Properties

Label 2100.2.ce.b.493.2
Level $2100$
Weight $2$
Character 2100.493
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(157,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.157");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.ce (of order \(12\), degree \(4\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 493.2
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2100.493
Dual form 2100.2.ce.b.1657.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.258819 + 0.965926i) q^{3} +(-1.15539 + 2.38014i) q^{7} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\) \(q+(0.258819 + 0.965926i) q^{3} +(-1.15539 + 2.38014i) q^{7} +(-0.866025 + 0.500000i) q^{9} +(-1.36603 + 2.36603i) q^{11} +(-0.707107 + 0.707107i) q^{13} +(6.50266 - 1.74238i) q^{17} +(4.23205 + 7.33013i) q^{19} +(-2.59808 - 0.500000i) q^{21} +(1.93185 - 7.20977i) q^{23} +(-0.707107 - 0.707107i) q^{27} +6.92820i q^{29} +(-1.73205 - 1.00000i) q^{31} +(-2.63896 - 0.707107i) q^{33} +(-10.8147 - 2.89778i) q^{37} +(-0.866025 - 0.500000i) q^{39} -4.19615i q^{41} +(-0.378937 - 0.378937i) q^{43} +(-2.63896 + 9.84873i) q^{47} +(-4.33013 - 5.50000i) q^{49} +(3.36603 + 5.83013i) q^{51} +(-8.43451 + 2.26002i) q^{53} +(-5.98502 + 5.98502i) q^{57} +(-4.63397 + 8.02628i) q^{59} +(-8.42820 + 4.86603i) q^{61} +(-0.189469 - 2.63896i) q^{63} +(1.81173 + 6.76148i) q^{67} +7.46410 q^{69} -8.00000 q^{71} +(-3.98382 - 14.8678i) q^{73} +(-4.05317 - 5.98502i) q^{77} +(1.96410 - 1.13397i) q^{79} +(0.500000 - 0.866025i) q^{81} +(-3.34607 + 3.34607i) q^{83} +(-6.69213 + 1.79315i) q^{87} +(3.63397 + 6.29423i) q^{89} +(-0.866025 - 2.50000i) q^{91} +(0.517638 - 1.93185i) q^{93} +(2.50026 + 2.50026i) q^{97} -2.73205i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{11} + 20 q^{19} + 20 q^{51} - 44 q^{59} - 12 q^{61} + 32 q^{69} - 64 q^{71} - 12 q^{79} + 4 q^{81} + 36 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 0.258819 + 0.965926i 0.149429 + 0.557678i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.15539 + 2.38014i −0.436698 + 0.899608i
\(8\) 0 0
\(9\) −0.866025 + 0.500000i −0.288675 + 0.166667i
\(10\) 0 0
\(11\) −1.36603 + 2.36603i −0.411872 + 0.713384i −0.995094 0.0989291i \(-0.968458\pi\)
0.583222 + 0.812313i \(0.301792\pi\)
\(12\) 0 0
\(13\) −0.707107 + 0.707107i −0.196116 + 0.196116i −0.798333 0.602217i \(-0.794284\pi\)
0.602217 + 0.798333i \(0.294284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.50266 1.74238i 1.57713 0.422590i 0.639092 0.769130i \(-0.279310\pi\)
0.938035 + 0.346540i \(0.112644\pi\)
\(18\) 0 0
\(19\) 4.23205 + 7.33013i 0.970899 + 1.68165i 0.692853 + 0.721079i \(0.256353\pi\)
0.278046 + 0.960568i \(0.410313\pi\)
\(20\) 0 0
\(21\) −2.59808 0.500000i −0.566947 0.109109i
\(22\) 0 0
\(23\) 1.93185 7.20977i 0.402819 1.50334i −0.405224 0.914217i \(-0.632807\pi\)
0.808043 0.589123i \(-0.200527\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −0.707107 0.707107i −0.136083 0.136083i
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) −1.73205 1.00000i −0.311086 0.179605i 0.336327 0.941745i \(-0.390815\pi\)
−0.647412 + 0.762140i \(0.724149\pi\)
\(32\) 0 0
\(33\) −2.63896 0.707107i −0.459384 0.123091i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.8147 2.89778i −1.77792 0.476392i −0.787717 0.616038i \(-0.788737\pi\)
−0.990201 + 0.139646i \(0.955403\pi\)
\(38\) 0 0
\(39\) −0.866025 0.500000i −0.138675 0.0800641i
\(40\) 0 0
\(41\) 4.19615i 0.655329i −0.944794 0.327664i \(-0.893738\pi\)
0.944794 0.327664i \(-0.106262\pi\)
\(42\) 0 0
\(43\) −0.378937 0.378937i −0.0577874 0.0577874i 0.677623 0.735410i \(-0.263010\pi\)
−0.735410 + 0.677623i \(0.763010\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.63896 + 9.84873i −0.384932 + 1.43658i 0.453343 + 0.891336i \(0.350231\pi\)
−0.838275 + 0.545248i \(0.816435\pi\)
\(48\) 0 0
\(49\) −4.33013 5.50000i −0.618590 0.785714i
\(50\) 0 0
\(51\) 3.36603 + 5.83013i 0.471338 + 0.816381i
\(52\) 0 0
\(53\) −8.43451 + 2.26002i −1.15857 + 0.310438i −0.786394 0.617725i \(-0.788055\pi\)
−0.372175 + 0.928163i \(0.621388\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −5.98502 + 5.98502i −0.792736 + 0.792736i
\(58\) 0 0
\(59\) −4.63397 + 8.02628i −0.603292 + 1.04493i 0.389027 + 0.921226i \(0.372811\pi\)
−0.992319 + 0.123706i \(0.960522\pi\)
\(60\) 0 0
\(61\) −8.42820 + 4.86603i −1.07912 + 0.623031i −0.930659 0.365887i \(-0.880766\pi\)
