Properties

Label 2016.2.s.x.865.3
Level $2016$
Weight $2$
Character 2016.865
Analytic conductor $16.098$
Analytic rank $0$
Dimension $8$
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] = [2016,2,Mod(289,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2016.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.s (of order \(3\), degree \(2\), 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: \(16.0978410475\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1445900625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 77x^{4} + 36x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 865.3
Root \(-1.46040 + 2.52950i\) of defining polynomial
Character \(\chi\) \(=\) 2016.865
Dual form 2016.2.s.x.289.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.342371 + 0.593004i) q^{5} +(-1.80278 - 1.93649i) q^{7} +O(q^{10})\) \(q+(0.342371 + 0.593004i) q^{5} +(-1.80278 - 1.93649i) q^{7} +(-1.76556 + 3.05805i) q^{11} -2.53113 q^{13} +(2.23607 - 3.87298i) q^{17} +(1.46040 + 2.52950i) q^{19} +(1.00000 + 1.73205i) q^{23} +(2.26556 - 3.92407i) q^{25} +6.52636 q^{29} +(-0.433292 + 0.750484i) q^{31} +(0.531129 - 1.73205i) q^{35} +(-1.26556 - 2.19202i) q^{37} +11.6832 q^{41} +7.39295 q^{43} +(-2.53113 - 4.38404i) q^{47} +(-0.500000 + 6.98212i) q^{49} +(4.12977 - 7.15296i) q^{53} -2.41791 q^{55} +(4.76556 - 8.25420i) q^{59} +(2.53113 + 4.38404i) q^{61} +(-0.866585 - 1.50097i) q^{65} +(-2.82989 + 4.90151i) q^{67} +9.06226 q^{71} +(1.26556 - 2.19202i) q^{73} +(9.10480 - 2.09397i) q^{77} +(-6.09307 - 10.5535i) q^{79} -9.53113 q^{83} +3.06226 q^{85} +(2.05422 + 3.55802i) q^{89} +(4.56306 + 4.90151i) q^{91} +(-1.00000 + 1.73205i) q^{95} -9.53113 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{11} + 12 q^{13} + 8 q^{23} + 2 q^{25} - 28 q^{35} + 6 q^{37} + 12 q^{47} - 4 q^{49} + 22 q^{59} - 12 q^{61} + 8 q^{71} - 6 q^{73} - 44 q^{83} - 40 q^{85} - 8 q^{95} - 44 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 0.342371 + 0.593004i 0.153113 + 0.265199i 0.932370 0.361505i \(-0.117737\pi\)
−0.779257 + 0.626704i \(0.784404\pi\)
\(6\) 0 0
\(7\) −1.80278 1.93649i −0.681385 0.731925i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.76556 + 3.05805i −0.532338 + 0.922036i 0.466949 + 0.884284i \(0.345353\pi\)
−0.999287 + 0.0377519i \(0.987980\pi\)
\(12\) 0 0
\(13\) −2.53113 −0.702009 −0.351004 0.936374i \(-0.614160\pi\)
−0.351004 + 0.936374i \(0.614160\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.23607 3.87298i 0.542326 0.939336i −0.456444 0.889752i \(-0.650877\pi\)
0.998770 0.0495842i \(-0.0157896\pi\)
\(18\) 0 0
\(19\) 1.46040 + 2.52950i 0.335040 + 0.580306i 0.983492 0.180949i \(-0.0579169\pi\)
−0.648453 + 0.761255i \(0.724584\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.00000 + 1.73205i 0.208514 + 0.361158i 0.951247 0.308431i \(-0.0998038\pi\)
−0.742732 + 0.669588i \(0.766471\pi\)
\(24\) 0 0
\(25\) 2.26556 3.92407i 0.453113 0.784815i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.52636 1.21191 0.605957 0.795497i \(-0.292790\pi\)
0.605957 + 0.795497i \(0.292790\pi\)
\(30\) 0 0
\(31\) −0.433292 + 0.750484i −0.0778216 + 0.134791i −0.902310 0.431088i \(-0.858130\pi\)
0.824488 + 0.565879i \(0.191463\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.531129 1.73205i 0.0897772 0.292770i
\(36\) 0 0
\(37\) −1.26556 2.19202i −0.208058 0.360366i 0.743045 0.669241i \(-0.233381\pi\)
−0.951103 + 0.308875i \(0.900047\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 11.6832 1.82462 0.912308 0.409505i \(-0.134299\pi\)
0.912308 + 0.409505i \(0.134299\pi\)
\(42\) 0 0
\(43\) 7.39295 1.12741 0.563707 0.825975i \(-0.309375\pi\)
0.563707 + 0.825975i \(0.309375\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.53113 4.38404i −0.369203 0.639479i 0.620238 0.784414i \(-0.287036\pi\)
−0.989441 + 0.144935i \(0.953703\pi\)
\(48\) 0 0
\(49\) −0.500000 + 6.98212i −0.0714286 + 0.997446i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.12977 7.15296i 0.567267 0.982535i −0.429568 0.903035i \(-0.641334\pi\)
0.996835 0.0795005i \(-0.0253325\pi\)
\(54\) 0 0
\(55\) −2.41791 −0.326031
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.76556 8.25420i 0.620424 1.07461i −0.368983 0.929436i \(-0.620294\pi\)
0.989407 0.145169i \(-0.0463727\pi\)
\(60\) 0 0
\(61\) 2.53113 + 4.38404i 0.324078 + 0.561319i 0.981325 0.192356i \(-0.0616127\pi\)
−0.657247 + 0.753675i \(0.728279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.866585 1.50097i −0.107487 0.186172i
\(66\) 0 0
\(67\) −2.82989 + 4.90151i −0.345726 + 0.598815i −0.985485 0.169760i \(-0.945701\pi\)
0.639760 + 0.768575i \(0.279034\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.06226 1.07549 0.537746 0.843107i \(-0.319276\pi\)
