Properties

Label 528.2.o.b.175.3
Level $528$
Weight $2$
Character 528.175
Analytic conductor $4.216$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,2,Mod(175,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("528.175");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.o (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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.454201344.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 175.3
Root \(1.02715 + 1.10132i\) of defining polynomial
Character \(\chi\) \(=\) 528.175
Dual form 528.2.o.b.175.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} +2.73205 q^{5} -5.13734 q^{7} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{3} +2.73205 q^{5} -5.13734 q^{7} -1.00000 q^{9} +(1.88040 - 2.73205i) q^{11} -5.13734i q^{13} -2.73205i q^{15} -3.76080i q^{17} +3.76080 q^{19} +5.13734i q^{21} -1.26795i q^{23} +2.46410 q^{25} +1.00000i q^{27} -2.00000i q^{31} +(-2.73205 - 1.88040i) q^{33} -14.0355 q^{35} +0.535898 q^{37} -5.13734 q^{39} +10.2747i q^{41} +3.76080 q^{43} -2.73205 q^{45} -4.19615i q^{47} +19.3923 q^{49} -3.76080 q^{51} -10.7321 q^{53} +(5.13734 - 7.46410i) q^{55} -3.76080i q^{57} -9.46410i q^{59} +2.38425i q^{61} +5.13734 q^{63} -14.0355i q^{65} +10.3923i q^{67} -1.26795 q^{69} +10.7321i q^{71} -10.2747i q^{73} -2.46410i q^{75} +(-9.66025 + 14.0355i) q^{77} +5.13734 q^{79} +1.00000 q^{81} +10.2747 q^{83} -10.2747i q^{85} -10.3923 q^{89} +26.3923i q^{91} -2.00000 q^{93} +10.2747 q^{95} +5.46410 q^{97} +(-1.88040 + 2.73205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{5} - 8 q^{9} - 8 q^{25} - 8 q^{33} + 32 q^{37} - 8 q^{45} + 72 q^{49} - 72 q^{53} - 24 q^{69} - 8 q^{77} + 8 q^{81} - 16 q^{93} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(145\) \(353\) \(463\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000i 0.577350i
\(4\) 0 0
\(5\) 2.73205 1.22181 0.610905 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 0 0
\(7\) −5.13734 −1.94173 −0.970867 0.239620i \(-0.922977\pi\)
−0.970867 + 0.239620i \(0.922977\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.88040 2.73205i 0.566961 0.823744i
\(12\) 0 0
\(13\) 5.13734i 1.42484i −0.701752 0.712421i \(-0.747598\pi\)
0.701752 0.712421i \(-0.252402\pi\)
\(14\) 0 0
\(15\) 2.73205i 0.705412i
\(16\) 0 0
\(17\) 3.76080i 0.912127i −0.889947 0.456064i \(-0.849259\pi\)
0.889947 0.456064i \(-0.150741\pi\)
\(18\) 0 0
\(19\) 3.76080 0.862786 0.431393 0.902164i \(-0.358022\pi\)
0.431393 + 0.902164i \(0.358022\pi\)
\(20\) 0 0
\(21\) 5.13734i 1.12106i
\(22\) 0 0
\(23\) 1.26795i 0.264386i −0.991224 0.132193i \(-0.957798\pi\)
0.991224 0.132193i \(-0.0422018\pi\)
\(24\) 0 0
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 2.00000i 0.359211i −0.983739 0.179605i \(-0.942518\pi\)
0.983739 0.179605i \(-0.0574821\pi\)
\(32\) 0 0
\(33\) −2.73205 1.88040i −0.475589 0.327335i
\(34\) 0 0
\(35\) −14.0355 −2.37243
\(36\) 0 0
\(37\) 0.535898 0.0881012 0.0440506 0.999029i \(-0.485974\pi\)
0.0440506 + 0.999029i \(0.485974\pi\)
\(38\) 0 0
\(39\) −5.13734 −0.822633
\(40\) 0 0
\(41\) 10.2747i 1.60464i 0.596896 + 0.802318i \(0.296400\pi\)
−0.596896 + 0.802318i \(0.703600\pi\)
\(42\) 0 0
\(43\) 3.76080 0.573516 0.286758 0.958003i \(-0.407422\pi\)
0.286758 + 0.958003i \(0.407422\pi\)
\(44\) 0 0
\(45\) −2.73205 −0.407270
\(46\) 0 0
\(47\) 4.19615i 0.612072i −0.952020 0.306036i \(-0.900997\pi\)
0.952020 0.306036i \(-0.0990028\pi\)
\(48\) 0 0
\(49\) 19.3923 2.77033