−0.148462 + 0.988918i \(0.547432\pi\)
\(62\) 0 0
\(63\) −0.189469 2.63896i −0.0238708 0.332478i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.81173 + 6.76148i 0.221338 + 0.826046i 0.983838 + 0.179058i \(0.0573050\pi\)
−0.762500 + 0.646988i \(0.776028\pi\)
\(68\) 0 0
\(69\) 7.46410 0.898572
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −3.98382 14.8678i −0.466271 1.74015i −0.652641 0.757668i \(-0.726339\pi\)
0.186370 0.982480i \(-0.440328\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.05317 5.98502i −0.461902 0.682057i
\(78\) 0 0
\(79\) 1.96410 1.13397i 0.220979 0.127582i −0.385425 0.922739i \(-0.625945\pi\)
0.606403 + 0.795157i \(0.292612\pi\)
\(80\) 0 0
\(81\) 0.500000 0.866025i 0.0555556 0.0962250i
\(82\) 0 0
\(83\) −3.34607 + 3.34607i −0.367278 + 0.367278i −0.866484 0.499205i \(-0.833625\pi\)
0.499205 + 0.866484i \(0.333625\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.69213 + 1.79315i −0.717472 + 0.192246i
\(88\) 0 0
\(89\) 3.63397 + 6.29423i 0.385201 + 0.667187i 0.991797 0.127823i \(-0.0407989\pi\)
−0.606596 + 0.795010i \(0.707466\pi\)
\(90\) 0 0
\(91\) −0.866025 2.50000i −0.0907841 0.262071i
\(92\) 0 0
\(93\) 0.517638 1.93185i 0.0536766 0.200324i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.50026 + 2.50026i 0.253863 + 0.253863i 0.822552 0.568690i \(-0.192549\pi\)
−0.568690 + 0.822552i \(0.692549\pi\)
\(98\) 0 0
\(99\) 2.73205i 0.274581i
\(100\) 0 0
\(101\) 5.83013 + 3.36603i 0.580119 + 0.334932i 0.761181 0.648540i \(-0.224620\pi\)
−0.181061 + 0.983472i \(0.557953\pi\)
\(102\) 0 0
\(103\) −4.50146 1.20616i −0.443542 0.118847i 0.0301341 0.999546i \(-0.490407\pi\)
−0.473676 + 0.880699i \(0.657073\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.2301 + 3.81294i 1.37567 + 0.368610i 0.869547 0.493850i \(-0.164411\pi\)
0.506125 + 0.862460i \(0.331077\pi\)
\(108\) 0 0
\(109\) 12.0622 + 6.96410i 1.15535 + 0.667040i 0.950185 0.311688i \(-0.100894\pi\)
0.205163 + 0.978728i \(0.434228\pi\)
\(110\) 0 0
\(111\) 11.1962i 1.06269i
\(112\) 0 0
\(113\) −6.03579 6.03579i −0.567800 0.567800i 0.363712 0.931511i \(-0.381509\pi\)
−0.931511 + 0.363712i \(0.881509\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.258819 0.965926i 0.0239278 0.0892999i
\(118\) 0 0
\(119\) −3.36603 + 17.4904i −0.308563 + 1.60334i
\(120\) 0 0
\(121\) 1.76795 + 3.06218i 0.160723 + 0.278380i
\(122\) 0 0
\(123\) 4.05317 1.08604i 0.365462 0.0979253i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −0.0507680 + 0.0507680i −0.00450493 + 0.00450493i −0.709356 0.704851i \(-0.751014\pi\)
0.704851 + 0.709356i \(0.251014\pi\)
\(128\) 0 0
\(129\) 0.267949 0.464102i 0.0235916 0.0408619i
\(130\) 0 0
\(131\) 14.6603 8.46410i 1.28087 0.739512i 0.303864 0.952715i \(-0.401723\pi\)
0.977008 + 0.213203i \(0.0683897\pi\)
\(132\) 0 0
\(133\) −22.3364 + 1.60368i −1.93681 + 0.139057i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1.55291 5.79555i −0.132674 0.495148i 0.867322 0.497747i \(-0.165839\pi\)
−0.999997 + 0.00259945i \(0.999173\pi\)
\(138\) 0 0
\(139\) −16.8564 −1.42974 −0.714871 0.699256i \(-0.753515\pi\)
−0.714871 + 0.699256i \(0.753515\pi\)
\(140\) 0 0
\(141\) −10.1962 −0.858671
\(142\) 0 0
\(143\) −0.707107 2.63896i −0.0591312 0.220681i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.19187 5.60609i 0.345740 0.462382i
\(148\) 0 0
\(149\) −7.90192 + 4.56218i −0.647351 + 0.373748i −0.787441 0.616391i \(-0.788594\pi\)
0.140090 + 0.990139i \(0.455261\pi\)
\(150\) 0 0
\(151\) 5.33013 9.23205i 0.433760 0.751294i −0.563434 0.826161i \(-0.690520\pi\)
0.997193 + 0.0748675i \(0.0238534\pi\)
\(152\) 0 0
\(153\) −4.76028 + 4.76028i −0.384846 + 0.384846i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −12.4183 + 3.32748i −0.991091 + 0.265562i −0.717709 0.696343i \(-0.754809\pi\)
−0.273382 + 0.961905i \(0.588142\pi\)
\(158\) 0 0
\(159\) −4.36603 7.56218i −0.346248 0.599720i
\(160\) 0 0
\(161\) 14.9282 + 12.9282i 1.17651 + 1.01889i
\(162\) 0 0
\(163\) 0.776457 2.89778i 0.0608168 0.226971i −0.928828 0.370512i \(-0.879182\pi\)
0.989644 + 0.143541i \(0.0458488\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.240237 0.240237i −0.0185901 0.0185901i 0.697751 0.716341i \(-0.254184\pi\)
−0.716341 + 0.697751i \(0.754184\pi\)
\(168\) 0 0
\(169\) 12.0000i 0.923077i
\(170\) 0 0
\(171\) −7.33013 4.23205i −0.560549 0.323633i
