 23.0206i 0.732011i
\(990\) 0 0
\(991\) 10.7764 0.342324 0.171162 0.985243i \(-0.445248\pi\)
0.171162 + 0.985243i \(0.445248\pi\)
\(992\) 0 0
\(993\) 27.3532i 0.868028i
\(994\) 0 0
\(995\) 1.85440i 0.0587884i
\(996\) 0 0
\(997\) 47.9952 1.52002 0.760011 0.649910i \(-0.225194\pi\)
0.760011 + 0.649910i \(0.225194\pi\)
\(998\) 0 0
\(999\) 31.0354i 0.981915i
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 4028.2.c.a.3497.18 82
53.52 even 2 inner 4028.2.c.a.3497.65 yes 82
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4028.2.c.a.3497.18 82 1.1 even 1 trivial
4028.2.c.a.3497.65 yes 82 53.52 even 2 inner