\(50\) 0 0
\(51\) −3.76080 −0.526617
\(52\) 0 0
\(53\) −10.7321 −1.47416 −0.737080 0.675805i \(-0.763796\pi\)
−0.737080 + 0.675805i \(0.763796\pi\)
\(54\) 0 0
\(55\) 5.13734 7.46410i 0.692719 1.00646i
\(56\) 0 0
\(57\) 3.76080i 0.498130i
\(58\) 0 0
\(59\) 9.46410i 1.23212i −0.787699 0.616061i \(-0.788728\pi\)
0.787699 0.616061i \(-0.211272\pi\)
\(60\) 0 0
\(61\) 2.38425i 0.305272i 0.988282 + 0.152636i \(0.0487762\pi\)
−0.988282 + 0.152636i \(0.951224\pi\)
\(62\) 0 0
\(63\) 5.13734 0.647245
\(64\) 0 0
\(65\) 14.0355i 1.74089i
\(66\) 0 0
\(67\) 10.3923i 1.26962i 0.772667 + 0.634811i \(0.218922\pi\)
−0.772667 + 0.634811i \(0.781078\pi\)
\(68\) 0 0
\(69\) −1.26795 −0.152643
\(70\) 0 0
\(71\) 10.7321i 1.27366i 0.771004 + 0.636830i \(0.219755\pi\)
−0.771004 + 0.636830i \(0.780245\pi\)
\(72\) 0 0
\(73\) 10.2747i 1.20256i −0.799038 0.601281i \(-0.794657\pi\)
0.799038 0.601281i \(-0.205343\pi\)
\(74\) 0 0
\(75\) 2.46410i 0.284530i
\(76\) 0 0
\(77\) −9.66025 + 14.0355i −1.10089 + 1.59949i
\(78\) 0 0
\(79\) 5.13734 0.577996 0.288998 0.957330i \(-0.406678\pi\)
0.288998 + 0.957330i \(0.406678\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 10.2747 1.12779 0.563897 0.825845i \(-0.309302\pi\)
0.563897 + 0.825845i \(0.309302\pi\)
\(84\) 0 0
\(85\) 10.2747i 1.11445i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.3923 −1.10158 −0.550791 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) 26.3923i 2.76667i
\(92\) 0 0
\(93\) −2.00000 −0.207390
\(94\) 0 0
\(95\) 10.2747 1.05416
\(96\) 0 0
\(97\) 5.46410 0.554795 0.277398 0.960755i \(-0.410528\pi\)
0.277398 + 0.960755i \(0.410528\pi\)
\(98\) 0 0
\(99\) −1.88040 + 2.73205i −0.188987 + 0.274581i
\(100\) 0 0
\(101\) 17.7963i 1.77080i 0.464833 + 0.885398i \(0.346114\pi\)
−0.464833 + 0.885398i \(0.653886\pi\)
\(102\) 0 0
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) 0 0
\(105\) 14.0355i 1.36972i
\(106\) 0 0
\(107\) −3.76080 −0.363570 −0.181785 0.983338i \(-0.558187\pi\)
−0.181785 + 0.983338i \(0.558187\pi\)
\(108\) 0 0
\(109\) 12.6589i 1.21251i 0.795272 + 0.606253i \(0.207328\pi\)
−0.795272 + 0.606253i \(0.792672\pi\)
\(110\) 0 0
\(111\) 0.535898i 0.0508652i
\(112\) 0 0
\(113\) 17.3205 1.62938 0.814688 0.579899i \(-0.196908\pi\)
0.814688 + 0.579899i \(0.196908\pi\)
\(114\) 0 0
\(115\) 3.46410i 0.323029i
\(116\) 0 0
\(117\) 5.13734i 0.474948i
\(118\) 0 0
\(119\) 19.3205i 1.77111i
\(120\) 0 0
\(121\) −3.92820 10.2747i −0.357109 0.934063i
\(122\) 0 0
\(123\) 10.2747 0.926437
\(124\) 0 0
\(125\) −6.92820 −0.619677
\(126\) 0 0
\(127\) −5.13734 −0.455866 −0.227933 0.973677i \(-0.573197\pi\)
−0.227933 + 0.973677i \(0.573197\pi\)
\(128\) 0 0
\(129\) 3.76080i 0.331120i
\(130\) 0 0
\(131\) 2.75309 0.240539 0.120269 0.992741i \(-0.461624\pi\)
0.120269 + 0.992741i \(0.461624\pi\)
\(132\) 0 0
\(133\) −19.3205 −1.67530
\(134\) 0 0
\(135\) 2.73205i 0.235137i
\(136\) 0 0
\(137\) 4.92820 0.421045 0.210522 0.977589i \(-0.432484\pi\)
0.210522 + 0.977589i \(0.432484\pi\)
\(138\) 0 0
\(139\) 14.0355 1.19047 0.595237 0.803550i \(-0.297058\pi\)
0.595237 + 0.803550i \(0.297058\pi\)
\(140\) 0 0
\(141\) −4.19615 −0.353380
\(142\) 0 0
\(143\) −14.0355 9.66025i −1.17371 0.807831i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 19.3923i 1.59945i
\(148\) 0 0
\(149\) 17.7963i 1.45793i −0.684552 0.728964i \(-0.740002\pi\)
0.684552 0.728964i \(-0.259998\pi\)
\(150\) 0 0
\(151\) −2.38425 −0.194027 −0.0970137 0.995283i \(-0.530929\pi\)