\(172\) 0 0
\(173\) 12.1087 + 3.24453i 0.920611 + 0.246677i 0.687847 0.725856i \(-0.258556\pi\)
0.232765 + 0.972533i \(0.425223\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.95215 2.39872i −0.672885 0.180299i
\(178\) 0 0
\(179\) 8.19615 + 4.73205i 0.612609 + 0.353690i 0.773986 0.633203i \(-0.218260\pi\)
−0.161377 + 0.986893i \(0.551593\pi\)
\(180\) 0 0
\(181\) 6.39230i 0.475136i −0.971371 0.237568i \(-0.923650\pi\)
0.971371 0.237568i \(-0.0763503\pi\)
\(182\) 0 0
\(183\) −6.88160 6.88160i −0.508702 0.508702i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −4.76028 + 17.7656i −0.348106 + 1.29915i
\(188\) 0 0
\(189\) 2.50000 0.866025i 0.181848 0.0629941i
\(190\) 0 0
\(191\) 12.0981 + 20.9545i 0.875386 + 1.51621i 0.856351 + 0.516394i \(0.172726\pi\)
0.0190352 + 0.999819i \(0.493941\pi\)
\(192\) 0 0
\(193\) 12.8666 3.44760i 0.926160 0.248164i 0.235943 0.971767i \(-0.424182\pi\)
0.690216 + 0.723603i \(0.257515\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.58630 3.58630i 0.255513 0.255513i −0.567713 0.823226i \(-0.692172\pi\)
0.823226 + 0.567713i \(0.192172\pi\)
\(198\) 0 0
\(199\) 4.23205 7.33013i 0.300002 0.519619i −0.676134 0.736779i \(-0.736346\pi\)
0.976136 + 0.217160i \(0.0696793\pi\)
\(200\) 0 0
\(201\) −6.06218 + 3.50000i −0.427593 + 0.246871i
\(202\) 0 0
\(203\) −16.4901 8.00481i −1.15738 0.561827i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.93185 + 7.20977i 0.134273 + 0.501114i
\(208\) 0 0
\(209\) −23.1244 −1.59955
\(210\) 0 0
\(211\) −15.7321 −1.08304 −0.541520 0.840688i \(-0.682151\pi\)
−0.541520 + 0.840688i \(0.682151\pi\)
\(212\) 0 0
\(213\) −2.07055 7.72741i −0.141872 0.529473i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 4.38134 2.96713i 0.297425 0.201422i
\(218\) 0 0
\(219\) 13.3301 7.69615i 0.900767 0.520058i
\(220\) 0 0
\(221\) −3.36603 + 5.83013i −0.226423 + 0.392177i
\(222\) 0 0
\(223\) −0.947343 + 0.947343i −0.0634388 + 0.0634388i −0.738114 0.674676i \(-0.764284\pi\)
0.674676 + 0.738114i \(0.264284\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 13.3843 3.58630i 0.888345 0.238031i 0.214341 0.976759i \(-0.431240\pi\)
0.674004 + 0.738728i \(0.264573\pi\)
\(228\) 0 0
\(229\) −5.86603 10.1603i −0.387638 0.671408i 0.604493 0.796610i \(-0.293376\pi\)
−0.992131 + 0.125202i \(0.960042\pi\)
\(230\) 0 0
\(231\) 4.73205 5.46410i 0.311346 0.359511i
\(232\) 0 0
\(233\) −2.77766 + 10.3664i −0.181971 + 0.679123i 0.813288 + 0.581861i \(0.197675\pi\)
−0.995259 + 0.0972622i \(0.968991\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.60368 + 1.60368i 0.104170 + 0.104170i
\(238\) 0 0
\(239\) 23.3205i 1.50848i 0.656600 + 0.754239i \(0.271994\pi\)
−0.656600 + 0.754239i \(0.728006\pi\)
\(240\) 0 0
\(241\) −9.69615 5.59808i −0.624584 0.360604i 0.154068 0.988060i \(-0.450763\pi\)
−0.778652 + 0.627457i \(0.784096\pi\)
\(242\) 0 0
\(243\) 0.965926 + 0.258819i 0.0619642 + 0.0166032i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.17569 2.19067i −0.520207 0.139389i
\(248\) 0 0
\(249\) −4.09808 2.36603i −0.259705 0.149941i
\(250\) 0 0
\(251\) 10.5885i 0.668337i −0.942513 0.334169i \(-0.891544\pi\)
0.942513 0.334169i \(-0.108456\pi\)
\(252\) 0 0
\(253\) 14.4195 + 14.4195i 0.906549 + 0.906549i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.56892 20.7835i 0.347380 1.29644i −0.542428 0.840103i \(-0.682495\pi\)
0.889807 0.456336i \(-0.150839\pi\)
\(258\) 0 0
\(259\) 19.3923 22.3923i 1.20498 1.39139i
\(260\) 0 0
\(261\) −3.46410 6.00000i −0.214423 0.371391i
\(262\) 0 0
\(263\) 8.76268 2.34795i 0.540330 0.144781i 0.0216758 0.999765i \(-0.493100\pi\)
0.518654 + 0.854984i \(0.326433\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.13922 + 5.13922i −0.314515 + 0.314515i
\(268\) 0 0
\(269\) 4.29423 7.43782i 0.261824 0.453492i −0.704903 0.709304i \(-0.749009\pi\)
0.966727 + 0.255812i \(0.0823428\pi\)
\(270\) 0 0
\(271\) 12.9282 7.46410i 0.785332 0.453412i −0.0529843 0.998595i \(-0.516873\pi\)
0.838317 + 0.545183i \(0.183540\pi\)
\(272\) 0 0
\(273\) 2.19067 1.48356i 0.132585 0.0897894i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −5.48597 20.4739i −0.329620 1.23016i −0.909585 0.415517i \(-0.863601\pi\)
0.579966 0.814641i \(-0.303066\pi\)
\(278\) 0 0
\(279\) 2.00000 0.119737
\(280\) 0 0
\(281\) −8.87564 −0.529477 −0.264738 0.964320i \(-0.585286\pi\)
−0.264738 + 0.964320i \(0.585286\pi\)