0.537746 + 0.843107i \(0.319276\pi\)
\(72\) 0 0
\(73\) 1.26556 2.19202i 0.148123 0.256557i −0.782411 0.622763i \(-0.786010\pi\)
0.930534 + 0.366206i \(0.119343\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9.10480 2.09397i 1.03759 0.238630i
\(78\) 0 0
\(79\) −6.09307 10.5535i −0.685524 1.18736i −0.973272 0.229656i \(-0.926240\pi\)
0.287748 0.957706i \(-0.407093\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.53113 −1.04618 −0.523089 0.852278i \(-0.675220\pi\)
−0.523089 + 0.852278i \(0.675220\pi\)
\(84\) 0 0
\(85\) 3.06226 0.332148
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.05422 + 3.55802i 0.217747 + 0.377150i 0.954119 0.299428i \(-0.0967958\pi\)
−0.736372 + 0.676577i \(0.763462\pi\)
\(90\) 0 0
\(91\) 4.56306 + 4.90151i 0.478338 + 0.513818i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.00000 + 1.73205i −0.102598 + 0.177705i
\(96\) 0 0
\(97\) −9.53113 −0.967740 −0.483870 0.875140i \(-0.660769\pi\)
−0.483870 + 0.875140i \(0.660769\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.15688 8.93197i 0.513129 0.888765i −0.486756 0.873538i \(-0.661820\pi\)
0.999884 0.0152264i \(-0.00484691\pi\)
\(102\) 0 0
\(103\) −3.69647 6.40248i −0.364224 0.630855i 0.624427 0.781083i \(-0.285333\pi\)
−0.988651 + 0.150228i \(0.951999\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.23444 + 5.60221i 0.312685 + 0.541586i 0.978943 0.204136i \(-0.0654384\pi\)
−0.666258 + 0.745721i \(0.732105\pi\)
\(108\) 0 0
\(109\) −0.734436 + 1.27208i −0.0703462 + 0.121843i −0.899053 0.437840i \(-0.855744\pi\)
0.828707 + 0.559683i \(0.189077\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.7859 1.39094 0.695470 0.718555i \(-0.255196\pi\)
0.695470 + 0.718555i \(0.255196\pi\)
\(114\) 0 0
\(115\) −0.684742 + 1.18601i −0.0638525 + 0.110596i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11.5311 + 2.65199i −1.05706 + 0.243108i
\(120\) 0 0
\(121\) −0.734436 1.27208i −0.0667669 0.115644i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.52636 0.583735
\(126\) 0 0
\(127\) −15.6525 −1.38893 −0.694466 0.719525i \(-0.744359\pi\)
−0.694466 + 0.719525i \(0.744359\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.29669 + 12.6382i 0.637515 + 1.10421i 0.985976 + 0.166885i \(0.0533710\pi\)
−0.348461 + 0.937323i \(0.613296\pi\)
\(132\) 0 0
\(133\) 2.26556 7.38817i 0.196449 0.640636i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.52636 11.3040i 0.557585 0.965765i −0.440113 0.897943i \(-0.645062\pi\)
0.997697 0.0678224i \(-0.0216051\pi\)
\(138\) 0 0
\(139\) 20.4457 1.73418 0.867089 0.498152i \(-0.165988\pi\)
0.867089 + 0.498152i \(0.165988\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.46887 7.74031i 0.373706 0.647277i
\(144\) 0 0
\(145\) 2.23444 + 3.87016i 0.185560 + 0.321399i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.52636 + 11.3040i 0.534660 + 0.926059i 0.999180 + 0.0404959i \(0.0128938\pi\)
−0.464519 + 0.885563i \(0.653773\pi\)
\(150\) 0 0
\(151\) 3.26318 5.65199i 0.265554 0.459953i −0.702155 0.712024i \(-0.747779\pi\)
0.967709 + 0.252072i \(0.0811119\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −0.593387 −0.0476620
\(156\) 0 0
\(157\) −2.53113 + 4.38404i −0.202006 + 0.349885i −0.949175 0.314750i \(-0.898079\pi\)
0.747169 + 0.664635i \(0.231413\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.55133 5.05899i 0.122262 0.398704i
\(162\) 0 0
\(163\) 8.25953 + 14.3059i 0.646936 + 1.12053i 0.983851 + 0.178991i \(0.0572834\pi\)
−0.336914 + 0.941535i \(0.609383\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −14.0000 −1.08335 −0.541676 0.840587i \(-0.682210\pi\)
−0.541676 + 0.840587i \(0.682210\pi\)
\(168\) 0 0
\(169\) −6.59339 −0.507184
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3.60555 6.24500i −0.274125 0.474798i 0.695789 0.718246i \(-0.255055\pi\)
−0.969914 + 0.243448i \(0.921722\pi\)
\(174\) 0 0
\(175\) −11.6832 + 2.68698i −0.883170 + 0.203116i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.46887 + 2.54416i −0.109789 + 0.190159i −0.915685 0.401898i \(-0.868351\pi\)
0.805896 + 0.592057i \(0.201684\pi\)
\(180\) 0 0
\(181\) 16.5311 1.22875 0.614375 0.789015i \(-0.289408\pi\)
0.614375 + 0.789015i \(0.289408\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.866585 1.50097i 0.0637126 0.110353i
\(186\) 0 0
\(187\) 7.89584 + 13.6760i 0.577401 + 1.00009i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 10.5311 + 18.2405i 0.762006 + 1.31983i 0.941815 + 0.336131i \(0.109119\pi\)
−0.179809 + 0.983701i \(0.557548\pi\)
\(192\) 0 0
\(193\) 0.968871 1.67813i 0.0697409 0.120795i −0.829046 0.559180i \(-0.811116\pi\)
0.898787 + 0.438385i \(0.144449\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.73317 −0.123483 −0.0617416 0.998092i \(-0.519665\pi\)
