Properties

Label 1568.2.i.m.961.2
Level $1568$
Weight $2$
Character 1568.961
Analytic conductor $12.521$
Analytic rank $0$
Dimension $4$
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] = [1568,2,Mod(961,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1568.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.i (of order \(3\), degree \(2\), not 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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 961.2
Root \(-0.309017 - 0.535233i\) of defining polynomial
Character \(\chi\) \(=\) 1568.961
Dual form 1568.2.i.m.1537.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.618034 + 1.07047i) q^{3} +(-1.61803 + 2.80252i) q^{5} +(0.736068 - 1.27491i) q^{9} +O(q^{10})\) \(q+(0.618034 + 1.07047i) q^{3} +(-1.61803 + 2.80252i) q^{5} +(0.736068 - 1.27491i) q^{9} +(-3.23607 - 5.60503i) q^{11} +0.763932 q^{13} -4.00000 q^{15} +(-2.23607 - 3.87298i) q^{17} +(-0.618034 + 1.07047i) q^{19} +(2.00000 - 3.46410i) q^{23} +(-2.73607 - 4.73901i) q^{25} +5.52786 q^{27} -4.47214 q^{29} +(-1.23607 - 2.14093i) q^{31} +(4.00000 - 6.92820i) q^{33} +(2.23607 - 3.87298i) q^{37} +(0.472136 + 0.817763i) q^{39} -8.47214 q^{41} -6.47214 q^{43} +(2.38197 + 4.12569i) q^{45} +(5.23607 - 9.06914i) q^{47} +(2.76393 - 4.78727i) q^{51} +(5.00000 + 8.66025i) q^{53} +20.9443 q^{55} -1.52786 q^{57} +(-4.61803 - 7.99867i) q^{59} +(-5.61803 + 9.73072i) q^{61} +(-1.23607 + 2.14093i) q^{65} +(2.00000 + 3.46410i) q^{67} +4.94427 q^{69} +4.94427 q^{71} +(1.47214 + 2.54981i) q^{73} +(3.38197 - 5.85774i) q^{75} +(6.47214 - 11.2101i) q^{79} +(1.20820 + 2.09267i) q^{81} +9.23607 q^{83} +14.4721 q^{85} +(-2.76393 - 4.78727i) q^{87} +(3.00000 - 5.19615i) q^{89} +(1.52786 - 2.64634i) q^{93} +(-2.00000 - 3.46410i) q^{95} +12.4721 q^{97} -9.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} - 6 q^{9} - 4 q^{11} + 12 q^{13} - 16 q^{15} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 40 q^{27} + 4 q^{31} + 16 q^{33} - 16 q^{39} - 16 q^{41} - 8 q^{43} + 14 q^{45} + 12 q^{47} + 20 q^{51} + 20 q^{53} + 48 q^{55} - 24 q^{57} - 14 q^{59} - 18 q^{61} + 4 q^{65} + 8 q^{67} - 16 q^{69} - 16 q^{71} - 12 q^{73} + 18 q^{75} + 8 q^{79} - 22 q^{81} + 28 q^{83} + 40 q^{85} - 20 q^{87} + 12 q^{89} + 24 q^{93} - 8 q^{95} + 32 q^{97} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0.618034 + 1.07047i 0.356822 + 0.618034i 0.987428 0.158069i \(-0.0505269\pi\)
−0.630606 + 0.776103i \(0.717194\pi\)
\(4\) 0 0
\(5\) −1.61803 + 2.80252i −0.723607 + 1.25332i 0.235938 + 0.971768i \(0.424184\pi\)
−0.959545 + 0.281556i \(0.909150\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0.736068 1.27491i 0.245356 0.424969i
\(10\) 0 0
\(11\) −3.23607 5.60503i −0.975711 1.68998i −0.677568 0.735460i \(-0.736966\pi\)
−0.298143 0.954521i \(-0.596367\pi\)
\(12\) 0 0
\(13\) 0.763932 0.211877 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 0 0
\(17\) −2.23607 3.87298i −0.542326 0.939336i −0.998770 0.0495842i \(-0.984210\pi\)
0.456444 0.889752i \(-0.349123\pi\)
\(18\) 0 0
\(19\) −0.618034 + 1.07047i −0.141787 + 0.245582i −0.928170 0.372158i \(-0.878618\pi\)
0.786383 + 0.617740i \(0.211951\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.00000 3.46410i 0.417029 0.722315i −0.578610 0.815604i \(-0.696405\pi\)
0.995639 + 0.0932891i \(0.0297381\pi\)
\(24\) 0 0
\(25\) −2.73607 4.73901i −0.547214 0.947802i
\(26\) 0 0
\(27\) 5.52786 1.06384
\(28\) 0 0
\(29\) −4.47214 −0.830455 −0.415227 0.909718i \(-0.636298\pi\)
−0.415227 + 0.909718i \(0.636298\pi\)
\(30\) 0 0
\(31\) −1.23607 2.14093i −0.222004 0.384523i 0.733412 0.679784i \(-0.237927\pi\)
−0.955417 + 0.295261i \(0.904593\pi\)
\(32\) 0 0
\(33\) 4.00000 6.92820i 0.696311 1.20605i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.23607 3.87298i 0.367607 0.636715i −0.621584 0.783348i \(-0.713510\pi\)
0.989191 + 0.146633i \(0.0468437\pi\)
\(38\) 0 0
\(39\) 0.472136 + 0.817763i 0.0756023 + 0.130947i
\(40\) 0 0
\(41\) −8.47214 −1.32313 −0.661563 0.749890i \(-0.730106\pi\)
−0.661563 + 0.749890i \(0.730106\pi\)
\(42\) 0 0
\(43\) −6.47214 −0.986991 −0.493496 0.869748i \(-0.664281\pi\)
−0.493496 + 0.869748i \(0.664281\pi\)
\(44\) 0 0
\(45\) 2.38197 + 4.12569i 0.355083 + 0.615021i
\(46\) 0 0
\(47\) 5.23607 9.06914i 0.763759 1.32287i −0.177141 0.984185i \(-0.556685\pi\)
0.940900 0.338684i \(-0.109982\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 2.76393 4.78727i 0.387028 0.670352i
\(52\) 0 0
\(53\) 5.00000 + 8.66025i 0.686803 + 1.18958i 0.972867 + 0.231367i \(0.0743197\pi\)
−0.286064 + 0.958211i \(0.592347\pi\)
\(54\) 0 0
\(55\) 20.9443 2.82413
\(56\) 0 0
\(57\) −1.52786 −0.202371
\(58\) 0 0
\(59\) −4.61803 7.99867i −0.601217 1.04134i −0.992637 0.121126i \(-0.961349\pi\)
0.391420 0.920212i \(-0.371984\pi\)
\(60\) 0 0
\(61\) −5.61803 + 9.73072i −0.719316 + 1.24589i 0.241956 + 0.970287i \(0.422211\pi\)
−0.961271 + 0.275604i \(0.911122\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.23607 + 2.14093i −0.153315 + 0.265550i
\(66\) 0 0
\(67\) 2.00000 + 3.46410i 0.244339 + 0.423207i 0.961946 0.273241i \(-0.0880957\pi\)