−0.0970137 + 0.995283i \(0.530929\pi\)
\(152\) 0 0
\(153\) 3.76080i 0.304042i
\(154\) 0 0
\(155\) 5.46410i 0.438887i
\(156\) 0 0
\(157\) 11.8564 0.946244 0.473122 0.880997i \(-0.343127\pi\)
0.473122 + 0.880997i \(0.343127\pi\)
\(158\) 0 0
\(159\) 10.7321i 0.851107i
\(160\) 0 0
\(161\) 6.51389i 0.513367i
\(162\) 0 0
\(163\) 2.92820i 0.229355i −0.993403 0.114677i \(-0.963417\pi\)
0.993403 0.114677i \(-0.0365834\pi\)
\(164\) 0 0
\(165\) −7.46410 5.13734i −0.581080 0.399942i
\(166\) 0 0
\(167\) 17.7963 1.37712 0.688559 0.725180i \(-0.258244\pi\)
0.688559 + 0.725180i \(0.258244\pi\)
\(168\) 0 0
\(169\) −13.3923 −1.03018
\(170\) 0 0
\(171\) −3.76080 −0.287595
\(172\) 0 0
\(173\) 13.0278i 0.990484i −0.868755 0.495242i \(-0.835079\pi\)
0.868755 0.495242i \(-0.164921\pi\)
\(174\) 0 0
\(175\) −12.6589 −0.956926
\(176\) 0 0
\(177\) −9.46410 −0.711365
\(178\) 0 0
\(179\) 0.392305i 0.0293222i −0.999893 0.0146611i \(-0.995333\pi\)
0.999893 0.0146611i \(-0.00466695\pi\)
\(180\) 0 0
\(181\) 2.39230 0.177819 0.0889093 0.996040i \(-0.471662\pi\)
0.0889093 + 0.996040i \(0.471662\pi\)
\(182\) 0 0
\(183\) 2.38425 0.176249
\(184\) 0 0
\(185\) 1.46410 0.107643
\(186\) 0 0
\(187\) −10.2747 7.07180i −0.751360 0.517141i
\(188\) 0 0
\(189\) 5.13734i 0.373687i
\(190\) 0 0
\(191\) 11.1244i 0.804930i 0.915435 + 0.402465i \(0.131847\pi\)
−0.915435 + 0.402465i \(0.868153\pi\)
\(192\) 0 0
\(193\) 13.0278i 0.937760i 0.883262 + 0.468880i \(0.155342\pi\)
−0.883262 + 0.468880i \(0.844658\pi\)
\(194\) 0 0
\(195\) −14.0355 −1.00510
\(196\) 0 0
\(197\) 2.75309i 0.196150i 0.995179 + 0.0980749i \(0.0312685\pi\)
−0.995179 + 0.0980749i \(0.968732\pi\)
\(198\) 0 0
\(199\) 23.4641i 1.66333i −0.555281 0.831663i \(-0.687389\pi\)
0.555281 0.831663i \(-0.312611\pi\)
\(200\) 0 0
\(201\) 10.3923 0.733017
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 28.0710i 1.96056i
\(206\) 0 0
\(207\) 1.26795i 0.0881286i
\(208\) 0 0
\(209\) 7.07180 10.2747i 0.489166 0.710715i
\(210\) 0 0
\(211\) 21.5571 1.48405 0.742025 0.670372i \(-0.233865\pi\)
0.742025 + 0.670372i \(0.233865\pi\)
\(212\) 0 0
\(213\) 10.7321 0.735348
\(214\) 0 0
\(215\) 10.2747 0.700728
\(216\) 0 0
\(217\) 10.2747i 0.697491i
\(218\) 0 0
\(219\) −10.2747 −0.694299
\(220\) 0 0
\(221\) −19.3205 −1.29964
\(222\) 0 0
\(223\) 11.4641i 0.767693i 0.923397 + 0.383847i \(0.125401\pi\)
−0.923397 + 0.383847i \(0.874599\pi\)
\(224\) 0 0
\(225\) −2.46410 −0.164273
\(226\) 0 0
\(227\) −16.7886 −1.11430 −0.557149 0.830413i \(-0.688105\pi\)
−0.557149 + 0.830413i \(0.688105\pi\)
\(228\) 0 0
\(229\) 23.8564 1.57648 0.788238 0.615371i \(-0.210994\pi\)
0.788238 + 0.615371i \(0.210994\pi\)
\(230\) 0 0
\(231\) 14.0355 + 9.66025i 0.923467 + 0.635598i
\(232\) 0 0
\(233\) 10.2747i 0.673117i 0.941663 + 0.336559i \(0.109263\pi\)
−0.941663 + 0.336559i \(0.890737\pi\)
\(234\) 0 0
\(235\) 11.4641i 0.747836i
\(236\) 0 0
\(237\) 5.13734i 0.333706i
\(238\) 0 0
\(239\) −23.3025 −1.50731 −0.753656 0.657269i \(-0.771711\pi\)
−0.753656 + 0.657269i \(0.771711\pi\)
\(240\) 0 0
\(241\) 28.0710i 1.80821i −0.427310 0.904105i \(-0.640539\pi\)
0.427310 0.904105i \(-0.359461\pi\)
\(242\) 0 0
\(243\) 1.00000i 0.0641500i
\(244\) 0 0
\(245\) 52.9808 3.38482
\(246\) 0 0
\(247\) 19.3205i 1.22933i
\(248\) 0 0
\(249\) 10.2747i 0.651132i
\(250\) 0 0
\(251\) 21.8564i 1.37956i 0.724017 + 0.689782i \(0.242294\pi\)
−0.724017 + 0.689782i \(0.757706\pi\)
\(252\) 0 0
\(253\) −3.46410 2.38425i −0.217786 0.149896i
\(254\) 0 0