−0.0617416 + 0.998092i \(0.519665\pi\)
\(198\) 0 0
\(199\) 7.21110 12.4900i 0.511182 0.885392i −0.488735 0.872433i \(-0.662541\pi\)
0.999916 0.0129598i \(-0.00412534\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −11.7656 12.6382i −0.825781 0.887031i
\(204\) 0 0
\(205\) 4.00000 + 6.92820i 0.279372 + 0.483887i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.3138 −0.713417
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.53113 + 4.38404i 0.172622 + 0.298989i
\(216\) 0 0
\(217\) 2.23444 0.513888i 0.151683 0.0348850i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.65978 + 9.80302i −0.380718 + 0.659422i
\(222\) 0 0
\(223\) −14.1011 −0.944283 −0.472141 0.881523i \(-0.656519\pi\)
−0.472141 + 0.881523i \(0.656519\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.7656 20.3786i 0.780908 1.35257i −0.150505 0.988609i \(-0.548090\pi\)
0.931413 0.363963i \(-0.118577\pi\)
\(228\) 0 0
\(229\) −13.3278 23.0845i −0.880727 1.52546i −0.850534 0.525920i \(-0.823721\pi\)
−0.0301934 0.999544i \(-0.509612\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.866585 + 1.50097i 0.0567718 + 0.0983317i 0.893015 0.450028i \(-0.148586\pi\)
−0.836243 + 0.548359i \(0.815253\pi\)
\(234\) 0 0
\(235\) 1.73317 3.00194i 0.113060 0.195825i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.93774 0.319396 0.159698 0.987166i \(-0.448948\pi\)
0.159698 + 0.987166i \(0.448948\pi\)
\(240\) 0 0
\(241\) 4.76556 8.25420i 0.306977 0.531700i −0.670723 0.741708i \(-0.734016\pi\)
0.977700 + 0.210009i \(0.0673492\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.31161 + 2.09397i −0.275459 + 0.133779i
\(246\) 0 0
\(247\) −3.69647 6.40248i −0.235201 0.407380i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0.593387 0.0374542 0.0187271 0.999825i \(-0.494039\pi\)
0.0187271 + 0.999825i \(0.494039\pi\)
\(252\) 0 0
\(253\) −7.06226 −0.444000
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.60555 + 6.24500i 0.224908 + 0.389552i 0.956292 0.292414i \(-0.0944585\pi\)
−0.731384 + 0.681966i \(0.761125\pi\)
\(258\) 0 0
\(259\) −1.96330 + 6.40248i −0.121994 + 0.397831i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.06226 1.83988i 0.0655016 0.113452i −0.831415 0.555652i \(-0.812469\pi\)
0.896916 + 0.442200i \(0.145802\pi\)
\(264\) 0 0
\(265\) 5.65564 0.347423
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.36583 + 11.0259i −0.388132 + 0.672264i −0.992198 0.124670i \(-0.960213\pi\)
0.604067 + 0.796934i \(0.293546\pi\)
\(270\) 0 0
\(271\) −9.78954 16.9560i −0.594672 1.03000i −0.993593 0.113017i \(-0.963948\pi\)
0.398921 0.916985i \(-0.369385\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.00000 + 13.8564i 0.482418 + 0.835573i
\(276\) 0 0
\(277\) −13.3278 + 23.0845i −0.800791 + 1.38701i 0.118305 + 0.992977i \(0.462254\pi\)
−0.919096 + 0.394034i \(0.871079\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −27.8386 −1.66071 −0.830356 0.557233i \(-0.811863\pi\)
−0.830356 + 0.557233i \(0.811863\pi\)
\(282\) 0 0
\(283\) 5.24780 9.08945i 0.311949 0.540312i −0.666835 0.745205i \(-0.732351\pi\)
0.978784 + 0.204893i \(0.0656848\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −21.0623 22.6245i −1.24327 1.33548i
\(288\) 0 0
\(289\) −1.50000 2.59808i −0.0882353 0.152828i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.69054 −0.0987623 −0.0493812 0.998780i \(-0.515725\pi\)
−0.0493812 + 0.998780i \(0.515725\pi\)
\(294\) 0 0
\(295\) 6.52636 0.379979
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2.53113 4.38404i −0.146379 0.253536i
\(300\) 0 0
\(301\) −13.3278 14.3164i −0.768203 0.825182i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.73317 + 3.00194i −0.0992410 + 0.171890i
\(306\) 0 0
\(307\) −11.5014 −0.656419 −0.328210 0.944605i \(-0.606445\pi\)
−0.328210 + 0.944605i \(0.606445\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.0000 24.2487i 0.793867 1.37502i −0.129689 0.991555i \(-0.541398\pi\)
0.923556 0.383464i \(-0.125269\pi\)
\(312\) 0 0
\(313\) 6.03113 + 10.4462i 0.340900 + 0.590455i 0.984600 0.174822i \(-0.0559351\pi\)
−0.643700 + 0.765278i \(0.722602\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.3159 28.2600i −0.916392 1.58724i −0.804850 0.593478i \(-0.797754\pi\)
−0.111543 0.993760i \(-0.535579\pi\)
\(318\) 0 0
\(319\) −11.5227 + 19.9579i −0.645148 + 1.11743i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.0623 0.726803
\(324\) 0 0
\(325\) −5.73444 + 9.93233i −0.318089 + 0.550947i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.92661 + 12.8050i −0.216481 + 0.705960i
\(330\) 0 0
\(331\) 1.96330 + 3.40054i 0.107913 + 0.186911i 0.914925 0.403625i \(-0.132250\pi\)