−0.717607 + 0.696449i \(0.754762\pi\)
\(68\) 0 0
\(69\) 4.94427 0.595220
\(70\) 0 0
\(71\) 4.94427 0.586777 0.293389 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 0 0
\(73\) 1.47214 + 2.54981i 0.172300 + 0.298433i 0.939224 0.343306i \(-0.111547\pi\)
−0.766923 + 0.641739i \(0.778213\pi\)
\(74\) 0 0
\(75\) 3.38197 5.85774i 0.390516 0.676393i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.47214 11.2101i 0.728172 1.26123i −0.229483 0.973313i \(-0.573703\pi\)
0.957655 0.287918i \(-0.0929632\pi\)
\(80\) 0 0
\(81\) 1.20820 + 2.09267i 0.134245 + 0.232519i
\(82\) 0 0
\(83\) 9.23607 1.01379 0.506895 0.862008i \(-0.330793\pi\)
0.506895 + 0.862008i \(0.330793\pi\)
\(84\) 0 0
\(85\) 14.4721 1.56972
\(86\) 0 0
\(87\) −2.76393 4.78727i −0.296325 0.513249i
\(88\) 0 0
\(89\) 3.00000 5.19615i 0.317999 0.550791i −0.662071 0.749441i \(-0.730322\pi\)
0.980071 + 0.198650i \(0.0636557\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.52786 2.64634i 0.158432 0.274412i
\(94\) 0 0
\(95\) −2.00000 3.46410i −0.205196 0.355409i
\(96\) 0 0
\(97\) 12.4721 1.26635 0.633177 0.774007i \(-0.281751\pi\)
0.633177 + 0.774007i \(0.281751\pi\)
\(98\) 0 0
\(99\) −9.52786 −0.957586
\(100\) 0 0
\(101\) 0.854102 + 1.47935i 0.0849863 + 0.147201i 0.905385 0.424591i \(-0.139582\pi\)
−0.820399 + 0.571791i \(0.806249\pi\)
\(102\) 0 0
\(103\) −2.76393 + 4.78727i −0.272338 + 0.471704i −0.969460 0.245249i \(-0.921130\pi\)
0.697122 + 0.716953i \(0.254464\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.47214 7.74597i 0.432338 0.748831i −0.564736 0.825271i \(-0.691022\pi\)
0.997074 + 0.0764405i \(0.0243555\pi\)
\(108\) 0 0
\(109\) −4.23607 7.33708i −0.405742 0.702765i 0.588666 0.808377i \(-0.299653\pi\)
−0.994407 + 0.105611i \(0.966320\pi\)
\(110\) 0 0
\(111\) 5.52786 0.524682
\(112\) 0 0
\(113\) −12.4721 −1.17328 −0.586640 0.809848i \(-0.699550\pi\)
−0.586640 + 0.809848i \(0.699550\pi\)
\(114\) 0 0
\(115\) 6.47214 + 11.2101i 0.603530 + 1.04534i
\(116\) 0 0
\(117\) 0.562306 0.973942i 0.0519852 0.0900410i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −15.4443 + 26.7503i −1.40402 + 2.43184i
\(122\) 0 0
\(123\) −5.23607 9.06914i −0.472120 0.817736i
\(124\) 0 0
\(125\) 1.52786 0.136656
\(126\) 0 0
\(127\) −8.94427 −0.793676 −0.396838 0.917889i \(-0.629892\pi\)
−0.396838 + 0.917889i \(0.629892\pi\)
\(128\) 0 0
\(129\) −4.00000 6.92820i −0.352180 0.609994i
\(130\) 0 0
\(131\) 5.85410 10.1396i 0.511475 0.885901i −0.488436 0.872600i \(-0.662433\pi\)
0.999912 0.0133016i \(-0.00423415\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −8.94427 + 15.4919i −0.769800 + 1.33333i
\(136\) 0 0
\(137\) −7.47214 12.9421i −0.638388 1.10572i −0.985787 0.168002i \(-0.946268\pi\)
0.347399 0.937717i \(-0.387065\pi\)
\(138\) 0 0
\(139\) −1.23607 −0.104842 −0.0524210 0.998625i \(-0.516694\pi\)
−0.0524210 + 0.998625i \(0.516694\pi\)
\(140\) 0 0
\(141\) 12.9443 1.09010
\(142\) 0 0
\(143\) −2.47214 4.28187i −0.206730 0.358068i
\(144\) 0 0
\(145\) 7.23607 12.5332i 0.600923 1.04083i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.47214 + 2.54981i −0.120602 + 0.208889i −0.920005 0.391906i \(-0.871816\pi\)
0.799403 + 0.600795i \(0.205149\pi\)
\(150\) 0 0
\(151\) −4.47214 7.74597i −0.363937 0.630358i 0.624668 0.780891i \(-0.285234\pi\)
−0.988605 + 0.150533i \(0.951901\pi\)
\(152\) 0 0
\(153\) −6.58359 −0.532252
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −0.381966 0.661585i −0.0304842 0.0528002i 0.850381 0.526168i \(-0.176372\pi\)
−0.880865 + 0.473368i \(0.843038\pi\)
\(158\) 0 0
\(159\) −6.18034 + 10.7047i −0.490133 + 0.848935i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.70820 + 2.95870i −0.133797 + 0.231743i −0.925137 0.379633i \(-0.876050\pi\)
0.791340 + 0.611376i \(0.209384\pi\)
\(164\) 0 0
\(165\) 12.9443 + 22.4201i 1.00771 + 1.74541i
\(166\) 0 0
\(167\) −23.4164 −1.81202 −0.906008 0.423261i \(-0.860886\pi\)
−0.906008 + 0.423261i \(0.860886\pi\)
\(168\) 0 0
\(169\) −12.4164 −0.955108
\(170\) 0 0
\(171\) 0.909830 + 1.57587i 0.0695764 + 0.120510i
\(172\) 0 0
\(173\) −2.85410 + 4.94345i −0.216993 + 0.375844i −0.953887 0.300165i \(-0.902958\pi\)
0.736894 + 0.676008i \(0.236292\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.70820 9.88690i 0.429055 0.743145i
\(178\) 0 0
\(179\) −3.52786 6.11044i −0.263685 0.456716i 0.703533 0.710662i \(-0.251605\pi\)
−0.967218 + 0.253947i \(0.918271\pi\)
\(180\) 0 0
\(181\) −12.1803 −0.905358 −0.452679 0.891674i \(-0.649532\pi\)
−0.452679 + 0.891674i \(0.649532\pi\)
\(182\) 0 0
\(183\) −13.8885 −1.02667
\(184\) 0 0
\(185\) 7.23607 + 12.5332i 0.532006 + 0.921462i
\(186\) 0 0
\(187\) −14.4721 + 25.0665i −1.05831 + 1.83304i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.47214 4.28187i 0.178877 0.309825i −0.762619 0.646848i \(-0.776087\pi\)
0.941496 + 0.337023i \(0.109420\pi\)
\(192\) 0 0
\(193\) −0.236068 0.408882i −0.0169925 0.0294320i 0.857404 0.514644i \(-0.172076\pi\)
−0.874397 + 0.485212i \(0.838742\pi\)
\(194\) 0 0
\(195\) −3.05573 −0.218825
\(196\) 0 0
\(197\) 10.9443 0.779747 0.389874 0.920868i \(-0.372519\pi\)