\(255\) −10.2747 −0.643426
\(256\) 0 0
\(257\) −15.4641 −0.964624 −0.482312 0.875999i \(-0.660203\pi\)
−0.482312 + 0.875999i \(0.660203\pi\)
\(258\) 0 0
\(259\) −2.75309 −0.171069
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.76850 0.294038 0.147019 0.989134i \(-0.453032\pi\)
0.147019 + 0.989134i \(0.453032\pi\)
\(264\) 0 0
\(265\) −29.3205 −1.80114
\(266\) 0 0
\(267\) 10.3923i 0.635999i
\(268\) 0 0
\(269\) −0.875644 −0.0533890 −0.0266945 0.999644i \(-0.508498\pi\)
−0.0266945 + 0.999644i \(0.508498\pi\)
\(270\) 0 0
\(271\) −2.38425 −0.144833 −0.0724164 0.997374i \(-0.523071\pi\)
−0.0724164 + 0.997374i \(0.523071\pi\)
\(272\) 0 0
\(273\) 26.3923 1.59733
\(274\) 0 0
\(275\) 4.63349 6.73205i 0.279410 0.405958i
\(276\) 0 0
\(277\) 9.90584i 0.595184i −0.954693 0.297592i \(-0.903816\pi\)
0.954693 0.297592i \(-0.0961836\pi\)
\(278\) 0 0
\(279\) 2.00000i 0.119737i
\(280\) 0 0
\(281\) 9.26699i 0.552822i −0.961040 0.276411i \(-0.910855\pi\)
0.961040 0.276411i \(-0.0891451\pi\)
\(282\) 0 0
\(283\) −14.0355 −0.834323 −0.417161 0.908832i \(-0.636975\pi\)
−0.417161 + 0.908832i \(0.636975\pi\)
\(284\) 0 0
\(285\) 10.2747i 0.608620i
\(286\) 0 0
\(287\) 52.7846i 3.11578i
\(288\) 0 0
\(289\) 2.85641 0.168024
\(290\) 0 0
\(291\) 5.46410i 0.320311i
\(292\) 0 0
\(293\) 20.5494i 1.20051i −0.799810 0.600254i \(-0.795066\pi\)
0.799810 0.600254i \(-0.204934\pi\)
\(294\) 0 0
\(295\) 25.8564i 1.50542i
\(296\) 0 0
\(297\) 2.73205 + 1.88040i 0.158530 + 0.109112i
\(298\) 0 0
\(299\) −6.51389 −0.376708
\(300\) 0 0
\(301\) −19.3205 −1.11362
\(302\) 0 0
\(303\) 17.7963 1.02237
\(304\) 0 0
\(305\) 6.51389i 0.372984i
\(306\) 0 0
\(307\) −6.51389 −0.371767 −0.185884 0.982572i \(-0.559515\pi\)
−0.185884 + 0.982572i \(0.559515\pi\)
\(308\) 0 0
\(309\) 10.0000 0.568880
\(310\) 0 0
\(311\) 20.9808i 1.18971i −0.803833 0.594855i \(-0.797209\pi\)
0.803833 0.594855i \(-0.202791\pi\)
\(312\) 0 0
\(313\) 11.8564 0.670164 0.335082 0.942189i \(-0.391236\pi\)
0.335082 + 0.942189i \(0.391236\pi\)
\(314\) 0 0
\(315\) 14.0355 0.790810
\(316\) 0 0
\(317\) 9.66025 0.542574 0.271287 0.962499i \(-0.412551\pi\)
0.271287 + 0.962499i \(0.412551\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 3.76080i 0.209907i
\(322\) 0 0
\(323\) 14.1436i 0.786971i
\(324\) 0 0
\(325\) 12.6589i 0.702192i
\(326\) 0 0
\(327\) 12.6589 0.700041
\(328\) 0 0
\(329\) 21.5571i 1.18848i
\(330\) 0 0
\(331\) 1.07180i 0.0589113i 0.999566 + 0.0294556i \(0.00937738\pi\)
−0.999566 + 0.0294556i \(0.990623\pi\)
\(332\) 0 0
\(333\) −0.535898 −0.0293671
\(334\) 0 0
\(335\) 28.3923i 1.55124i
\(336\) 0 0
\(337\) 2.75309i 0.149971i 0.997185 + 0.0749853i \(0.0238910\pi\)
−0.997185 + 0.0749853i \(0.976109\pi\)
\(338\) 0 0
\(339\) 17.3205i 0.940721i
\(340\) 0 0
\(341\) −5.46410 3.76080i −0.295898 0.203659i
\(342\) 0 0
\(343\) −63.6635 −3.43751
\(344\) 0 0
\(345\) −3.46410 −0.186501
\(346\) 0 0
\(347\) −30.8241 −1.65472 −0.827361 0.561670i \(-0.810159\pi\)
−0.827361 + 0.561670i \(0.810159\pi\)
\(348\) 0 0
\(349\) 7.89044i 0.422365i 0.977447 + 0.211183i \(0.0677315\pi\)
−0.977447 + 0.211183i \(0.932268\pi\)
\(350\) 0 0
\(351\) 5.13734 0.274211
\(352\) 0 0
\(353\) 8.14359 0.433440 0.216720 0.976234i \(-0.430464\pi\)
0.216720 + 0.976234i \(0.430464\pi\)
\(354\) 0 0
\(355\) 29.3205i 1.55617i
\(356\) 0 0
\(357\) 19.3205 1.02255
\(358\) 0 0
\(359\) −7.52159 −0.396975 −0.198487 0.980103i \(-0.563603\pi\)
−0.198487 + 0.980103i \(0.563603\pi\)
\(360\) 0 0