−0.807012 + 0.590536i \(0.798917\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.87548 −0.211740
\(336\) 0 0
\(337\) 19.1245 1.04178 0.520889 0.853624i \(-0.325600\pi\)
0.520889 + 0.853624i \(0.325600\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.53001 2.65006i −0.0828547 0.143509i
\(342\) 0 0
\(343\) 14.4222 11.6190i 0.778726 0.627364i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.53113 + 11.3122i −0.350609 + 0.607273i −0.986356 0.164624i \(-0.947359\pi\)
0.635747 + 0.771898i \(0.280692\pi\)
\(348\) 0 0
\(349\) 24.1245 1.29136 0.645678 0.763610i \(-0.276575\pi\)
0.645678 + 0.763610i \(0.276575\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −10.4956 + 18.1789i −0.558624 + 0.967566i 0.438987 + 0.898493i \(0.355337\pi\)
−0.997612 + 0.0690725i \(0.977996\pi\)
\(354\) 0 0
\(355\) 3.10265 + 5.37395i 0.164672 + 0.285220i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 10.4689 + 18.1326i 0.552526 + 0.957003i 0.998091 + 0.0617536i \(0.0196693\pi\)
−0.445566 + 0.895249i \(0.646997\pi\)
\(360\) 0 0
\(361\) 5.23444 9.06631i 0.275497 0.477174i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.73317 0.0907182
\(366\) 0 0
\(367\) −8.51098 + 14.7414i −0.444270 + 0.769497i −0.998001 0.0631979i \(-0.979870\pi\)
0.553731 + 0.832695i \(0.313203\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −21.2967 + 4.89793i −1.10567 + 0.254288i
\(372\) 0 0
\(373\) 9.32782 + 16.1563i 0.482976 + 0.836540i 0.999809 0.0195469i \(-0.00622238\pi\)
−0.516833 + 0.856086i \(0.672889\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −16.5191 −0.850775
\(378\) 0 0
\(379\) 22.1788 1.13925 0.569625 0.821905i \(-0.307088\pi\)
0.569625 + 0.821905i \(0.307088\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.00000 + 12.1244i 0.357683 + 0.619526i 0.987573 0.157159i \(-0.0502334\pi\)
−0.629890 + 0.776684i \(0.716900\pi\)
\(384\) 0 0
\(385\) 4.35895 + 4.68226i 0.222153 + 0.238630i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −15.6525 + 27.1109i −0.793612 + 1.37458i 0.130105 + 0.991500i \(0.458469\pi\)
−0.923717 + 0.383076i \(0.874865\pi\)
\(390\) 0 0
\(391\) 8.94427 0.452331
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.17218 7.22642i 0.209925 0.363601i
\(396\) 0 0
\(397\) −5.73444 9.93233i −0.287803 0.498490i 0.685482 0.728090i \(-0.259592\pi\)
−0.973285 + 0.229600i \(0.926258\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.39295 + 12.8050i 0.369186 + 0.639449i 0.989439 0.144953i \(-0.0463032\pi\)
−0.620252 + 0.784402i \(0.712970\pi\)
\(402\) 0 0
\(403\) 1.09672 1.89957i 0.0546315 0.0946245i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 8.93774 0.443027
\(408\) 0 0
\(409\) −11.0934 + 19.2143i −0.548533 + 0.950086i 0.449843 + 0.893108i \(0.351480\pi\)
−0.998375 + 0.0569787i \(0.981853\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −24.5754 + 5.65199i −1.20928 + 0.278116i
\(414\) 0 0
\(415\) −3.26318 5.65199i −0.160183 0.277445i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −38.1245 −1.86250 −0.931252 0.364375i \(-0.881283\pi\)
−0.931252 + 0.364375i \(0.881283\pi\)
\(420\) 0 0
\(421\) −21.5934 −1.05240 −0.526199 0.850362i \(-0.676383\pi\)
−0.526199 + 0.850362i \(0.676383\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −10.1319 17.5490i −0.491470 0.851251i
\(426\) 0 0
\(427\) 3.92661 12.8050i 0.190022 0.619675i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −17.5934 + 30.4726i −0.847444 + 1.46782i 0.0360384 + 0.999350i \(0.488526\pi\)
−0.883482 + 0.468465i \(0.844807\pi\)
\(432\) 0 0
\(433\) −35.5934 −1.71051 −0.855255 0.518208i \(-0.826599\pi\)
−0.855255 + 0.518208i \(0.826599\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.92081 + 5.05899i −0.139721 + 0.242004i
\(438\) 0 0
\(439\) 4.63266 + 8.02401i 0.221105 + 0.382965i 0.955144 0.296142i \(-0.0957003\pi\)
−0.734039 + 0.679108i \(0.762367\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.8278 + 30.8787i 0.847025 + 1.46709i 0.883851 + 0.467769i \(0.154942\pi\)
−0.0368252 + 0.999322i \(0.511724\pi\)
\(444\) 0 0
\(445\) −1.40661 + 2.43633i −0.0666799 + 0.115493i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −24.3723 −1.15020 −0.575099 0.818084i \(-0.695037\pi\)
−0.575099 + 0.818084i \(0.695037\pi\)
\(450\) 0 0
\(451\) −20.6275 + 35.7279i −0.971312 + 1.68236i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.34436 + 4.38404i −0.0630244 + 0.205527i
\(456\) 0 0
\(457\) −10.0311 17.3744i −0.469236 0.812741i 0.530145 0.847907i \(-0.322137\pi\)
−0.999381 + 0.0351656i \(0.988804\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.9413 −1.44108 −0.720539 0.693414i \(-0.756106\pi\)
−0.720539 + 0.693414i \(0.756106\pi\)
\(462\) 0 0
\(463\) −3.92661 −0.182485 −0.0912424 0.995829i \(-0.529084\pi\)