\(282\) 0 0
\(283\) −1.89967 7.08965i −0.112923 0.421436i 0.886200 0.463303i \(-0.153336\pi\)
−0.999123 + 0.0418674i \(0.986669\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.98743 + 4.84821i 0.589539 + 0.286181i
\(288\) 0 0
\(289\) 24.5263 14.1603i 1.44272 0.832956i
\(290\) 0 0
\(291\) −1.76795 + 3.06218i −0.103639 + 0.179508i
\(292\) 0 0
\(293\) −11.5911 + 11.5911i −0.677160 + 0.677160i −0.959357 0.282197i \(-0.908937\pi\)
0.282197 + 0.959357i \(0.408937\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.63896 0.707107i 0.153128 0.0410305i
\(298\) 0 0
\(299\) 3.73205 + 6.46410i 0.215830 + 0.373829i
\(300\) 0 0
\(301\) 1.33975 0.464102i 0.0772217 0.0267504i
\(302\) 0 0
\(303\) −1.74238 + 6.50266i −0.100097 + 0.373568i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −1.79315 1.79315i −0.102341 0.102341i 0.654083 0.756423i \(-0.273055\pi\)
−0.756423 + 0.654083i \(0.773055\pi\)
\(308\) 0 0
\(309\) 4.66025i 0.265113i
\(310\) 0 0
\(311\) 5.19615 + 3.00000i 0.294647 + 0.170114i 0.640036 0.768345i \(-0.278920\pi\)
−0.345389 + 0.938460i \(0.612253\pi\)
\(312\) 0 0
\(313\) −2.44949 0.656339i −0.138453 0.0370985i 0.188927 0.981991i \(-0.439499\pi\)
−0.327380 + 0.944893i \(0.606166\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.31079 0.619174i −0.129787 0.0347763i 0.193341 0.981132i \(-0.438068\pi\)
−0.323128 + 0.946355i \(0.604734\pi\)
\(318\) 0 0
\(319\) −16.3923 9.46410i −0.917793 0.529888i
\(320\) 0 0
\(321\) 14.7321i 0.822263i
\(322\) 0 0
\(323\) 40.2915 + 40.2915i 2.24188 + 2.24188i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −3.60488 + 13.4536i −0.199351 + 0.743986i
\(328\) 0 0
\(329\) −20.3923 17.6603i −1.12426 0.973641i
\(330\) 0 0
\(331\) 6.66987 + 11.5526i 0.366609 + 0.634986i 0.989033 0.147694i \(-0.0471852\pi\)
−0.622424 + 0.782681i \(0.713852\pi\)
\(332\) 0 0
\(333\) 10.8147 2.89778i 0.592639 0.158797i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.656339 + 0.656339i −0.0357531 + 0.0357531i −0.724757 0.689004i \(-0.758048\pi\)
0.689004 + 0.724757i \(0.258048\pi\)
\(338\) 0 0
\(339\) 4.26795 7.39230i 0.231803 0.401495i
\(340\) 0 0
\(341\) 4.73205 2.73205i 0.256255 0.147949i
\(342\) 0 0
\(343\) 18.0938 3.95164i 0.976972 0.213368i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.19239 + 34.3065i 0.493473 + 1.84167i 0.538418 + 0.842678i \(0.319022\pi\)
−0.0449449 + 0.998989i \(0.514311\pi\)
\(348\) 0 0
\(349\) 12.9282 0.692031 0.346015 0.938229i \(-0.387534\pi\)
0.346015 + 0.938229i \(0.387534\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 3.01790 + 11.2629i 0.160626 + 0.599466i 0.998558 + 0.0536901i \(0.0170983\pi\)
−0.837931 + 0.545776i \(0.816235\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −17.7656 + 1.27551i −0.940255 + 0.0675073i
\(358\) 0 0
\(359\) −24.2942 + 14.0263i −1.28220 + 0.740279i −0.977250 0.212090i \(-0.931973\pi\)
−0.304950 + 0.952368i \(0.598640\pi\)
\(360\) 0 0
\(361\) −26.3205 + 45.5885i −1.38529 + 2.39939i
\(362\) 0 0
\(363\) −2.50026 + 2.50026i −0.131229 + 0.131229i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 7.20977 1.93185i 0.376347 0.100842i −0.0656868 0.997840i \(-0.520924\pi\)
0.442034 + 0.896998i \(0.354257\pi\)
\(368\) 0 0
\(369\) 2.09808 + 3.63397i 0.109221 + 0.189177i
\(370\) 0 0
\(371\) 4.36603 22.6865i 0.226673 1.17783i
\(372\) 0 0
\(373\) 3.89589 14.5397i 0.201721 0.752835i −0.788702 0.614775i \(-0.789247\pi\)
0.990424 0.138060i \(-0.0440866\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.89898 4.89898i −0.252310 0.252310i
\(378\) 0 0
\(379\) 23.4449i 1.20428i −0.798390 0.602141i \(-0.794315\pi\)
0.798390 0.602141i \(-0.205685\pi\)
\(380\) 0 0
\(381\) −0.0621778 0.0358984i −0.00318547 0.00183913i
\(382\) 0 0
\(383\) 28.7004 + 7.69024i 1.46652 + 0.392953i 0.901737 0.432286i \(-0.142293\pi\)
0.564784 + 0.825239i \(0.308960\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.517638 + 0.138701i 0.0263130 + 0.00705055i
\(388\) 0 0
\(389\) 15.4186 + 8.90192i 0.781753 + 0.451345i 0.837051 0.547124i \(-0.184278\pi\)
−0.0552980 + 0.998470i \(0.517611\pi\)
\(390\) 0 0
\(391\) 50.2487i 2.54119i
\(392\) 0 0
\(393\) 11.9700 + 11.9700i 0.603809 + 0.603809i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.69161 + 6.31319i −0.0848997 + 0.316850i −0.995295 0.0968891i \(-0.969111\pi\)
0.910395 + 0.413739i \(0.135777\pi\)
\(398\) 0 0