\(361\) −4.85641 −0.255600
\(362\) 0 0
\(363\) −10.2747 + 3.92820i −0.539281 + 0.206177i
\(364\) 0 0
\(365\) 28.0710i 1.46930i
\(366\) 0 0
\(367\) 10.3923i 0.542474i −0.962513 0.271237i \(-0.912567\pi\)
0.962513 0.271237i \(-0.0874327\pi\)
\(368\) 0 0
\(369\) 10.2747i 0.534879i
\(370\) 0 0
\(371\) 55.1342 2.86243
\(372\) 0 0
\(373\) 18.1651i 0.940555i 0.882519 + 0.470277i \(0.155846\pi\)
−0.882519 + 0.470277i \(0.844154\pi\)
\(374\) 0 0
\(375\) 6.92820i 0.357771i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2.92820i 0.150412i 0.997168 + 0.0752058i \(0.0239614\pi\)
−0.997168 + 0.0752058i \(0.976039\pi\)
\(380\) 0 0
\(381\) 5.13734i 0.263194i
\(382\) 0 0
\(383\) 20.9808i 1.07207i 0.844197 + 0.536033i \(0.180078\pi\)
−0.844197 + 0.536033i \(0.819922\pi\)
\(384\) 0 0
\(385\) −26.3923 + 38.3457i −1.34508 + 1.95428i
\(386\) 0 0
\(387\) −3.76080 −0.191172
\(388\) 0 0
\(389\) −1.26795 −0.0642876 −0.0321438 0.999483i \(-0.510233\pi\)
−0.0321438 + 0.999483i \(0.510233\pi\)
\(390\) 0 0
\(391\) −4.76850 −0.241153
\(392\) 0 0
\(393\) 2.75309i 0.138875i
\(394\) 0 0
\(395\) 14.0355 0.706202
\(396\) 0 0
\(397\) −21.3205 −1.07005 −0.535023 0.844838i \(-0.679697\pi\)
−0.535023 + 0.844838i \(0.679697\pi\)
\(398\) 0 0
\(399\) 19.3205i 0.967235i
\(400\) 0 0
\(401\) 3.46410 0.172989 0.0864945 0.996252i \(-0.472434\pi\)
0.0864945 + 0.996252i \(0.472434\pi\)
\(402\) 0 0
\(403\) −10.2747 −0.511819
\(404\) 0 0
\(405\) 2.73205 0.135757
\(406\) 0 0
\(407\) 1.00770 1.46410i 0.0499500 0.0725728i
\(408\) 0 0
\(409\) 25.3179i 1.25189i 0.779868 + 0.625944i \(0.215286\pi\)
−0.779868 + 0.625944i \(0.784714\pi\)
\(410\) 0 0
\(411\) 4.92820i 0.243090i
\(412\) 0 0
\(413\) 48.6203i 2.39245i
\(414\) 0 0
\(415\) 28.0710 1.37795
\(416\) 0 0
\(417\) 14.0355i 0.687321i
\(418\) 0 0
\(419\) 6.14359i 0.300134i 0.988676 + 0.150067i \(0.0479490\pi\)
−0.988676 + 0.150067i \(0.952051\pi\)
\(420\) 0 0
\(421\) 9.32051 0.454254 0.227127 0.973865i \(-0.427067\pi\)
0.227127 + 0.973865i \(0.427067\pi\)
\(422\) 0 0
\(423\) 4.19615i 0.204024i
\(424\) 0 0
\(425\) 9.26699i 0.449515i
\(426\) 0 0
\(427\) 12.2487i 0.592757i
\(428\) 0 0
\(429\) −9.66025 + 14.0355i −0.466401 + 0.677640i
\(430\) 0 0
\(431\) 23.3025 1.12244 0.561220 0.827666i \(-0.310332\pi\)
0.561220 + 0.827666i \(0.310332\pi\)
\(432\) 0 0
\(433\) −16.9282 −0.813518 −0.406759 0.913536i \(-0.633341\pi\)
−0.406759 + 0.913536i \(0.633341\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.76850i 0.228108i
\(438\) 0 0
\(439\) 28.4398 1.35736 0.678679 0.734435i \(-0.262553\pi\)
0.678679 + 0.734435i \(0.262553\pi\)
\(440\) 0 0
\(441\) −19.3923 −0.923443
\(442\) 0 0
\(443\) 2.14359i 0.101845i 0.998703 + 0.0509226i \(0.0162162\pi\)
−0.998703 + 0.0509226i \(0.983784\pi\)
\(444\) 0 0
\(445\) −28.3923 −1.34592
\(446\) 0 0
\(447\) −17.7963 −0.841735
\(448\) 0 0
\(449\) −15.0718 −0.711282 −0.355641 0.934623i \(-0.615737\pi\)
−0.355641 + 0.934623i \(0.615737\pi\)
\(450\) 0 0
\(451\) 28.0710 + 19.3205i 1.32181 + 0.909767i
\(452\) 0 0
\(453\) 2.38425i 0.112022i
\(454\) 0 0
\(455\) 72.1051i 3.38034i
\(456\) 0 0
\(457\) 2.75309i 0.128784i −0.997925 0.0643922i \(-0.979489\pi\)
0.997925 0.0643922i \(-0.0205109\pi\)
\(458\) 0 0
\(459\) 3.76080 0.175539
\(460\) 0 0
\(461\) 5.50619i 0.256449i −0.991745 0.128224i \(-0.959072\pi\)
0.991745 0.128224i \(-0.0409278\pi\)
\(462\) 0 0
\(463\) 32.9282i 1.53030i −0.643850 0.765152i \(-0.722664\pi\)
0.643850 0.765152i \(-0.277336\pi\)
\(464\) 0 0