−0.0912424 + 0.995829i \(0.529084\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −2.53113 4.38404i −0.117127 0.202869i 0.801501 0.597993i \(-0.204035\pi\)
−0.918628 + 0.395124i \(0.870702\pi\)
\(468\) 0 0
\(469\) 14.5934 3.35627i 0.673860 0.154978i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −13.0527 + 22.6080i −0.600165 + 1.03952i
\(474\) 0 0
\(475\) 13.2346 0.607243
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 21.5934 37.4008i 0.986627 1.70889i 0.352156 0.935941i \(-0.385449\pi\)
0.634471 0.772947i \(-0.281218\pi\)
\(480\) 0 0
\(481\) 3.20331 + 5.54829i 0.146058 + 0.252980i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.26318 5.65199i −0.148173 0.256644i
\(486\) 0 0
\(487\) 15.2192 26.3604i 0.689647 1.19450i −0.282305 0.959325i \(-0.591099\pi\)
0.971952 0.235179i \(-0.0755676\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 29.6556 1.33834 0.669170 0.743109i \(-0.266650\pi\)
0.669170 + 0.743109i \(0.266650\pi\)
\(492\) 0 0
\(493\) 14.5934 25.2765i 0.657253 1.13840i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −16.3372 17.5490i −0.732824 0.787180i
\(498\) 0 0
\(499\) −11.9560 20.7084i −0.535224 0.927036i −0.999152 0.0411627i \(-0.986894\pi\)
0.463928 0.885873i \(-0.346440\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −39.3113 −1.75280 −0.876402 0.481580i \(-0.840063\pi\)
−0.876402 + 0.481580i \(0.840063\pi\)
\(504\) 0 0
\(505\) 7.06226 0.314266
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.4435 + 25.0169i 0.640198 + 1.10886i 0.985388 + 0.170322i \(0.0544809\pi\)
−0.345191 + 0.938533i \(0.612186\pi\)
\(510\) 0 0
\(511\) −6.52636 + 1.50097i −0.288709 + 0.0663989i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.53113 4.38404i 0.111535 0.193184i
\(516\) 0 0
\(517\) 17.8755 0.786163
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −3.10265 + 5.37395i −0.135930 + 0.235437i −0.925952 0.377641i \(-0.876735\pi\)
0.790023 + 0.613078i \(0.210069\pi\)
\(522\) 0 0
\(523\) −2.82989 4.90151i −0.123742 0.214328i 0.797498 0.603321i \(-0.206156\pi\)
−0.921241 + 0.388993i \(0.872823\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.93774 + 3.35627i 0.0844094 + 0.146201i
\(528\) 0 0
\(529\) 9.50000 16.4545i 0.413043 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −29.5718 −1.28090
\(534\) 0 0
\(535\) −2.21475 + 3.83606i −0.0957521 + 0.165848i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −20.4689 13.8564i −0.881657 0.596838i
\(540\) 0 0
\(541\) 5.73444 + 9.93233i 0.246543 + 0.427024i 0.962564 0.271054i \(-0.0873721\pi\)
−0.716022 + 0.698078i \(0.754039\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.00580 −0.0430836
\(546\) 0 0
\(547\) −39.1582 −1.67428 −0.837141 0.546987i \(-0.815775\pi\)
−0.837141 + 0.546987i \(0.815775\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.53113 + 16.5084i 0.406040 + 0.703281i
\(552\) 0 0
\(553\) −9.45234 + 30.8248i −0.401954 + 1.31080i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.26318 5.65199i 0.138265 0.239483i −0.788575 0.614939i \(-0.789181\pi\)
0.926840 + 0.375456i \(0.122514\pi\)
\(558\) 0 0
\(559\) −18.7125 −0.791454
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 18.7656 32.5029i 0.790874 1.36983i −0.134552 0.990907i \(-0.542960\pi\)
0.925426 0.378928i \(-0.123707\pi\)
\(564\) 0 0
\(565\) 5.06226 + 8.76809i 0.212971 + 0.368876i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.79319 8.30205i −0.200941 0.348040i 0.747891 0.663822i \(-0.231067\pi\)
−0.948832 + 0.315782i \(0.897733\pi\)
\(570\) 0 0
\(571\) 9.35625 16.2055i 0.391547 0.678179i −0.601107 0.799169i \(-0.705273\pi\)
0.992654 + 0.120990i \(0.0386068\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 9.06226 0.377922
\(576\) 0 0
\(577\) 10.5000 18.1865i 0.437121 0.757115i −0.560345 0.828259i \(-0.689332\pi\)
0.997466 + 0.0711438i \(0.0226649\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 17.1825 + 18.4570i 0.712850 + 0.765723i
\(582\) 0 0
\(583\) 14.5827 + 25.2580i 0.603955 + 1.04608i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.4066 −0.553350 −0.276675 0.960964i \(-0.589233\pi\)
−0.276675 + 0.960964i \(0.589233\pi\)
\(588\) 0 0
\(589\) −2.53113 −0.104293
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −2.05422 3.55802i −0.0843569 0.146110i 0.820760 0.571273i \(-0.193550\pi\)
−0.905117 + 0.425163i \(0.860217\pi\)
\(594\) 0 0
\(595\) −5.52056 5.93004i −0.226321 0.243108i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.468871 0.812109i 0.0191576 0.0331819i −0.856288 0.516499i \(-0.827235\pi\)
0.875445 + 0.483317i \(0.160568\pi\)
\(600\) 0 0
\(601\) −41.2490 −1.68258 −0.841292 0.540582i \(-0.818204\pi\)