\(399\) −7.33013 21.1603i −0.366965 1.05934i
\(400\) 0 0
\(401\) 13.2942 + 23.0263i 0.663882 + 1.14988i 0.979587 + 0.201020i \(0.0644257\pi\)
−0.315705 + 0.948857i \(0.602241\pi\)
\(402\) 0 0
\(403\) 1.93185 0.517638i 0.0962324 0.0257854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 21.6293 21.6293i 1.07212 1.07212i
\(408\) 0 0
\(409\) −5.79423 + 10.0359i −0.286506 + 0.496243i −0.972973 0.230918i \(-0.925827\pi\)
0.686467 + 0.727161i \(0.259161\pi\)
\(410\) 0 0
\(411\) 5.19615 3.00000i 0.256307 0.147979i
\(412\) 0 0
\(413\) −13.7496 20.3030i −0.676573 0.999047i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4.36276 16.2820i −0.213645 0.797335i
\(418\) 0 0
\(419\) −0.0525589 −0.00256767 −0.00128383 0.999999i \(-0.500409\pi\)
−0.00128383 + 0.999999i \(0.500409\pi\)
\(420\) 0 0
\(421\) −31.3923 −1.52997 −0.764984 0.644050i \(-0.777253\pi\)
−0.764984 + 0.644050i \(0.777253\pi\)
\(422\) 0 0
\(423\) −2.63896 9.84873i −0.128311 0.478861i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.84392 25.6825i −0.0892335 1.24286i
\(428\) 0 0
\(429\) 2.36603 1.36603i 0.114233 0.0659523i
\(430\) 0 0
\(431\) −5.53590 + 9.58846i −0.266655 + 0.461860i −0.967996 0.250966i \(-0.919252\pi\)
0.701341 + 0.712826i \(0.252585\pi\)
\(432\) 0 0
\(433\) −21.5921 + 21.5921i −1.03765 + 1.03765i −0.0383892 + 0.999263i \(0.512223\pi\)
−0.999263 + 0.0383892i \(0.987777\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 61.0242 16.3514i 2.91918 0.782193i
\(438\) 0 0
\(439\) 6.89230 + 11.9378i 0.328952 + 0.569761i 0.982304 0.187293i \(-0.0599713\pi\)
−0.653352 + 0.757054i \(0.726638\pi\)
\(440\) 0 0
\(441\) 6.50000 + 2.59808i 0.309524 + 0.123718i
\(442\) 0 0
\(443\) 5.99863 22.3872i 0.285003 1.06365i −0.663834 0.747880i \(-0.731072\pi\)
0.948837 0.315767i \(-0.102262\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −6.45189 6.45189i −0.305164 0.305164i
\(448\) 0 0
\(449\) 10.3397i 0.487963i 0.969780 + 0.243981i \(0.0784536\pi\)
−0.969780 + 0.243981i \(0.921546\pi\)
\(450\) 0 0
\(451\) 9.92820 + 5.73205i 0.467501 + 0.269912i
\(452\) 0 0
\(453\) 10.2970 + 2.75908i 0.483796 + 0.129633i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15.5749 + 4.17329i 0.728565 + 0.195218i 0.603990 0.796992i \(-0.293577\pi\)
0.124575 + 0.992210i \(0.460243\pi\)
\(458\) 0 0
\(459\) −5.83013 3.36603i −0.272127 0.157113i
\(460\) 0 0
\(461\) 17.7128i 0.824968i −0.910965 0.412484i \(-0.864661\pi\)
0.910965 0.412484i \(-0.135339\pi\)
\(462\) 0 0
\(463\) 22.0962 + 22.0962i 1.02690 + 1.02690i 0.999628 + 0.0272682i \(0.00868082\pi\)
0.0272682 + 0.999628i \(0.491319\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.20977 26.9072i 0.333628 1.24512i −0.571720 0.820449i \(-0.693724\pi\)
0.905349 0.424669i \(-0.139610\pi\)
\(468\) 0 0
\(469\) −18.1865 3.50000i −0.839776 0.161615i
\(470\) 0 0
\(471\) −6.42820 11.1340i −0.296196 0.513026i
\(472\) 0 0
\(473\) 1.41421 0.378937i 0.0650256 0.0174236i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 6.17449 6.17449i 0.282711 0.282711i
\(478\) 0 0
\(479\) 13.2224 22.9019i 0.604148 1.04642i −0.388037 0.921644i \(-0.626847\pi\)
0.992185 0.124772i \(-0.0398199\pi\)
\(480\) 0 0
\(481\) 9.69615 5.59808i 0.442106 0.255250i
\(482\) 0 0
\(483\) −8.62398 + 17.7656i −0.392405 + 0.808363i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.55343 + 24.4577i 0.296964 + 1.10829i 0.939645 + 0.342151i \(0.111156\pi\)
−0.642681 + 0.766134i \(0.722178\pi\)
\(488\) 0 0
\(489\) 3.00000 0.135665
\(490\) 0 0
\(491\) 14.3923 0.649516 0.324758 0.945797i \(-0.394717\pi\)
0.324758 + 0.945797i \(0.394717\pi\)
\(492\) 0 0
\(493\) 12.0716 + 45.0518i 0.543677 + 2.02903i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.24316 19.0411i 0.414612 0.854111i
\(498\) 0 0
\(499\) 1.03590 0.598076i 0.0463732 0.0267736i −0.476634 0.879102i \(-0.658143\pi\)
0.523007 + 0.852328i \(0.324810\pi\)
\(500\) 0 0
\(501\) 0.169873 0.294229i 0.00758937 0.0131452i
\(502\) 0 0
\(503\) 14.6969 14.6969i 0.655304 0.655304i −0.298961 0.954265i \(-0.596640\pi\)
0.954265 + 0.298961i \(0.0966401\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −11.5911 + 3.10583i −0.514779 + 0.137935i
\(508\) 0 0
\(509\) −10.7321 18.5885i −0.475690 0.823919i 0.523922 0.851766i \(-0.324468\pi\)
−0.999612 + 0.0278471i \(0.991135\pi\)
\(510\) 0 0
\(511\) 39.9904 + 7.69615i 1.76907 + 0.340458i
\(512\) 0 0