\(465\) −5.46410 −0.253392
\(466\) 0 0
\(467\) 9.46410i 0.437946i 0.975731 + 0.218973i \(0.0702707\pi\)
−0.975731 + 0.218973i \(0.929729\pi\)
\(468\) 0 0
\(469\) 53.3888i 2.46527i
\(470\) 0 0
\(471\) 11.8564i 0.546314i
\(472\) 0 0
\(473\) 7.07180 10.2747i 0.325162 0.472431i
\(474\) 0 0
\(475\) 9.26699 0.425198
\(476\) 0 0
\(477\) 10.7321 0.491387
\(478\) 0 0
\(479\) −10.2747 −0.469462 −0.234731 0.972060i \(-0.575421\pi\)
−0.234731 + 0.972060i \(0.575421\pi\)
\(480\) 0 0
\(481\) 2.75309i 0.125530i
\(482\) 0 0
\(483\) 6.51389 0.296392
\(484\) 0 0
\(485\) 14.9282 0.677855
\(486\) 0 0
\(487\) 18.3923i 0.833435i −0.909036 0.416717i \(-0.863180\pi\)
0.909036 0.416717i \(-0.136820\pi\)
\(488\) 0 0
\(489\) −2.92820 −0.132418
\(490\) 0 0
\(491\) −11.2824 −0.509167 −0.254584 0.967051i \(-0.581938\pi\)
−0.254584 + 0.967051i \(0.581938\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −5.13734 + 7.46410i −0.230906 + 0.335486i
\(496\) 0 0
\(497\) 55.1342i 2.47311i
\(498\) 0 0
\(499\) 17.0718i 0.764239i −0.924113 0.382119i \(-0.875194\pi\)
0.924113 0.382119i \(-0.124806\pi\)
\(500\) 0 0
\(501\) 17.7963i 0.795079i
\(502\) 0 0
\(503\) 2.75309 0.122754 0.0613772 0.998115i \(-0.480451\pi\)
0.0613772 + 0.998115i \(0.480451\pi\)
\(504\) 0 0
\(505\) 48.6203i 2.16358i
\(506\) 0 0
\(507\) 13.3923i 0.594773i
\(508\) 0 0
\(509\) 26.7321 1.18488 0.592439 0.805616i \(-0.298165\pi\)
0.592439 + 0.805616i \(0.298165\pi\)
\(510\) 0 0
\(511\) 52.7846i 2.33505i
\(512\) 0 0
\(513\) 3.76080i 0.166043i
\(514\) 0 0
\(515\) 27.3205i 1.20389i
\(516\) 0 0
\(517\) −11.4641 7.89044i −0.504191 0.347021i
\(518\) 0 0
\(519\) −13.0278 −0.571856
\(520\) 0 0
\(521\) −41.3205 −1.81028 −0.905142 0.425109i \(-0.860236\pi\)
−0.905142 + 0.425109i \(0.860236\pi\)
\(522\) 0 0
\(523\) 37.3380 1.63267 0.816337 0.577575i \(-0.196001\pi\)
0.816337 + 0.577575i \(0.196001\pi\)
\(524\) 0 0
\(525\) 12.6589i 0.552481i
\(526\) 0 0
\(527\) −7.52159 −0.327646
\(528\) 0 0
\(529\) 21.3923 0.930100
\(530\) 0 0
\(531\) 9.46410i 0.410707i
\(532\) 0 0
\(533\) 52.7846 2.28636
\(534\) 0 0
\(535\) −10.2747 −0.444214
\(536\) 0 0
\(537\) −0.392305 −0.0169292
\(538\) 0 0
\(539\) 36.4653 52.9808i 1.57067 2.28204i
\(540\) 0 0
\(541\) 27.7021i 1.19101i 0.803353 + 0.595504i \(0.203047\pi\)
−0.803353 + 0.595504i \(0.796953\pi\)
\(542\) 0 0
\(543\) 2.39230i 0.102664i
\(544\) 0 0
\(545\) 34.5849i 1.48145i
\(546\) 0 0
\(547\) −9.26699 −0.396228 −0.198114 0.980179i \(-0.563482\pi\)
−0.198114 + 0.980179i \(0.563482\pi\)
\(548\) 0 0
\(549\) 2.38425i 0.101757i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −26.3923 −1.12231
\(554\) 0 0
\(555\) 1.46410i 0.0621477i
\(556\) 0 0
\(557\) 20.5494i 0.870705i 0.900260 + 0.435353i \(0.143376\pi\)
−0.900260 + 0.435353i \(0.856624\pi\)
\(558\) 0 0
\(559\) 19.3205i 0.817170i
\(560\) 0 0
\(561\) −7.07180 + 10.2747i −0.298571 + 0.433798i
\(562\) 0 0
\(563\) −31.8318 −1.34155 −0.670775 0.741661i \(-0.734038\pi\)
−0.670775 + 0.741661i \(0.734038\pi\)
\(564\) 0 0
\(565\) 47.3205 1.99079
\(566\) 0 0
\(567\) −5.13734 −0.215748
\(568\) 0 0
\(569\) 9.26699i 0.388492i −0.980953 0.194246i \(-0.937774\pi\)
0.980953 0.194246i \(-0.0622260\pi\)
\(570\) 0 0
\(571\) −14.0355 −0.587367 −0.293683 0.955903i \(-0.594881\pi\)
−0.293683 + 0.955903i \(0.594881\pi\)
\(572\) 0 0
\(573\) 11.1244 0.464727
\(574\) 0 0
\(575\) 3.12436i 0.130295i
\(576\) 0 0
\(577\) 40.3923 1.68155 0.840777 0.541382i \(-0.182099\pi\)
0.840777 + 0.541382i \(0.182099\pi\)