−0.841292 + 0.540582i \(0.818204\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.502899 0.871046i 0.0204457 0.0354131i
\(606\) 0 0
\(607\) 13.8497 + 23.9884i 0.562142 + 0.973658i 0.997309 + 0.0733089i \(0.0233559\pi\)
−0.435167 + 0.900350i \(0.643311\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.40661 + 11.0966i 0.259184 + 0.448920i
\(612\) 0 0
\(613\) −5.00000 + 8.66025i −0.201948 + 0.349784i −0.949156 0.314806i \(-0.898061\pi\)
0.747208 + 0.664590i \(0.231394\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.1054 1.05097 0.525483 0.850804i \(-0.323885\pi\)
0.525483 + 0.850804i \(0.323885\pi\)
\(618\) 0 0
\(619\) 18.4824 32.0124i 0.742869 1.28669i −0.208315 0.978062i \(-0.566798\pi\)
0.951184 0.308625i \(-0.0998688\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.18677 10.3923i 0.127675 0.416359i
\(624\) 0 0
\(625\) −9.09339 15.7502i −0.363735 0.630008i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −11.3196 −0.451340
\(630\) 0 0
\(631\) 32.6318 1.29905 0.649526 0.760340i \(-0.274967\pi\)
0.649526 + 0.760340i \(0.274967\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.35895 9.28198i −0.212663 0.368344i
\(636\) 0 0
\(637\) 1.26556 17.6726i 0.0501435 0.700216i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 5.65978 9.80302i 0.223548 0.387196i −0.732335 0.680944i \(-0.761569\pi\)
0.955883 + 0.293748i \(0.0949028\pi\)
\(642\) 0 0
\(643\) −40.7095 −1.60543 −0.802713 0.596366i \(-0.796611\pi\)
−0.802713 + 0.596366i \(0.796611\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.0623 33.0168i 0.749415 1.29802i −0.198689 0.980063i \(-0.563668\pi\)
0.948104 0.317962i \(-0.102998\pi\)
\(648\) 0 0
\(649\) 16.8278 + 29.1466i 0.660550 + 1.14411i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 18.0491 + 31.2619i 0.706315 + 1.22337i 0.966215 + 0.257738i \(0.0829770\pi\)
−0.259900 + 0.965635i \(0.583690\pi\)
\(654\) 0 0
\(655\) −4.99635 + 8.65393i −0.195224 + 0.338137i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 29.0623 1.13210 0.566052 0.824369i \(-0.308470\pi\)
0.566052 + 0.824369i \(0.308470\pi\)
\(660\) 0 0
\(661\) −5.73444 + 9.93233i −0.223044 + 0.386323i −0.955731 0.294243i \(-0.904933\pi\)
0.732687 + 0.680566i \(0.238266\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.15688 1.18601i 0.199975 0.0459914i
\(666\) 0 0
\(667\) 6.52636 + 11.3040i 0.252702 + 0.437692i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −17.8755 −0.690075
\(672\) 0 0
\(673\) 26.0623 1.00463 0.502313 0.864686i \(-0.332482\pi\)
0.502313 + 0.864686i \(0.332482\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.8365 37.8219i −0.839244 1.45361i −0.890528 0.454928i \(-0.849665\pi\)
0.0512845 0.998684i \(-0.483668\pi\)
\(678\) 0 0
\(679\) 17.1825 + 18.4570i 0.659403 + 0.708313i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14.7656 + 25.5747i −0.564989 + 0.978589i 0.432062 + 0.901844i \(0.357786\pi\)
−0.997051 + 0.0767452i \(0.975547\pi\)
\(684\) 0 0
\(685\) 8.93774 0.341494
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −10.4530 + 18.1051i −0.398226 + 0.689748i
\(690\) 0 0
\(691\) −24.1421 41.8154i −0.918410 1.59073i −0.801830 0.597552i \(-0.796140\pi\)
−0.116580 0.993181i \(-0.537193\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.00000 + 12.1244i 0.265525 + 0.459903i
\(696\) 0 0
\(697\) 26.1245 45.2490i 0.989537 1.71393i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −8.25953 −0.311958 −0.155979 0.987760i \(-0.549853\pi\)
−0.155979 + 0.987760i \(0.549853\pi\)
\(702\) 0 0
\(703\) 3.69647 6.40248i 0.139415 0.241474i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −26.5934 + 6.11609i −1.00015 + 0.230019i
\(708\) 0 0
\(709\) −13.1245 22.7323i −0.492902 0.853730i 0.507065 0.861908i \(-0.330730\pi\)
−0.999967 + 0.00817731i \(0.997397\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.73317 −0.0649077
\(714\) 0 0
\(715\) 6.12004 0.228877
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.46887 7.74031i −0.166661 0.288665i 0.770583 0.637340i \(-0.219965\pi\)
−0.937244 + 0.348675i \(0.886632\pi\)
\(720\) 0 0
\(721\) −5.73444 + 18.7004i −0.213562 + 0.696440i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 14.7859 25.6099i 0.549134 0.951128i
\(726\) 0 0
\(727\) 45.2243 1.67727 0.838637 0.544690i \(-0.183353\pi\)
0.838637 + 0.544690i \(0.183353\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 16.5311 28.6328i 0.611426 1.05902i
\(732\) 0 0
\(733\) −8.26556 14.3164i −0.305296 0.528787i 0.672032 0.740523i \(-0.265422\pi\)
−0.977327 + 0.211735i \(0.932089\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.99270 17.3079i −0.368086 0.637543i
\(738\) 0 0
\(739\) −25.0087 + 43.3164i −0.919961 + 1.59342i −0.120490 + 0.992715i \(0.538446\pi\)