\(513\) 2.19067 8.17569i 0.0967205 0.360966i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −19.6975 19.6975i −0.866293 0.866293i
\(518\) 0 0
\(519\) 12.5359i 0.550265i
\(520\) 0 0
\(521\) 32.3205 + 18.6603i 1.41599 + 0.817521i 0.995943 0.0899832i \(-0.0286813\pi\)
0.420044 + 0.907504i \(0.362015\pi\)
\(522\) 0 0
\(523\) 18.6622 + 5.00052i 0.816040 + 0.218657i 0.642614 0.766190i \(-0.277850\pi\)
0.173426 + 0.984847i \(0.444516\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −13.0053 3.48477i −0.566521 0.151799i
\(528\) 0 0
\(529\) −28.3301 16.3564i −1.23174 0.711148i
\(530\) 0 0
\(531\) 9.26795i 0.402195i
\(532\) 0 0
\(533\) 2.96713 + 2.96713i 0.128521 + 0.128521i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −2.44949 + 9.14162i −0.105703 + 0.394490i
\(538\) 0 0
\(539\) 18.9282 2.73205i 0.815295 0.117678i
\(540\) 0 0
\(541\) 19.1603 + 33.1865i 0.823764 + 1.42680i 0.902861 + 0.429934i \(0.141463\pi\)
−0.0790969 + 0.996867i \(0.525204\pi\)
\(542\) 0 0
\(543\) 6.17449 1.65445i 0.264973 0.0709993i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.3538 + 20.3538i −0.870265 + 0.870265i −0.992501 0.122236i \(-0.960994\pi\)
0.122236 + 0.992501i \(0.460994\pi\)
\(548\) 0 0
\(549\) 4.86603 8.42820i 0.207677 0.359707i
\(550\) 0 0
\(551\) −50.7846 + 29.3205i −2.16350 + 1.24910i
\(552\) 0 0
\(553\) 0.429705 + 5.98502i 0.0182729 + 0.254509i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.13681 + 4.24264i 0.0481683 + 0.179766i 0.985819 0.167813i \(-0.0536704\pi\)
−0.937651 + 0.347579i \(0.887004\pi\)
\(558\) 0 0
\(559\) 0.535898 0.0226661
\(560\) 0 0
\(561\) −18.3923 −0.776524
\(562\) 0 0
\(563\) 10.2277 + 38.1702i 0.431045 + 1.60868i 0.750358 + 0.661031i \(0.229881\pi\)
−0.319314 + 0.947649i \(0.603452\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.48356 + 2.19067i 0.0623038 + 0.0919995i
\(568\) 0 0
\(569\) 33.4186 19.2942i 1.40098 0.808856i 0.406487 0.913657i \(-0.366754\pi\)
0.994493 + 0.104801i \(0.0334204\pi\)
\(570\) 0 0
\(571\) 13.4545 23.3038i 0.563053 0.975236i −0.434175 0.900828i \(-0.642960\pi\)
0.997228 0.0744073i \(-0.0237065\pi\)
\(572\) 0 0
\(573\) −17.1093 + 17.1093i −0.714750 + 0.714750i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −9.65926 + 2.58819i −0.402120 + 0.107748i −0.454210 0.890895i \(-0.650078\pi\)
0.0520899 + 0.998642i \(0.483412\pi\)
\(578\) 0 0
\(579\) 6.66025 + 11.5359i 0.276791 + 0.479416i
\(580\) 0 0
\(581\) −4.09808 11.8301i −0.170017 0.490796i
\(582\) 0 0
\(583\) 6.17449 23.0435i 0.255721 0.954365i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.0101 + 21.0101i 0.867181 + 0.867181i 0.992159 0.124979i \(-0.0398862\pi\)
−0.124979 + 0.992159i \(0.539886\pi\)
\(588\) 0 0
\(589\) 16.9282i 0.697514i
\(590\) 0 0
\(591\) 4.39230 + 2.53590i 0.180675 + 0.104313i
\(592\) 0 0
\(593\) −33.9783 9.10446i −1.39532 0.373875i −0.518659 0.854981i \(-0.673568\pi\)
−0.876662 + 0.481106i \(0.840235\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.17569 + 2.19067i 0.334609 + 0.0896582i
\(598\) 0 0
\(599\) 11.9545 + 6.90192i 0.488447 + 0.282005i 0.723930 0.689874i \(-0.242334\pi\)
−0.235483 + 0.971878i \(0.575667\pi\)
\(600\) 0 0
\(601\) 12.5167i 0.510565i −0.966866 0.255283i \(-0.917831\pi\)
0.966866 0.255283i \(-0.0821685\pi\)
\(602\) 0 0
\(603\) −4.94975 4.94975i −0.201569 0.201569i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.79435 14.1607i 0.154008 0.574766i −0.845180 0.534481i \(-0.820507\pi\)
0.999188 0.0402844i \(-0.0128264\pi\)
\(608\) 0 0
\(609\) 3.46410 18.0000i 0.140372 0.729397i
\(610\) 0 0
\(611\) −5.09808 8.83013i −0.206246 0.357229i
\(612\) 0 0
\(613\) 26.9072 7.20977i 1.08677 0.291200i 0.329405 0.944189i \(-0.393152\pi\)
0.757367 + 0.652989i \(0.226485\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.14110 + 4.14110i −0.166715 + 0.166715i −0.785534 0.618819i \(-0.787611\pi\)
0.618819 + 0.785534i \(0.287611\pi\)
\(618\) 0 0
\(619\) 19.1962 33.2487i 0.771559 1.33638i −0.165150 0.986268i \(-0.552811\pi\)
0.936709 0.350110i \(-0.113856\pi\)
\(620\) 0 0
\(621\) −6.46410 + 3.73205i −0.259395 + 0.149762i
\(622\) 0 0
\(623\) −19.1798 + 1.37705i −0.768423 + 0.0551703i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.98502 22.3364i −0.239019 0.892030i
\(628\) 0 0
\(629\) −75.3731 −3.00532
\(630\) 0 0
\(631\) −26.8038 −1.06704 −0.533522 0.845786i \(-0.679132\pi\)
−0.533522 + 0.845786i \(0.679132\pi\)