\(578\) 0 0
\(579\) 13.0278 0.541416
\(580\) 0 0
\(581\) −52.7846 −2.18987
\(582\) 0 0
\(583\) −20.1805 + 29.3205i −0.835792 + 1.21433i
\(584\) 0 0
\(585\) 14.0355i 0.580296i
\(586\) 0 0
\(587\) 33.1769i 1.36936i 0.728845 + 0.684679i \(0.240058\pi\)
−0.728845 + 0.684679i \(0.759942\pi\)
\(588\) 0 0
\(589\) 7.52159i 0.309922i
\(590\) 0 0
\(591\) 2.75309 0.113247
\(592\) 0 0
\(593\) 24.3102i 0.998299i 0.866516 + 0.499150i \(0.166354\pi\)
−0.866516 + 0.499150i \(0.833646\pi\)
\(594\) 0 0
\(595\) 52.7846i 2.16396i
\(596\) 0 0
\(597\) −23.4641 −0.960322
\(598\) 0 0
\(599\) 15.8038i 0.645728i −0.946445 0.322864i \(-0.895354\pi\)
0.946445 0.322864i \(-0.104646\pi\)
\(600\) 0 0
\(601\) 32.8395i 1.33955i 0.742564 + 0.669775i \(0.233609\pi\)
−0.742564 + 0.669775i \(0.766391\pi\)
\(602\) 0 0
\(603\) 10.3923i 0.423207i
\(604\) 0 0
\(605\) −10.7321 28.0710i −0.436320 1.14125i
\(606\) 0 0
\(607\) 27.7021 1.12439 0.562197 0.827003i \(-0.309956\pi\)
0.562197 + 0.827003i \(0.309956\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −21.5571 −0.872106
\(612\) 0 0
\(613\) 33.2083i 1.34127i 0.741787 + 0.670636i \(0.233979\pi\)
−0.741787 + 0.670636i \(0.766021\pi\)
\(614\) 0 0
\(615\) 28.0710 1.13193
\(616\) 0 0
\(617\) 8.53590 0.343642 0.171821 0.985128i \(-0.445035\pi\)
0.171821 + 0.985128i \(0.445035\pi\)
\(618\) 0 0
\(619\) 8.53590i 0.343087i −0.985177 0.171543i \(-0.945125\pi\)
0.985177 0.171543i \(-0.0548754\pi\)
\(620\) 0 0
\(621\) 1.26795 0.0508810
\(622\) 0 0
\(623\) 53.3888 2.13898
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) 0 0
\(627\) −10.2747 7.07180i −0.410332 0.282420i
\(628\) 0 0
\(629\) 2.01540i 0.0803595i
\(630\) 0 0
\(631\) 6.78461i 0.270091i −0.990839 0.135046i \(-0.956882\pi\)
0.990839 0.135046i \(-0.0431181\pi\)
\(632\) 0 0
\(633\) 21.5571i 0.856817i
\(634\) 0 0
\(635\) −14.0355 −0.556981
\(636\) 0 0
\(637\) 99.6249i 3.94728i
\(638\) 0 0
\(639\) 10.7321i 0.424553i
\(640\) 0 0
\(641\) −4.53590 −0.179157 −0.0895786 0.995980i \(-0.528552\pi\)
−0.0895786 + 0.995980i \(0.528552\pi\)
\(642\) 0 0
\(643\) 15.4641i 0.609845i 0.952377 + 0.304922i \(0.0986305\pi\)
−0.952377 + 0.304922i \(0.901369\pi\)
\(644\) 0 0
\(645\) 10.2747i 0.404565i
\(646\) 0 0
\(647\) 12.8756i 0.506194i −0.967441 0.253097i \(-0.918551\pi\)
0.967441 0.253097i \(-0.0814492\pi\)
\(648\) 0 0
\(649\) −25.8564 17.7963i −1.01495 0.698565i
\(650\) 0 0
\(651\) 10.2747 0.402697
\(652\) 0 0
\(653\) −21.2679 −0.832279 −0.416140 0.909301i \(-0.636617\pi\)
−0.416140 + 0.909301i \(0.636617\pi\)
\(654\) 0 0
\(655\) 7.52159 0.293893
\(656\) 0 0
\(657\) 10.2747i 0.400854i
\(658\) 0 0
\(659\) −37.3380 −1.45448 −0.727240 0.686383i \(-0.759197\pi\)
−0.727240 + 0.686383i \(0.759197\pi\)
\(660\) 0 0
\(661\) 31.8564 1.23907 0.619535 0.784969i \(-0.287321\pi\)
0.619535 + 0.784969i \(0.287321\pi\)
\(662\) 0 0
\(663\) 19.3205i 0.750346i
\(664\) 0 0
\(665\) −52.7846 −2.04690
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 11.4641 0.443228
\(670\) 0 0
\(671\) 6.51389 + 4.48334i 0.251466 + 0.173077i
\(672\) 0 0
\(673\) 32.8395i 1.26587i −0.774206 0.632934i \(-0.781850\pi\)
0.774206 0.632934i \(-0.218150\pi\)
\(674\) 0 0
\(675\) 2.46410i 0.0948433i
\(676\) 0 0
\(677\) 4.76850i 0.183268i 0.995793 + 0.0916342i \(0.0292090\pi\)
−0.995793 + 0.0916342i \(0.970791\pi\)
\(678\) 0 0
\(679\) −28.0710 −1.07726
\(680\) 0 0
\(681\) 16.7886i 0.643340i
\(682\) 0 0
\(683\) 19.7128i 0.754290i −0.926154 0.377145i \(-0.876906\pi\)