−0.799471 + 0.600704i \(0.794887\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.93774 0.181148 0.0905741 0.995890i \(-0.471130\pi\)
0.0905741 + 0.995890i \(0.471130\pi\)
\(744\) 0 0
\(745\) −4.46887 + 7.74031i −0.163727 + 0.283583i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.01767 16.3630i 0.183341 0.597890i
\(750\) 0 0
\(751\) 20.8790 + 36.1634i 0.761884 + 1.31962i 0.941878 + 0.335954i \(0.109059\pi\)
−0.179994 + 0.983668i \(0.557608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 4.46887 0.162639
\(756\) 0 0
\(757\) −10.9377 −0.397539 −0.198770 0.980046i \(-0.563695\pi\)
−0.198770 + 0.980046i \(0.563695\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.6583 + 28.8530i 0.603862 + 1.04592i 0.992230 + 0.124415i \(0.0397055\pi\)
−0.388368 + 0.921504i \(0.626961\pi\)
\(762\) 0 0
\(763\) 3.78739 0.871046i 0.137113 0.0315340i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0623 + 20.8924i −0.435543 + 0.754382i
\(768\) 0 0
\(769\) −29.9377 −1.07958 −0.539791 0.841799i \(-0.681497\pi\)
−0.539791 + 0.841799i \(0.681497\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.42371 + 5.93004i −0.123142 + 0.213289i −0.921005 0.389550i \(-0.872630\pi\)
0.797863 + 0.602839i \(0.205964\pi\)
\(774\) 0 0
\(775\) 1.96330 + 3.40054i 0.0705239 + 0.122151i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 17.0623 + 29.5527i 0.611319 + 1.05884i
\(780\) 0 0
\(781\) −16.0000 + 27.7128i −0.572525 + 0.991642i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.46634 −0.123719
\(786\) 0 0
\(787\) 11.3196 19.6060i 0.403498 0.698880i −0.590647 0.806930i \(-0.701127\pi\)
0.994145 + 0.108050i \(0.0344607\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −26.6556 28.6328i −0.947766 1.01806i
\(792\) 0 0
\(793\) −6.40661 11.0966i −0.227506 0.394051i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 42.5819 1.50833 0.754164 0.656686i \(-0.228043\pi\)
0.754164 + 0.656686i \(0.228043\pi\)
\(798\) 0 0
\(799\) −22.6391 −0.800914
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.46887 + 7.74031i 0.157703 + 0.273150i
\(804\) 0 0
\(805\) 3.53113 0.812109i 0.124456 0.0286231i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.65978 + 9.80302i −0.198987 + 0.344656i −0.948200 0.317673i \(-0.897099\pi\)
0.749213 + 0.662329i \(0.230432\pi\)
\(810\) 0 0
\(811\) −13.0527 −0.458343 −0.229172 0.973386i \(-0.573602\pi\)
−0.229172 + 0.973386i \(0.573602\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.65564 + 9.79586i −0.198109 + 0.343134i
\(816\) 0 0
\(817\) 10.7967 + 18.7004i 0.377728 + 0.654245i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.39660 + 4.15103i 0.0836418 + 0.144872i 0.904812 0.425812i \(-0.140012\pi\)
−0.821170 + 0.570684i \(0.806678\pi\)
\(822\) 0 0
\(823\) 1.73317 3.00194i 0.0604145 0.104641i −0.834236 0.551407i \(-0.814091\pi\)
0.894651 + 0.446766i \(0.147424\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −5.53113 −0.192336 −0.0961681 0.995365i \(-0.530659\pi\)
−0.0961681 + 0.995365i \(0.530659\pi\)
\(828\) 0 0
\(829\) −8.85895 + 15.3442i −0.307684 + 0.532924i −0.977855 0.209282i \(-0.932887\pi\)
0.670171 + 0.742207i \(0.266221\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25.9236 + 17.5490i 0.898200 + 0.608036i
\(834\) 0 0
\(835\) −4.79319 8.30205i −0.165875 0.287304i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −8.93774 −0.308565 −0.154283 0.988027i \(-0.549307\pi\)
−0.154283 + 0.988027i \(0.549307\pi\)
\(840\) 0 0
\(841\) 13.5934 0.468737
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.25738 3.90990i −0.0776563 0.134505i
\(846\) 0 0
\(847\) −1.13935 + 3.71550i −0.0391485 + 0.127666i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.53113 4.38404i 0.0867660 0.150283i
\(852\) 0 0
\(853\) 20.4066 0.698709 0.349355 0.936991i \(-0.386401\pi\)
0.349355 + 0.936991i \(0.386401\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.4749 47.5880i 0.938526 1.62557i 0.170303 0.985392i \(-0.445525\pi\)
0.768223 0.640182i \(-0.221141\pi\)
\(858\) 0 0
\(859\) −17.8885 30.9839i −0.610349 1.05716i −0.991181 0.132512i \(-0.957696\pi\)
0.380832 0.924644i \(-0.375638\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.6556 + 47.9010i 0.941409 + 1.63057i 0.762786 + 0.646651i \(0.223831\pi\)
0.178623 + 0.983918i \(0.442836\pi\)
\(864\) 0 0
\(865\) 2.46887 4.27621i 0.0839441 0.145396i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 43.0308 1.45972
\(870\) 0 0
\(871\) 7.16281 12.4064i 0.242703 0.420373i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −11.7656 12.6382i −0.397749 0.427251i
\(876\) 0 0
\(877\) 25.5934 + 44.3290i 0.864227 + 1.49689i 0.867812 + 0.496892i \(0.165525\pi\)