0.389874 + 0.920868i \(0.372519\pi\)
\(198\) 0 0
\(199\) 7.70820 + 13.3510i 0.546420 + 0.946427i 0.998516 + 0.0544581i \(0.0173431\pi\)
−0.452096 + 0.891969i \(0.649324\pi\)
\(200\) 0 0
\(201\) −2.47214 + 4.28187i −0.174371 + 0.302019i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 13.7082 23.7433i 0.957422 1.65830i
\(206\) 0 0
\(207\) −2.94427 5.09963i −0.204641 0.354449i
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 3.05573 + 5.29268i 0.209375 + 0.362648i
\(214\) 0 0
\(215\) 10.4721 18.1383i 0.714194 1.23702i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.81966 + 3.15174i −0.122961 + 0.212975i
\(220\) 0 0
\(221\) −1.70820 2.95870i −0.114906 0.199023i
\(222\) 0 0
\(223\) −12.9443 −0.866813 −0.433406 0.901199i \(-0.642688\pi\)
−0.433406 + 0.901199i \(0.642688\pi\)
\(224\) 0 0
\(225\) −8.05573 −0.537049
\(226\) 0 0
\(227\) −8.61803 14.9269i −0.571999 0.990731i −0.996361 0.0852385i \(-0.972835\pi\)
0.424362 0.905493i \(-0.360499\pi\)
\(228\) 0 0
\(229\) −11.7984 + 20.4354i −0.779658 + 1.35041i 0.152480 + 0.988307i \(0.451274\pi\)
−0.932139 + 0.362102i \(0.882059\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.94427 13.7599i 0.520447 0.901440i −0.479271 0.877667i \(-0.659099\pi\)
0.999717 0.0237728i \(-0.00756784\pi\)
\(234\) 0 0
\(235\) 16.9443 + 29.3483i 1.10532 + 1.91447i
\(236\) 0 0
\(237\) 16.0000 1.03931
\(238\) 0 0
\(239\) 13.8885 0.898375 0.449188 0.893437i \(-0.351713\pi\)
0.449188 + 0.893437i \(0.351713\pi\)
\(240\) 0 0
\(241\) −6.23607 10.8012i −0.401700 0.695766i 0.592231 0.805768i \(-0.298247\pi\)
−0.993931 + 0.110003i \(0.964914\pi\)
\(242\) 0 0
\(243\) 6.79837 11.7751i 0.436116 0.755375i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.472136 + 0.817763i −0.0300413 + 0.0520330i
\(248\) 0 0
\(249\) 5.70820 + 9.88690i 0.361743 + 0.626557i
\(250\) 0 0
\(251\) −4.29180 −0.270896 −0.135448 0.990784i \(-0.543247\pi\)
−0.135448 + 0.990784i \(0.543247\pi\)
\(252\) 0 0
\(253\) −25.8885 −1.62760
\(254\) 0 0
\(255\) 8.94427 + 15.4919i 0.560112 + 0.970143i
\(256\) 0 0
\(257\) 7.00000 12.1244i 0.436648 0.756297i −0.560781 0.827964i \(-0.689499\pi\)
0.997429 + 0.0716680i \(0.0228322\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −3.29180 + 5.70156i −0.203757 + 0.352918i
\(262\) 0 0
\(263\) 5.52786 + 9.57454i 0.340863 + 0.590392i 0.984593 0.174861i \(-0.0559475\pi\)
−0.643730 + 0.765252i \(0.722614\pi\)
\(264\) 0 0
\(265\) −32.3607 −1.98790
\(266\) 0 0
\(267\) 7.41641 0.453877
\(268\) 0 0
\(269\) 2.09017 + 3.62028i 0.127440 + 0.220732i 0.922684 0.385557i \(-0.125991\pi\)
−0.795244 + 0.606289i \(0.792657\pi\)
\(270\) 0 0
\(271\) −12.0000 + 20.7846i −0.728948 + 1.26258i 0.228380 + 0.973572i \(0.426657\pi\)
−0.957328 + 0.289003i \(0.906676\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −17.7082 + 30.6715i −1.06784 + 1.84956i
\(276\) 0 0
\(277\) −3.94427 6.83168i −0.236988 0.410476i 0.722860 0.690994i \(-0.242827\pi\)
−0.959849 + 0.280518i \(0.909494\pi\)
\(278\) 0 0
\(279\) −3.63932 −0.217880
\(280\) 0 0
\(281\) 26.0000 1.55103 0.775515 0.631329i \(-0.217490\pi\)
0.775515 + 0.631329i \(0.217490\pi\)
\(282\) 0 0
\(283\) −3.09017 5.35233i −0.183692 0.318163i 0.759443 0.650574i \(-0.225471\pi\)
−0.943135 + 0.332410i \(0.892138\pi\)
\(284\) 0 0
\(285\) 2.47214 4.28187i 0.146437 0.253636i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.50000 + 2.59808i −0.0882353 + 0.152828i
\(290\) 0 0
\(291\) 7.70820 + 13.3510i 0.451863 + 0.782650i
\(292\) 0 0
\(293\) −12.7639 −0.745677 −0.372838 0.927896i \(-0.621615\pi\)
−0.372838 + 0.927896i \(0.621615\pi\)
\(294\) 0 0
\(295\) 29.8885 1.74018
\(296\) 0 0
\(297\) −17.8885 30.9839i −1.03800 1.79787i
\(298\) 0 0
\(299\) 1.52786 2.64634i 0.0883587 0.153042i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −1.05573 + 1.82857i −0.0606500 + 0.105049i
\(304\) 0 0
\(305\) −18.1803 31.4893i −1.04100 1.80307i
\(306\) 0 0
\(307\) −1.81966 −0.103853 −0.0519267 0.998651i \(-0.516536\pi\)
−0.0519267 + 0.998651i \(0.516536\pi\)
\(308\) 0 0
\(309\) −6.83282 −0.388705
\(310\) 0 0
\(311\) 4.00000 + 6.92820i 0.226819 + 0.392862i 0.956864 0.290537i \(-0.0938340\pi\)
−0.730044 + 0.683400i \(0.760501\pi\)
\(312\) 0 0
\(313\) 4.23607 7.33708i 0.239437 0.414717i −0.721116 0.692814i \(-0.756371\pi\)
0.960553 + 0.278098i \(0.0897039\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.52786 + 7.84249i −0.254310 + 0.440478i −0.964708 0.263322i \(-0.915182\pi\)
0.710398 + 0.703800i \(0.248515\pi\)
\(318\) 0 0
\(319\) 14.4721 + 25.0665i 0.810284 + 1.40345i
\(320\) 0 0
\(321\) 11.0557 0.617071
\(322\) 0 0
\(323\) 5.52786 0.307579
\(324\) 0 0
\(325\) −2.09017 3.62028i −0.115942 0.200817i
\(326\) 0 0
\(327\) 5.23607 9.06914i 0.289555 0.501524i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 11.2361 19.4614i 0.617590 1.06970i −0.372334 0.928099i \(-0.621442\pi\)
0.989924 0.141599i \(-0.0452243\pi\)
\(332\) 0 0
\(333\) −3.29180 5.70156i −0.180389 0.312443i
\(334\) 0 0
\(335\) −12.9443 −0.707221
\(336\) 0 0
\(337\) 10.3607 0.564382 0.282191 0.959358i \(-0.408939\pi\)
0.282191 + 0.959358i \(0.408939\pi\)