0.926154 0.377145i \(-0.123094\pi\)
\(684\) 0 0
\(685\) 13.4641 0.514437
\(686\) 0 0
\(687\) 23.8564i 0.910179i
\(688\) 0 0
\(689\) 55.1342i 2.10045i
\(690\) 0 0
\(691\) 30.3923i 1.15618i 0.815974 + 0.578089i \(0.196201\pi\)
−0.815974 + 0.578089i \(0.803799\pi\)
\(692\) 0 0
\(693\) 9.66025 14.0355i 0.366963 0.533164i
\(694\) 0 0
\(695\) 38.3457 1.45453
\(696\) 0 0
\(697\) 38.6410 1.46363
\(698\) 0 0
\(699\) 10.2747 0.388624
\(700\) 0 0
\(701\) 13.0278i 0.492053i 0.969263 + 0.246026i \(0.0791250\pi\)
−0.969263 + 0.246026i \(0.920875\pi\)
\(702\) 0 0
\(703\) 2.01540 0.0760124
\(704\) 0 0
\(705\) −11.4641 −0.431763
\(706\) 0 0
\(707\) 91.4256i 3.43841i
\(708\) 0 0
\(709\) −22.7846 −0.855694 −0.427847 0.903851i \(-0.640728\pi\)
−0.427847 + 0.903851i \(0.640728\pi\)
\(710\) 0 0
\(711\) −5.13734 −0.192665
\(712\) 0 0
\(713\) −2.53590 −0.0949701
\(714\) 0 0
\(715\) −38.3457 26.3923i −1.43405 0.987016i
\(716\) 0 0
\(717\) 23.3025i 0.870247i
\(718\) 0 0
\(719\) 5.66025i 0.211092i −0.994414 0.105546i \(-0.966341\pi\)
0.994414 0.105546i \(-0.0336590\pi\)
\(720\) 0 0
\(721\) 51.3734i 1.91325i
\(722\) 0 0
\(723\) −28.0710 −1.04397
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.248711i 0.00922419i 0.999989 + 0.00461210i \(0.00146808\pi\)
−0.999989 + 0.00461210i \(0.998532\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 14.1436i 0.523120i
\(732\) 0 0
\(733\) 12.6589i 0.467569i 0.972288 + 0.233784i \(0.0751109\pi\)
−0.972288 + 0.233784i \(0.924889\pi\)
\(734\) 0 0
\(735\) 52.9808i 1.95422i
\(736\) 0 0
\(737\) 28.3923 + 19.5417i 1.04584 + 0.719827i
\(738\) 0 0
\(739\) 1.00770 0.0370689 0.0185345 0.999828i \(-0.494100\pi\)
0.0185345 + 0.999828i \(0.494100\pi\)
\(740\) 0 0
\(741\) −19.3205 −0.709757
\(742\) 0 0
\(743\) 22.5648 0.827822 0.413911 0.910317i \(-0.364163\pi\)
0.413911 + 0.910317i \(0.364163\pi\)
\(744\) 0 0
\(745\) 48.6203i 1.78131i
\(746\) 0 0
\(747\) −10.2747 −0.375931
\(748\) 0 0
\(749\) 19.3205 0.705956
\(750\) 0 0
\(751\) 31.5692i 1.15198i 0.817458 + 0.575989i \(0.195383\pi\)
−0.817458 + 0.575989i \(0.804617\pi\)
\(752\) 0 0
\(753\) 21.8564 0.796492
\(754\) 0 0
\(755\) −6.51389 −0.237065
\(756\) 0 0
\(757\) 43.1769 1.56929 0.784646 0.619944i \(-0.212845\pi\)
0.784646 + 0.619944i \(0.212845\pi\)
\(758\) 0 0
\(759\) −2.38425 + 3.46410i −0.0865428 + 0.125739i
\(760\) 0 0
\(761\) 37.3380i 1.35350i 0.736213 + 0.676750i \(0.236612\pi\)
−0.736213 + 0.676750i \(0.763388\pi\)
\(762\) 0 0
\(763\) 65.0333i 2.35436i
\(764\) 0 0
\(765\) 10.2747i 0.371482i
\(766\) 0 0
\(767\) −48.6203 −1.75558
\(768\) 0 0
\(769\) 5.50619i 0.198558i −0.995060 0.0992791i \(-0.968346\pi\)
0.995060 0.0992791i \(-0.0316537\pi\)
\(770\) 0 0
\(771\) 15.4641i 0.556926i
\(772\) 0 0
\(773\) −34.8372 −1.25301 −0.626503 0.779419i \(-0.715514\pi\)
−0.626503 + 0.779419i \(0.715514\pi\)
\(774\) 0 0
\(775\) 4.92820i 0.177026i
\(776\) 0 0
\(777\) 2.75309i 0.0987667i
\(778\) 0 0
\(779\) 38.6410i 1.38446i
\(780\) 0 0
\(781\) 29.3205 + 20.1805i 1.04917 + 0.722116i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 32.3923 1.15613
\(786\) 0 0
\(787\) −9.26699 −0.330332 −0.165166 0.986266i \(-0.552816\pi\)
−0.165166 + 0.986266i \(0.552816\pi\)
\(788\) 0 0
\(789\) 4.76850i 0.169763i
\(790\) 0 0
\(791\) −88.9814 −3.16381
\(792\) 0 0
\(793\) 12.2487 0.434964
\(794\) 0 0
\(795\) 29.3205i 1.03989i
\(796\) 0 0
\(797\) 17.6603 0.625558 0.312779 0.949826i \(-0.398740\pi\)
0.312779 + 0.949826i \(0.398740\pi\)
\(798\) 0