−0.00358516 + 0.999994i \(0.501141\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 19.6217 0.661072 0.330536 0.943793i \(-0.392770\pi\)
0.330536 + 0.943793i \(0.392770\pi\)
\(882\) 0 0
\(883\) −36.9647 −1.24396 −0.621981 0.783032i \(-0.713672\pi\)
−0.621981 + 0.783032i \(0.713672\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.53113 + 16.5084i 0.320024 + 0.554298i 0.980493 0.196556i \(-0.0629757\pi\)
−0.660469 + 0.750854i \(0.729642\pi\)
\(888\) 0 0
\(889\) 28.2179 + 30.3109i 0.946398 + 1.01659i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.39295 12.8050i 0.247395 0.428502i
\(894\) 0 0
\(895\) −2.01159 −0.0672402
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.82782 + 4.89793i −0.0943132 + 0.163355i
\(900\) 0 0
\(901\) −18.4689 31.9890i −0.615287 1.06571i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.65978 + 9.80302i 0.188137 + 0.325863i
\(906\) 0 0
\(907\) 23.2756 40.3144i 0.772852 1.33862i −0.163142 0.986603i \(-0.552163\pi\)
0.935994 0.352017i \(-0.114504\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −11.1868 −0.370634 −0.185317 0.982679i \(-0.559331\pi\)
−0.185317 + 0.982679i \(0.559331\pi\)
\(912\) 0 0
\(913\) 16.8278 29.1466i 0.556920 0.964613i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 11.3196 36.9139i 0.373805 1.21900i
\(918\) 0 0
\(919\) 1.96330 + 3.40054i 0.0647634 + 0.112174i 0.896589 0.442864i \(-0.146037\pi\)
−0.831826 + 0.555037i \(0.812704\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −22.9377 −0.755005
\(924\) 0 0
\(925\) −11.4689 −0.377094
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.18764 2.05705i −0.0389652 0.0674897i 0.845885 0.533365i \(-0.179073\pi\)
−0.884850 + 0.465875i \(0.845739\pi\)
\(930\) 0 0
\(931\) −18.3914 + 8.93197i −0.602755 + 0.292734i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.40661 + 9.36453i −0.176815 + 0.306253i
\(936\) 0 0
\(937\) −10.8755 −0.355287 −0.177643 0.984095i \(-0.556847\pi\)
−0.177643 + 0.984095i \(0.556847\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13.7588 23.8309i 0.448523 0.776865i −0.549767 0.835318i \(-0.685283\pi\)
0.998290 + 0.0584529i \(0.0186168\pi\)
\(942\) 0 0
\(943\) 11.6832 + 20.2360i 0.380459 + 0.658974i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.00000 + 10.3923i 0.194974 + 0.337705i 0.946892 0.321552i \(-0.104204\pi\)
−0.751918 + 0.659256i \(0.770871\pi\)
\(948\) 0 0
\(949\) −3.20331 + 5.54829i −0.103984 + 0.180105i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 40.8913 1.32460 0.662300 0.749239i \(-0.269580\pi\)
0.662300 + 0.749239i \(0.269580\pi\)
\(954\) 0 0
\(955\) −7.21110 + 12.4900i −0.233346 + 0.404167i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −33.6556 + 7.74031i −1.08680 + 0.249948i
\(960\) 0 0
\(961\) 15.1245 + 26.1964i 0.487888 + 0.845046i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.32685 0.0427129
\(966\) 0 0
\(967\) 15.6525 0.503350 0.251675 0.967812i \(-0.419019\pi\)
0.251675 + 0.967812i \(0.419019\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 5.35895 + 9.28198i 0.171977 + 0.297873i 0.939111 0.343614i \(-0.111651\pi\)
−0.767134 + 0.641487i \(0.778318\pi\)
\(972\) 0 0
\(973\) −36.8590 39.5929i −1.18164 1.26929i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 16.5191 28.6119i 0.528492 0.915374i −0.470957 0.882156i \(-0.656091\pi\)
0.999448 0.0332178i \(-0.0105755\pi\)
\(978\) 0 0
\(979\) −14.5075 −0.463661
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 11.4689 19.8647i 0.365800 0.633584i −0.623104 0.782139i \(-0.714129\pi\)
0.988904 + 0.148554i \(0.0474620\pi\)
\(984\) 0 0
\(985\) −0.593387 1.02778i −0.0189069 0.0327477i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.39295 + 12.8050i 0.235082 + 0.407174i
\(990\) 0 0
\(991\) 1.29988 2.25145i 0.0412920 0.0715198i −0.844641 0.535333i \(-0.820186\pi\)
0.885933 + 0.463814i \(0.153519\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.87548 0.313074
\(996\) 0 0
\(997\) −15.2656 + 26.4407i −0.483465 + 0.837386i −0.999820 0.0189886i \(-0.993955\pi\)
0.516354 + 0.856375i \(0.327289\pi\)
\(998\) 0 0
\(999\) 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 2016.2.s.x.865.3 yes 8
3.2 odd 2 2016.2.s.w.865.2 yes 8
4.3 odd 2 2016.2.s.w.865.3 yes 8
7.2 even 3 inner 2016.2.s.x.289.3 yes 8
12.11 even 2 inner 2016.2.s.x.865.2 yes 8
21.2 odd 6 2016.2.s.w.289.2 8
28.23 odd 6 2016.2.s.w.289.3 yes 8
84.23 even 6 inner 2016.2.s.x.289.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.s.w.289.2 8 21.2 odd 6
2016.2.s.w.289.3 yes 8 28.23 odd 6
2016.2.s.w.865.2 yes 8 3.2 odd 2
2016.2.s.w.865.3 yes 8 4.3 odd 2
2016.2.s.x.289.2 yes 8 84.23 even 6 inner
2016.2.s.x.289.3 yes 8 7.2 even 3 inner
2016.2.s.x.865.2 yes 8 12.11 even 2 inner
2016.2.s.x.865.3 yes 8 1.1 even 1 trivial