\(338\) 0 0
\(339\) −7.70820 13.3510i −0.418652 0.725127i
\(340\) 0 0
\(341\) −8.00000 + 13.8564i −0.433224 + 0.750366i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −8.00000 + 13.8564i −0.430706 + 0.746004i
\(346\) 0 0
\(347\) 3.23607 + 5.60503i 0.173721 + 0.300894i 0.939718 0.341950i \(-0.111087\pi\)
−0.765997 + 0.642844i \(0.777754\pi\)
\(348\) 0 0
\(349\) 26.6525 1.42667 0.713337 0.700821i \(-0.247183\pi\)
0.713337 + 0.700821i \(0.247183\pi\)
\(350\) 0 0
\(351\) 4.22291 0.225402
\(352\) 0 0
\(353\) 7.94427 + 13.7599i 0.422831 + 0.732365i 0.996215 0.0869220i \(-0.0277031\pi\)
−0.573384 + 0.819287i \(0.694370\pi\)
\(354\) 0 0
\(355\) −8.00000 + 13.8564i −0.424596 + 0.735422i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.47214 + 14.6742i −0.447142 + 0.774473i −0.998199 0.0599949i \(-0.980892\pi\)
0.551056 + 0.834468i \(0.314225\pi\)
\(360\) 0 0
\(361\) 8.73607 + 15.1313i 0.459793 + 0.796385i
\(362\) 0 0
\(363\) −38.1803 −2.00395
\(364\) 0 0
\(365\) −9.52786 −0.498711
\(366\) 0 0
\(367\) −11.4164 19.7738i −0.595932 1.03218i −0.993415 0.114574i \(-0.963450\pi\)
0.397483 0.917610i \(-0.369884\pi\)
\(368\) 0 0
\(369\) −6.23607 + 10.8012i −0.324637 + 0.562287i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.47214 + 2.54981i −0.0762243 + 0.132024i −0.901618 0.432533i \(-0.857620\pi\)
0.825394 + 0.564558i \(0.190953\pi\)
\(374\) 0 0
\(375\) 0.944272 + 1.63553i 0.0487620 + 0.0844582i
\(376\) 0 0
\(377\) −3.41641 −0.175954
\(378\) 0 0
\(379\) 4.58359 0.235443 0.117722 0.993047i \(-0.462441\pi\)
0.117722 + 0.993047i \(0.462441\pi\)
\(380\) 0 0
\(381\) −5.52786 9.57454i −0.283201 0.490519i
\(382\) 0 0
\(383\) −7.70820 + 13.3510i −0.393871 + 0.682204i −0.992956 0.118481i \(-0.962198\pi\)
0.599086 + 0.800685i \(0.295531\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.76393 + 8.25137i −0.242164 + 0.419441i
\(388\) 0 0
\(389\) 2.23607 + 3.87298i 0.113373 + 0.196368i 0.917128 0.398592i \(-0.130501\pi\)
−0.803755 + 0.594960i \(0.797168\pi\)
\(390\) 0 0
\(391\) −17.8885 −0.904663
\(392\) 0 0
\(393\) 14.4721 0.730023
\(394\) 0 0
\(395\) 20.9443 + 36.2765i 1.05382 + 1.82527i
\(396\) 0 0
\(397\) 7.61803 13.1948i 0.382338 0.662229i −0.609058 0.793126i \(-0.708452\pi\)
0.991396 + 0.130897i \(0.0417856\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.7639 20.3757i 0.587463 1.01752i −0.407101 0.913383i \(-0.633460\pi\)
0.994563 0.104132i \(-0.0332065\pi\)
\(402\) 0 0
\(403\) −0.944272 1.63553i −0.0470375 0.0814714i
\(404\) 0 0
\(405\) −7.81966 −0.388562
\(406\) 0 0
\(407\) −28.9443 −1.43471
\(408\) 0 0
\(409\) 10.7082 + 18.5472i 0.529487 + 0.917098i 0.999408 + 0.0343897i \(0.0109487\pi\)
−0.469922 + 0.882708i \(0.655718\pi\)
\(410\) 0 0
\(411\) 9.23607 15.9973i 0.455582 0.789091i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −14.9443 + 25.8842i −0.733585 + 1.27061i
\(416\) 0 0
\(417\) −0.763932 1.32317i −0.0374099 0.0647959i
\(418\) 0 0
\(419\) −22.1803 −1.08358 −0.541790 0.840514i \(-0.682253\pi\)
−0.541790 + 0.840514i \(0.682253\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) −7.70820 13.3510i −0.374786 0.649148i
\(424\) 0 0
\(425\) −12.2361 + 21.1935i −0.593536 + 1.02804i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 3.05573 5.29268i 0.147532 0.255533i
\(430\) 0 0
\(431\) −14.0000 24.2487i −0.674356 1.16802i −0.976657 0.214807i \(-0.931088\pi\)
0.302300 0.953213i \(-0.402245\pi\)
\(432\) 0 0
\(433\) 9.41641 0.452524 0.226262 0.974067i \(-0.427349\pi\)
0.226262 + 0.974067i \(0.427349\pi\)
\(434\) 0 0
\(435\) 17.8885 0.857690
\(436\) 0 0
\(437\) 2.47214 + 4.28187i 0.118258 + 0.204829i
\(438\) 0 0
\(439\) −16.0000 + 27.7128i −0.763638 + 1.32266i 0.177325 + 0.984152i \(0.443256\pi\)
−0.940963 + 0.338508i \(0.890078\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.94427 + 12.0278i −0.329932 + 0.571460i −0.982498 0.186273i \(-0.940359\pi\)
0.652566 + 0.757732i \(0.273693\pi\)
\(444\) 0 0
\(445\) 9.70820 + 16.8151i 0.460213 + 0.797112i
\(446\) 0 0
\(447\) −3.63932 −0.172134
\(448\) 0 0
\(449\) −7.88854 −0.372283 −0.186142 0.982523i \(-0.559598\pi\)
−0.186142 + 0.982523i \(0.559598\pi\)
\(450\) 0 0
\(451\) 27.4164 + 47.4866i 1.29099 + 2.23606i
\(452\) 0 0
\(453\) 5.52786 9.57454i 0.259722 0.449851i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.76393 6.51932i 0.176069 0.304961i −0.764462 0.644669i \(-0.776995\pi\)
0.940531 + 0.339708i \(0.110328\pi\)
\(458\) 0 0
\(459\) −12.3607 21.4093i −0.576947 0.999302i
\(460\) 0 0
\(461\) 21.7082 1.01105 0.505526 0.862811i \(-0.331298\pi\)
0.505526 + 0.862811i \(0.331298\pi\)
\(462\) 0 0
\(463\) 35.7771 1.66270 0.831351 0.555748i \(-0.187568\pi\)
0.831351 + 0.555748i \(0.187568\pi\)
\(464\) 0 0
\(465\) 4.94427 + 8.56373i 0.229285 + 0.397133i
\(466\) 0 0
\(467\) −16.0344 + 27.7725i −0.741985 + 1.28516i 0.209605 + 0.977786i \(0.432782\pi\)
−0.951590 + 0.307370i \(0.900551\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0.472136 0.817763i 0.0217549 0.0376806i
\(472\) 0 0
\(473\) 20.9443 + 36.2765i 0.963019 + 1.66800i
\(474\) 0 0
\(475\) 6.76393 0.310350