\(632\) 0 0
\(633\) −4.07175 15.1960i −0.161838 0.603987i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 6.95095 + 0.827225i 0.275407 + 0.0327759i
\(638\) 0 0
\(639\) 6.92820 4.00000i 0.274075 0.158238i
\(640\) 0 0
\(641\) 22.5885 39.1244i 0.892190 1.54532i 0.0549467 0.998489i \(-0.482501\pi\)
0.837244 0.546830i \(-0.184166\pi\)
\(642\) 0 0
\(643\) −33.5486 + 33.5486i −1.32303 + 1.32303i −0.411714 + 0.911313i \(0.635070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −34.3065 + 9.19239i −1.34873 + 0.361390i −0.859664 0.510859i \(-0.829327\pi\)
−0.489062 + 0.872249i \(0.662661\pi\)
\(648\) 0 0
\(649\) −12.6603 21.9282i −0.496958 0.860757i
\(650\) 0 0
\(651\) 4.00000 + 3.46410i 0.156772 + 0.135769i
\(652\) 0 0
\(653\) 0.152304 0.568406i 0.00596011 0.0222434i −0.962882 0.269924i \(-0.913002\pi\)
0.968842 + 0.247680i \(0.0796682\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 10.8840 + 10.8840i 0.424625 + 0.424625i
\(658\) 0 0
\(659\) 5.66025i 0.220492i −0.993904 0.110246i \(-0.964836\pi\)
0.993904 0.110246i \(-0.0351639\pi\)
\(660\) 0 0
\(661\) 1.37564 + 0.794229i 0.0535064 + 0.0308919i 0.526515 0.850166i \(-0.323499\pi\)
−0.473008 + 0.881058i \(0.656832\pi\)
\(662\) 0 0
\(663\) −6.50266 1.74238i −0.252542 0.0676685i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 49.9507 + 13.3843i 1.93410 + 0.518241i
\(668\) 0 0
\(669\) −1.16025 0.669873i −0.0448580 0.0258988i
\(670\) 0 0
\(671\) 26.5885i 1.02644i
\(672\) 0 0
\(673\) −8.29581 8.29581i −0.319780 0.319780i 0.528903 0.848683i \(-0.322604\pi\)
−0.848683 + 0.528903i \(0.822604\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.896575 + 3.34607i −0.0344582 + 0.128600i −0.981012 0.193945i \(-0.937872\pi\)
0.946554 + 0.322545i \(0.104538\pi\)
\(678\) 0 0
\(679\) −8.83975 + 3.06218i −0.339238 + 0.117516i
\(680\) 0 0
\(681\) 6.92820 + 12.0000i 0.265489 + 0.459841i
\(682\) 0 0
\(683\) 10.8840 2.91636i 0.416465 0.111591i −0.0445009 0.999009i \(-0.514170\pi\)
0.460966 + 0.887418i \(0.347503\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 8.29581 8.29581i 0.316505 0.316505i
\(688\) 0 0
\(689\) 4.36603 7.56218i 0.166332 0.288096i
\(690\) 0 0
\(691\) −11.4737 + 6.62436i −0.436481 + 0.252002i −0.702104 0.712075i \(-0.747756\pi\)
0.265623 + 0.964077i \(0.414422\pi\)
\(692\) 0 0
\(693\) 6.50266 + 3.15660i 0.247016 + 0.119909i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −7.31130 27.2862i −0.276935 1.03354i
\(698\) 0 0
\(699\) −10.7321 −0.405923
\(700\) 0 0
\(701\) −18.3397 −0.692683 −0.346341 0.938109i \(-0.612576\pi\)
−0.346341 + 0.938109i \(0.612576\pi\)
\(702\) 0 0
\(703\) −24.5271 91.5363i −0.925056 3.45236i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.7477 + 9.98743i −0.554645 + 0.375616i
\(708\) 0 0
\(709\) 4.45448 2.57180i 0.167292 0.0965859i −0.414017 0.910269i \(-0.635874\pi\)
0.581308 + 0.813684i \(0.302541\pi\)
\(710\) 0 0
\(711\) −1.13397 + 1.96410i −0.0425274 + 0.0736596i
\(712\) 0 0
\(713\) −10.5558 + 10.5558i −0.395319 + 0.395319i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −22.5259 + 6.03579i −0.841244 + 0.225411i
\(718\) 0 0
\(719\) −4.75833 8.24167i −0.177456 0.307362i 0.763553 0.645746i \(-0.223453\pi\)
−0.941008 + 0.338383i \(0.890120\pi\)
\(720\) 0 0
\(721\) 8.07180 9.32051i 0.300609 0.347114i
\(722\) 0 0
\(723\) 2.89778 10.8147i 0.107770 0.402201i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.568406 + 0.568406i 0.0210810 + 0.0210810i 0.717569 0.696488i \(-0.245255\pi\)
−0.696488 + 0.717569i \(0.745255\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) −3.12436 1.80385i −0.115558 0.0667177i
\(732\) 0 0
\(733\) 15.9031 + 4.26122i 0.587394 + 0.157392i 0.540262 0.841497i \(-0.318325\pi\)
0.0471323 + 0.998889i \(0.484992\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −18.4727 4.94975i −0.680451 0.182326i
\(738\) 0 0
\(739\) 25.0359 + 14.4545i 0.920960 + 0.531717i 0.883941 0.467598i \(-0.154880\pi\)
0.0370191 + 0.999315i \(0.488214\pi\)
\(740\) 0 0
\(741\) 8.46410i 0.310937i
\(742\) 0 0
\(743\) 24.1160 + 24.1160i 0.884729 + 0.884729i 0.994011 0.109282i \(-0.0348552\pi\)
−0.109282 + 0.994011i \(0.534855\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.22474 4.57081i 0.0448111 0.167237i
\(748\) 0 0
\(749\) −25.5167 + 29.4641i −0.932358 + 1.07659i
\(750\) 0 0
\(751\) −0.0621778 0.107695i −0.00226890 0.00392985i 0.864889 0.501964i \(-0.167389\pi\)