\(476\) 0 0
\(477\) 14.7214 0.674045
\(478\) 0 0
\(479\) −4.29180 7.43361i −0.196097 0.339650i 0.751162 0.660117i \(-0.229493\pi\)
−0.947260 + 0.320467i \(0.896160\pi\)
\(480\) 0 0
\(481\) 1.70820 2.95870i 0.0778874 0.134905i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −20.1803 + 34.9534i −0.916342 + 1.58715i
\(486\) 0 0
\(487\) 10.0000 + 17.3205i 0.453143 + 0.784867i 0.998579 0.0532853i \(-0.0169693\pi\)
−0.545436 + 0.838152i \(0.683636\pi\)
\(488\) 0 0
\(489\) −4.22291 −0.190967
\(490\) 0 0
\(491\) −37.8885 −1.70989 −0.854943 0.518722i \(-0.826408\pi\)
−0.854943 + 0.518722i \(0.826408\pi\)
\(492\) 0 0
\(493\) 10.0000 + 17.3205i 0.450377 + 0.780076i
\(494\) 0 0
\(495\) 15.4164 26.7020i 0.692916 1.20017i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 10.9443 18.9560i 0.489933 0.848589i −0.510000 0.860174i \(-0.670355\pi\)
0.999933 + 0.0115857i \(0.00368794\pi\)
\(500\) 0 0
\(501\) −14.4721 25.0665i −0.646567 1.11989i
\(502\) 0 0
\(503\) 4.94427 0.220454 0.110227 0.993906i \(-0.464842\pi\)
0.110227 + 0.993906i \(0.464842\pi\)
\(504\) 0 0
\(505\) −5.52786 −0.245987
\(506\) 0 0
\(507\) −7.67376 13.2913i −0.340804 0.590289i
\(508\) 0 0
\(509\) 20.5623 35.6150i 0.911408 1.57861i 0.0993316 0.995054i \(-0.468330\pi\)
0.812077 0.583551i \(-0.198337\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.41641 + 5.91739i −0.150838 + 0.261259i
\(514\) 0 0
\(515\) −8.94427 15.4919i −0.394132 0.682656i
\(516\) 0 0
\(517\) −67.7771 −2.98083
\(518\) 0 0
\(519\) −7.05573 −0.309712
\(520\) 0 0
\(521\) 3.29180 + 5.70156i 0.144216 + 0.249790i 0.929080 0.369878i \(-0.120601\pi\)
−0.784864 + 0.619668i \(0.787267\pi\)
\(522\) 0 0
\(523\) −2.14590 + 3.71680i −0.0938336 + 0.162525i −0.909121 0.416532i \(-0.863245\pi\)
0.815288 + 0.579056i \(0.196579\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.52786 + 9.57454i −0.240798 + 0.417074i
\(528\) 0 0
\(529\) 3.50000 + 6.06218i 0.152174 + 0.263573i
\(530\) 0 0
\(531\) −13.5967 −0.590049
\(532\) 0 0
\(533\) −6.47214 −0.280339
\(534\) 0 0
\(535\) 14.4721 + 25.0665i 0.625685 + 1.08372i
\(536\) 0 0
\(537\) 4.36068 7.55292i 0.188177 0.325933i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.52786 4.37839i 0.108681 0.188242i −0.806555 0.591159i \(-0.798671\pi\)
0.915236 + 0.402917i \(0.132004\pi\)
\(542\) 0 0
\(543\) −7.52786 13.0386i −0.323052 0.559542i
\(544\) 0 0
\(545\) 27.4164 1.17439
\(546\) 0 0
\(547\) 4.58359 0.195980 0.0979901 0.995187i \(-0.468759\pi\)
0.0979901 + 0.995187i \(0.468759\pi\)
\(548\) 0 0
\(549\) 8.27051 + 14.3249i 0.352977 + 0.611374i
\(550\) 0 0
\(551\) 2.76393 4.78727i 0.117747 0.203945i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −8.94427 + 15.4919i −0.379663 + 0.657596i
\(556\) 0 0
\(557\) −4.52786 7.84249i −0.191852 0.332297i 0.754012 0.656860i \(-0.228116\pi\)
−0.945864 + 0.324563i \(0.894783\pi\)
\(558\) 0 0
\(559\) −4.94427 −0.209120
\(560\) 0 0
\(561\) −35.7771 −1.51051
\(562\) 0 0
\(563\) −8.90983 15.4323i −0.375505 0.650393i 0.614898 0.788607i \(-0.289197\pi\)
−0.990402 + 0.138214i \(0.955864\pi\)
\(564\) 0 0
\(565\) 20.1803 34.9534i 0.848993 1.47050i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.1803 + 22.8290i −0.552549 + 0.957042i 0.445541 + 0.895261i \(0.353011\pi\)
−0.998090 + 0.0617808i \(0.980322\pi\)
\(570\) 0 0
\(571\) −7.23607 12.5332i −0.302820 0.524500i 0.673954 0.738774i \(-0.264595\pi\)
−0.976774 + 0.214274i \(0.931261\pi\)
\(572\) 0 0
\(573\) 6.11146 0.255310
\(574\) 0 0
\(575\) −21.8885 −0.912815
\(576\) 0 0
\(577\) 3.00000 + 5.19615i 0.124892 + 0.216319i 0.921691 0.387926i \(-0.126808\pi\)
−0.796799 + 0.604245i \(0.793475\pi\)
\(578\) 0 0
\(579\) 0.291796 0.505406i 0.0121266 0.0210039i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 32.3607 56.0503i 1.34024 2.32137i
\(584\) 0 0
\(585\) 1.81966 + 3.15174i 0.0752337 + 0.130309i
\(586\) 0 0
\(587\) 3.70820 0.153054 0.0765270 0.997068i \(-0.475617\pi\)
0.0765270 + 0.997068i \(0.475617\pi\)
\(588\) 0 0
\(589\) 3.05573 0.125909
\(590\) 0 0
\(591\) 6.76393 + 11.7155i 0.278231 + 0.481910i
\(592\) 0 0
\(593\) −16.4164 + 28.4341i −0.674141 + 1.16765i 0.302578 + 0.953125i \(0.402153\pi\)
−0.976719 + 0.214522i \(0.931181\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.52786 + 16.5027i −0.389950 + 0.675412i
\(598\) 0 0
\(599\) 8.94427 + 15.4919i 0.365453 + 0.632983i 0.988849 0.148923i \(-0.0475807\pi\)
−0.623396 + 0.781907i \(0.714247\pi\)
\(600\) 0 0
\(601\) 29.7771 1.21463 0.607316 0.794460i \(-0.292246\pi\)
0.607316 + 0.794460i \(0.292246\pi\)
\(602\) 0 0
\(603\) 5.88854 0.239800
\(604\) 0 0
\(605\) −49.9787 86.5657i −2.03192 3.51940i
\(606\) 0 0
\(607\) 4.94427 8.56373i 0.200682 0.347591i −0.748066 0.663624i \(-0.769018\pi\)
0.948748 + 0.316033i \(0.102351\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.00000 6.92820i 0.161823 0.280285i
\(612\) 0 0
\(613\) −14.7082 25.4754i −0.594059 1.02894i −0.993679 0.112259i \(-0.964191\pi\)
0.399620 0.916681i \(-0.369142\pi\)
\(614\) 0 0
\(615\) 33.8885 1.36652
\(616\) 0 0
\(617\) 34.3607 1.38331 0.691654 0.722229i \(-0.256882\pi\)
0.691654 + 0.722229i \(0.256882\pi\)
\(618\) 0 0
\(619\) −24.0344 41.6289i −0.966026 1.67321i −0.706833 0.707380i \(-0.749877\pi\)
−0.259192 0.965826i \(-0.583456\pi\)
\(620\) 0 0
\(621\) 11.0557 19.1491i 0.443651 0.768426i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 11.2082 19.4132i 0.448328 0.776527i
\(626\) 0 0
\(627\) 4.94427 + 8.56373i 0.197455 + 0.342002i
\(628\) 0 0
\(629\) −20.0000 −0.797452
\(630\) 0 0
\(631\) 44.9443 1.78920 0.894602 0.446865i \(-0.147459\pi\)
0.894602 + 0.446865i \(0.147459\pi\)
\(632\) 0 0
\(633\) 7.41641 + 12.8456i 0.294776 + 0.510567i
\(634\) 0 0
\(635\) 14.4721 25.0665i 0.574309 0.994733i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3.63932 6.30349i 0.143969 0.249362i
\(640\) 0 0
\(641\) −7.29180 12.6298i −0.288009 0.498846i 0.685326 0.728237i \(-0.259660\pi\)
−0.973334 + 0.229391i \(0.926327\pi\)
\(642\) 0 0
\(643\) 43.7082 1.72368 0.861842 0.507177i \(-0.169311\pi\)
0.861842 + 0.507177i \(0.169311\pi\)
\(644\) 0 0
\(645\) 25.8885 1.01936
\(646\) 0 0
\(647\) −10.1803 17.6329i −0.400230 0.693219i 0.593523 0.804817i \(-0.297737\pi\)
−0.993753 + 0.111598i \(0.964403\pi\)
\(648\) 0 0
\(649\) −29.8885 + 51.7685i −1.17323 + 2.03209i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.70820 8.15485i 0.184246 0.319124i −0.759076 0.651002i \(-0.774349\pi\)
0.943322 + 0.331878i \(0.107682\pi\)
\(654\) 0 0
\(655\) 18.9443 + 32.8124i 0.740214 + 1.28209i
\(656\) 0 0
\(657\) 4.33437 0.169100
\(658\) 0 0
\(659\) −37.3050 −1.45319 −0.726597 0.687064i \(-0.758899\pi\)
−0.726597 + 0.687064i \(0.758899\pi\)
\(660\) 0 0
\(661\) 11.3262 + 19.6176i 0.440540 + 0.763037i 0.997730 0.0673481i \(-0.0214538\pi\)
−0.557190 + 0.830385i \(0.688120\pi\)
\(662\) 0 0
\(663\) 2.11146 3.65715i 0.0820022 0.142032i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8.94427 + 15.4919i −0.346324 + 0.599850i
\(668\) 0 0
\(669\) −8.00000 13.8564i −0.309298 0.535720i
\(670\) 0 0
\(671\) 72.7214 2.80738
\(672\) 0 0
\(673\) −2.94427 −0.113493 −0.0567467 0.998389i \(-0.518073\pi\)
−0.0567467 + 0.998389i \(0.518073\pi\)
\(674\) 0 0
\(675\) −15.1246 26.1966i −0.582147 1.00831i
\(676\) 0 0
\(677\) −9.90983 + 17.1643i −0.380866 + 0.659679i −0.991186 0.132476i \(-0.957707\pi\)
0.610321 + 0.792155i \(0.291041\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.6525 18.4506i 0.408204 0.707030i
\(682\) 0 0
\(683\) 11.8885 + 20.5916i 0.454902 + 0.787914i 0.998683 0.0513135i \(-0.0163408\pi\)
−0.543780 + 0.839228i \(0.683007\pi\)
\(684\) 0 0
\(685\) 48.3607 1.84777
\(686\) 0 0
\(687\) −29.1672 −1.11280
\(688\) 0 0
\(689\) 3.81966 + 6.61585i 0.145517 + 0.252044i
\(690\) 0 0
\(691\) −7.09017 + 12.2805i −0.269723 + 0.467174i −0.968790 0.247882i \(-0.920265\pi\)
0.699067 + 0.715056i \(0.253599\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.00000 3.46410i 0.0758643 0.131401i
\(696\) 0 0
\(697\) 18.9443 + 32.8124i 0.717565 + 1.24286i
\(698\) 0 0
\(699\) 19.6393 0.742827
\(700\) 0 0
\(701\) −41.4164 −1.56428 −0.782138 0.623105i \(-0.785871\pi\)
−0.782138 + 0.623105i \(0.785871\pi\)
\(702\) 0 0
\(703\) 2.76393 + 4.78727i 0.104244 + 0.180555i
\(704\) 0 0
\(705\) −20.9443 + 36.2765i −0.788807 + 1.36625i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.819660 + 1.41969i −0.0307830 + 0.0533177i −0.881006 0.473104i \(-0.843133\pi\)
0.850223 + 0.526422i \(0.176467\pi\)
\(710\) 0 0
\(711\) −9.52786 16.5027i −0.357323 0.618901i
\(712\) 0 0
\(713\) −9.88854 −0.370329
\(714\) 0 0
\(715\) 16.0000 0.598366
\(716\) 0 0
\(717\) 8.58359 + 14.8672i 0.320560 + 0.555226i
\(718\) 0 0
\(719\) 25.5967 44.3349i 0.954598 1.65341i 0.219311 0.975655i \(-0.429619\pi\)
0.735286 0.677757i \(-0.237048\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 7.70820 13.3510i 0.286671 0.496529i
\(724\) 0 0
\(725\) 12.2361 + 21.1935i 0.454436 + 0.787107i
\(726\) 0 0
\(727\) −12.3607 −0.458432 −0.229216 0.973376i \(-0.573616\pi\)
−0.229216 + 0.973376i \(0.573616\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) 14.4721 + 25.0665i 0.535271 + 0.927117i
\(732\) 0 0
\(733\) 2.38197 4.12569i 0.0879799 0.152386i −0.818677 0.574254i \(-0.805292\pi\)
0.906657 + 0.421868i \(0.138626\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.9443 22.4201i 0.476808 0.825856i
\(738\) 0 0
\(739\) −1.70820 2.95870i −0.0628373 0.108837i 0.832895 0.553431i \(-0.186682\pi\)
−0.895733 + 0.444593i \(0.853348\pi\)
\(740\) 0 0
\(741\) −1.16718 −0.0428776
\(742\) 0 0
\(743\) −24.9443 −0.915117 −0.457558 0.889180i \(-0.651276\pi\)
−0.457558 + 0.889180i \(0.651276\pi\)
\(744\) 0 0
\(745\) −4.76393 8.25137i −0.174537 0.302307i
\(746\) 0 0
\(747\) 6.79837 11.7751i 0.248739 0.430829i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.0000 31.1769i 0.656829 1.13766i −0.324603 0.945851i \(-0.605231\pi\)
0.981432 0.191811i \(-0.0614361\pi\)
\(752\) 0 0
\(753\) −2.65248 4.59422i −0.0966616 0.167423i
\(754\) 0 0
\(755\) 28.9443 1.05339
\(756\) 0 0
\(757\) 39.3050 1.42856 0.714281 0.699859i \(-0.246754\pi\)
0.714281 + 0.699859i \(0.246754\pi\)
\(758\) 0 0
\(759\) −16.0000 27.7128i −0.580763 1.00591i
\(760\) 0 0
\(761\) 1.76393 3.05522i 0.0639425 0.110752i −0.832282 0.554353i \(-0.812966\pi\)
0.896224 + 0.443601i \(0.146299\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 10.6525 18.4506i 0.385141 0.667084i
\(766\) 0 0
\(767\) −3.52786 6.11044i −0.127384 0.220635i
\(768\) 0 0
\(769\) −18.3607 −0.662103 −0.331052 0.943613i \(-0.607403\pi\)
−0.331052 + 0.943613i \(0.607403\pi\)
\(770\) 0 0
\(771\) 17.3050 0.623223
\(772\) 0 0
\(773\) −20.0902 34.7972i −0.722593 1.25157i −0.959957 0.280147i \(-0.909617\pi\)
0.237364 0.971421i \(-0.423717\pi\)
\(774\) 0 0
\(775\) −6.76393 + 11.7155i −0.242968 + 0.420832i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.23607 9.06914i 0.187602 0.324936i
\(780\) 0 0
\(781\) −16.0000 27.7128i −0.572525 0.991642i
\(782\) 0 0
\(783\) −24.7214 −0.883469
\(784\) 0 0
\(785\) 2.47214 0.0882343
\(786\) 0 0
\(787\) −6.14590 10.6450i −0.219078 0.379454i 0.735449 0.677580i \(-0.236971\pi\)
−0.954526 + 0.298127i \(0.903638\pi\)
\(788\) 0 0
\(789\) −6.83282 + 11.8348i −0.243255 + 0.421329i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −4.29180 + 7.43361i −0.152406 + 0.263975i
\(794\) 0 0
\(795\) −20.0000 34.6410i −0.709327 1.22859i
\(796\) 0 0
\(797\) −52.1803 −1.84832 −0.924161 0.382003i \(-0.875235\pi\)
−0.924161 + 0.382003i \(0.875235\pi\)
\(798\) 0 0
\(799\) −46.8328 −1.65683
\(800\) 0 0
\(801\) −4.41641 7.64944i −0.156046 0.270280i
\(802\) 0 0
\(803\) 9.52786 16.5027i 0.336231 0.582369i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −2.58359 + 4.47491i −0.0909468 + 0.157524i
\(808\) 0 0
\(809\) 8.70820 + 15.0831i 0.306164 + 0.530292i 0.977520 0.210843i \(-0.0676210\pi\)
−0.671356 + 0.741135i \(0.734288\pi\)
\(810\) 0 0
\(811\) 53.0132 1.86154 0.930772 0.365601i \(-0.119136\pi\)
0.930772 + 0.365601i \(0.119136\pi\)
\(812\) 0 0
\(813\) −29.6656 −1.04042
\(814\) 0 0
\(815\) −5.52786 9.57454i −0.193633 0.335382i
\(816\) 0 0
\(817\) 4.00000 6.92820i 0.139942 0.242387i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.05573 + 3.56063i −0.0717454 + 0.124267i −0.899666 0.436578i \(-0.856190\pi\)
0.827921 + 0.560845i \(0.189524\pi\)
\(822\) 0 0
\(823\) −9.88854 17.1275i −0.344693 0.597026i 0.640605 0.767871i \(-0.278684\pi\)
−0.985298 + 0.170845i \(0.945350\pi\)
\(824\) 0 0
\(825\) −43.7771 −1.52412
\(826\) 0 0
\(827\) −15.0557 −0.523539 −0.261769 0.965130i \(-0.584306\pi\)
−0.261769 + 0.965130i \(0.584306\pi\)
\(828\) 0 0
\(829\) −5.61803 9.73072i −0.195122 0.337962i 0.751818 0.659370i \(-0.229177\pi\)
−0.946941 + 0.321409i \(0.895844\pi\)
\(830\) 0 0
\(831\) 4.87539 8.44442i 0.169125 0.292934i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 37.8885 65.6249i 1.31119 2.27104i
\(836\) 0 0
\(837\) −6.83282 11.8348i −0.236177 0.409070i
\(838\) 0 0
\(839\) 21.5279 0.743224 0.371612 0.928388i \(-0.378805\pi\)
0.371612 + 0.928388i \(0.378805\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 16.0689 + 27.8321i 0.553442 + 0.958589i
\(844\) 0 0
\(845\) 20.0902 34.7972i 0.691123 1.19706i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 3.81966 6.61585i 0.131090 0.227055i
\(850\) 0 0
\(851\) −8.94427 15.4919i −0.306606 0.531057i
\(852\) 0 0
\(853\) 13.7082 0.469360 0.234680 0.972073i \(-0.424596\pi\)
0.234680 + 0.972073i \(0.424596\pi\)
\(854\) 0 0
\(855\) −5.88854 −0.201384
\(856\) 0 0
\(857\) 13.7639 + 23.8398i 0.470167 + 0.814353i 0.999418 0.0341122i \(-0.0108604\pi\)
−0.529251 + 0.848465i \(0.677527\pi\)
\(858\) 0 0
\(859\) −16.9098 + 29.2887i −0.576956 + 0.999317i 0.418870 + 0.908046i \(0.362426\pi\)
−0.995826 + 0.0912709i \(0.970907\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −10.4721 + 18.1383i −0.356476 + 0.617434i −0.987369 0.158435i \(-0.949355\pi\)
0.630894 + 0.775869i \(0.282688\pi\)
\(864\) 0 0
\(865\) −9.23607 15.9973i −0.314036 0.543926i
\(866\) 0 0
\(867\) −3.70820 −0.125937
\(868\) 0 0
\(869\) −83.7771 −2.84194
\(870\) 0 0
\(871\) 1.52786 + 2.64634i 0.0517697 + 0.0896677i
\(872\) 0 0
\(873\) 9.18034 15.9008i 0.310707 0.538161i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6.70820 + 11.6190i −0.226520 + 0.392344i −0.956774 0.290831i \(-0.906068\pi\)
0.730254 + 0.683175i \(0.239402\pi\)
\(878\) 0 0
\(879\) −7.88854 13.6634i −0.266074 0.460854i
\(880\) 0 0
\(881\) 24.8328 0.836639 0.418319 0.908300i \(-0.362619\pi\)
0.418319 + 0.908300i \(0.362619\pi\)
\(882\) 0 0
\(883\) −23.0557 −0.775887 −0.387944 0.921683i \(-0.626814\pi\)
−0.387944 + 0.921683i \(0.626814\pi\)
\(884\) 0 0
\(885\) 18.4721 + 31.9947i 0.620934 + 1.07549i
\(886\) 0 0
\(887\) −16.6525 + 28.8429i −0.559135 + 0.968451i 0.438433 + 0.898764i \(0.355533\pi\)
−0.997569 + 0.0696873i \(0.977800\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 7.81966 13.5440i 0.261968 0.453743i
\(892\) 0 0
\(893\) 6.47214 + 11.2101i 0.216582 + 0.375131i
\(894\) 0 0
\(895\) 22.8328 0.763217
\(896\) 0 0
\(897\) 3.77709 0.126113
\(898\) 0 0
\(899\) 5.52786 + 9.57454i 0.184365 + 0.319329i
\(900\) 0 0
\(901\) 22.3607 38.7298i 0.744942 1.29028i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.7082 34.1356i 0.655123 1.13471i
\(906\) 0 0
\(907\) 0.472136 + 0.817763i 0.0156770 + 0.0271534i 0.873757 0.486362i \(-0.161676\pi\)
−0.858080 + 0.513515i \(0.828343\pi\)
\(908\) 0 0
\(909\) 2.51471 0.0834076
\(910\) 0 0
\(911\) 34.8328 1.15406 0.577031 0.816722i \(-0.304211\pi\)
0.577031 + 0.816722i \(0.304211\pi\)
\(912\) 0 0
\(913\) −29.8885 51.7685i −0.989166 1.71329i
\(914\) 0 0
\(915\) 22.4721 38.9229i 0.742906 1.28675i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −17.8885 + 30.9839i −0.590089 + 1.02206i 0.404131 + 0.914701i \(0.367574\pi\)
−0.994220 + 0.107362i \(0.965759\pi\)
\(920\) 0 0
\(921\) −1.12461 1.94788i −0.0370572 0.0641850i
\(922\) 0 0
\(923\) 3.77709 0.124324
\(924\) 0 0
\(925\) −24.4721 −0.804639
\(926\) 0 0
\(927\) 4.06888 + 7.04751i 0.133640 + 0.231471i
\(928\) 0 0
\(929\) 23.6525 40.9673i 0.776013 1.34409i −0.158211 0.987405i \(-0.550573\pi\)
0.934224 0.356688i \(-0.116094\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4.94427 + 8.56373i −0.161868 + 0.280364i
\(934\) 0 0
\(935\) −46.8328 81.1168i −1.53160 2.65280i
\(936\) 0 0
\(937\) −9.05573 −0.295838 −0.147919 0.988999i \(-0.547257\pi\)
−0.147919 + 0.988999i \(0.547257\pi\)
\(938\) 0 0
\(939\) 10.4721 0.341745
\(940\) 0 0
\(941\) 17.7984 + 30.8277i 0.580210 + 1.00495i 0.995454 + 0.0952436i \(0.0303630\pi\)
−0.415244 + 0.909710i \(0.636304\pi\)
\(942\) 0 0
\(943\) −16.9443 + 29.3483i −0.551781 + 0.955713i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.29180 + 3.96951i −0.0744734 + 0.128992i −0.900857 0.434116i \(-0.857061\pi\)
0.826384 + 0.563107i \(0.190394\pi\)
\(948\) 0 0
\(949\) 1.12461 + 1.94788i 0.0365064 + 0.0632310i
\(950\) 0 0
\(951\) −11.1935 −0.362974
\(952\) 0 0
\(953\) 51.8885 1.68083 0.840417 0.541940i \(-0.182310\pi\)
0.840417 + 0.541940i \(0.182310\pi\)
\(954\) 0 0
\(955\) 8.00000 + 13.8564i 0.258874 + 0.448383i
\(956\) 0 0
\(957\) −17.8885 + 30.9839i −0.578254 + 1.00157i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 12.4443 21.5541i 0.401428 0.695294i
\(962\) 0 0
\(963\) −6.58359 11.4031i −0.212153 0.367460i
\(964\) 0 0
\(965\) 1.52786 0.0491837
\(966\) 0 0
\(967\) −29.8885 −0.961151 −0.480575 0.876953i \(-0.659572\pi\)
−0.480575 + 0.876953i \(0.659572\pi\)
\(968\) 0 0
\(969\) 3.41641 + 5.91739i 0.109751 + 0.190094i
\(970\) 0 0
\(971\) 11.3820 19.7141i 0.365265 0.632657i −0.623554 0.781780i \(-0.714312\pi\)
0.988819 + 0.149123i \(0.0476452\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 2.58359 4.47491i 0.0827412 0.143312i
\(976\) 0 0
\(977\) 6.41641 + 11.1135i 0.205279 + 0.355554i 0.950222 0.311575i \(-0.100856\pi\)
−0.744943 + 0.667129i \(0.767523\pi\)
\(978\) 0 0
\(979\) −38.8328 −1.24110
\(980\) 0 0
\(981\) −12.4721 −0.398205
\(982\) 0 0
\(983\) −2.76393 4.78727i −0.0881557 0.152690i 0.818576 0.574398i \(-0.194764\pi\)
−0.906732 + 0.421708i \(0.861431\pi\)
\(984\) 0 0
\(985\) −17.7082 + 30.6715i −0.564230 + 0.977276i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −12.9443 + 22.4201i −0.411604 + 0.712919i
\(990\) 0 0
\(991\) −12.0000 20.7846i −0.381193 0.660245i 0.610040 0.792370i \(-0.291153\pi\)
−0.991233 + 0.132125i \(0.957820\pi\)
\(992\) 0 0
\(993\) 27.7771 0.881479
\(994\) 0 0
\(995\) −49.8885 −1.58157
\(996\) 0 0
\(997\) −9.03444 15.6481i −0.286124 0.495581i 0.686757 0.726887i \(-0.259033\pi\)
−0.972881 + 0.231306i \(0.925700\pi\)
\(998\) 0 0
\(999\) 12.3607 21.4093i 0.391075 0.677361i
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 1568.2.i.m.961.2 4
4.3 odd 2 1568.2.i.v.961.1 4
7.2 even 3 224.2.a.d.1.1 yes 2
7.3 odd 6 1568.2.i.w.1537.1 4
7.4 even 3 inner 1568.2.i.m.1537.2 4
7.5 odd 6 1568.2.a.k.1.2 2
7.6 odd 2 1568.2.i.w.961.1 4
21.2 odd 6 2016.2.a.o.1.1 2
28.3 even 6 1568.2.i.n.1537.2 4
28.11 odd 6 1568.2.i.v.1537.1 4
28.19 even 6 1568.2.a.v.1.1 2
28.23 odd 6 224.2.a.c.1.2 2
28.27 even 2 1568.2.i.n.961.2 4
35.9 even 6 5600.2.a.z.1.2 2
56.5 odd 6 3136.2.a.by.1.1 2
56.19 even 6 3136.2.a.bf.1.2 2
56.37 even 6 448.2.a.i.1.2 2
56.51 odd 6 448.2.a.j.1.1 2
84.23 even 6 2016.2.a.r.1.1 2
112.37 even 12 1792.2.b.m.897.3 4
112.51 odd 12 1792.2.b.k.897.3 4
112.93 even 12 1792.2.b.m.897.2 4
112.107 odd 12 1792.2.b.k.897.2 4
140.79 odd 6 5600.2.a.bk.1.1 2
168.107 even 6 4032.2.a.bw.1.2 2
168.149 odd 6 4032.2.a.bv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.2 2 28.23 odd 6
224.2.a.d.1.1 yes 2 7.2 even 3
448.2.a.i.1.2 2 56.37 even 6
448.2.a.j.1.1 2 56.51 odd 6
1568.2.a.k.1.2 2 7.5 odd 6
1568.2.a.v.1.1 2 28.19 even 6
1568.2.i.m.961.2 4 1.1 even 1 trivial
1568.2.i.m.1537.2 4 7.4 even 3 inner
1568.2.i.n.961.2 4 28.27 even 2
1568.2.i.n.1537.2 4 28.3 even 6
1568.2.i.v.961.1 4 4.3 odd 2
1568.2.i.v.1537.1 4 28.11 odd 6
1568.2.i.w.961.1 4 7.6 odd 2
1568.2.i.w.1537.1 4 7.3 odd 6
1792.2.b.k.897.2 4 112.107 odd 12
1792.2.b.k.897.3 4 112.51 odd 12
1792.2.b.m.897.2 4 112.93 even 12
1792.2.b.m.897.3 4 112.37 even 12
2016.2.a.o.1.1 2 21.2 odd 6
2016.2.a.r.1.1 2 84.23 even 6
3136.2.a.bf.1.2 2 56.19 even 6
3136.2.a.by.1.1 2 56.5 odd 6
4032.2.a.bv.1.2 2 168.149 odd 6
4032.2.a.bw.1.2 2 168.107 even 6
5600.2.a.z.1.2 2 35.9 even 6
5600.2.a.bk.1.1 2 140.79 odd 6