−0.867158 + 0.498034i \(0.834056\pi\)
\(752\) 0 0
\(753\) 10.2277 2.74049i 0.372717 0.0998692i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −5.74479 + 5.74479i −0.208798 + 0.208798i −0.803756 0.594959i \(-0.797168\pi\)
0.594959 + 0.803756i \(0.297168\pi\)
\(758\) 0 0
\(759\) −10.1962 + 17.6603i −0.370097 + 0.641027i
\(760\) 0 0
\(761\) 20.4904 11.8301i 0.742776 0.428842i −0.0803019 0.996771i \(-0.525588\pi\)
0.823078 + 0.567929i \(0.192255\pi\)
\(762\) 0 0
\(763\) −30.5121 + 20.6634i −1.10461 + 0.748065i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.39872 8.95215i −0.0866128 0.323243i
\(768\) 0 0
\(769\) 21.3205 0.768837 0.384419 0.923159i \(-0.374402\pi\)
0.384419 + 0.923159i \(0.374402\pi\)
\(770\) 0 0
\(771\) 21.5167 0.774904
\(772\) 0 0
\(773\) 0.832204 + 3.10583i 0.0299323 + 0.111709i 0.979276 0.202532i \(-0.0649169\pi\)
−0.949343 + 0.314240i \(0.898250\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 26.6484 + 12.9360i 0.956006 + 0.464075i
\(778\) 0 0
\(779\) 30.7583 17.7583i 1.10203 0.636258i
\(780\) 0 0
\(781\) 10.9282 18.9282i 0.391042 0.677304i
\(782\) 0 0
\(783\) 4.89898 4.89898i 0.175075 0.175075i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −0.397520 + 0.106515i −0.0141700 + 0.00379685i −0.265897 0.964001i \(-0.585668\pi\)
0.251727 + 0.967798i \(0.419001\pi\)
\(788\) 0 0
\(789\) 4.53590 + 7.85641i 0.161482 + 0.279695i
\(790\) 0 0
\(791\) 21.3397 7.39230i 0.758754 0.262840i
\(792\) 0 0
\(793\) 2.51884 9.40044i 0.0894466 0.333819i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.9759 29.9759i −1.06180 1.06180i −0.997960 0.0638402i \(-0.979665\pi\)
−0.0638402 0.997960i \(-0.520335\pi\)
\(798\) 0 0
\(799\) 68.6410i 2.42834i
\(800\) 0 0
\(801\) −6.29423 3.63397i −0.222396 0.128400i
\(802\) 0 0
\(803\) 40.6197 + 10.8840i 1.43344 + 0.384088i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 8.29581 + 2.22286i 0.292026 + 0.0782482i
\(808\) 0 0
\(809\) 44.5692 + 25.7321i 1.56697 + 0.904691i 0.996520 + 0.0833589i \(0.0265648\pi\)
0.570451 + 0.821332i \(0.306769\pi\)
\(810\) 0 0
\(811\) 27.1436i 0.953140i 0.879136 + 0.476570i \(0.158120\pi\)
−0.879136 + 0.476570i \(0.841880\pi\)
\(812\) 0 0
\(813\) 10.5558 + 10.5558i 0.370209 + 0.370209i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.17398 4.38134i 0.0410723 0.153284i
\(818\) 0 0
\(819\) 2.00000 + 1.73205i 0.0698857 + 0.0605228i
\(820\) 0 0
\(821\) 25.8301 + 44.7391i 0.901478 + 1.56141i 0.825577 + 0.564290i \(0.190850\pi\)
0.0759008 + 0.997115i \(0.475817\pi\)
\(822\) 0 0
\(823\) 0.309587 0.0829536i 0.0107915 0.00289158i −0.253419 0.967357i \(-0.581555\pi\)
0.264211 + 0.964465i \(0.414889\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.2784 + 10.2784i −0.357416 + 0.357416i −0.862860 0.505444i \(-0.831329\pi\)
0.505444 + 0.862860i \(0.331329\pi\)
\(828\) 0 0
\(829\) −22.7224 + 39.3564i −0.789183 + 1.36690i 0.137286 + 0.990532i \(0.456162\pi\)
−0.926468 + 0.376373i \(0.877171\pi\)
\(830\) 0 0
\(831\) 18.3564 10.5981i 0.636777 0.367643i
\(832\) 0 0
\(833\) −37.7405 28.2199i −1.30763 0.977762i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.517638 + 1.93185i 0.0178922 + 0.0667746i
\(838\) 0 0
\(839\) −51.9615 −1.79391 −0.896956 0.442121i \(-0.854226\pi\)
−0.896956 + 0.442121i \(0.854226\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) −2.29719 8.57321i −0.0791193 0.295277i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −9.33109 + 0.669942i −0.320620 + 0.0230195i
\(848\) 0 0
\(849\) 6.35641 3.66987i 0.218151 0.125950i
\(850\) 0 0
\(851\) −41.7846 + 72.3731i −1.43236 + 2.48092i
\(852\) 0 0
\(853\) −15.5563 + 15.5563i −0.532639 + 0.532639i −0.921357 0.388718i \(-0.872918\pi\)
0.388718 + 0.921357i \(0.372918\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −23.8014 + 6.37756i −0.813040 + 0.217853i −0.641301 0.767289i \(-0.721605\pi\)
−0.171739 + 0.985143i \(0.554938\pi\)
\(858\) 0 0
\(859\) 0.464102 + 0.803848i 0.0158349 + 0.0274269i 0.873834 0.486224i \(-0.161626\pi\)
−0.857999 + 0.513651i \(0.828293\pi\)
\(860\) 0 0
\(861\) −2.09808 + 10.9019i −0.0715022 + 0.371537i
\(862\) 0 0
\(863\) 11.5403 43.0691i 0.392838 1.46609i −0.432594 0.901589i \(-0.642402\pi\)
0.825432 0.564502i \(-0.190932\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 20.0256 + 20.0256i 0.680106 + 0.680106i
\(868\) 0 0
\(869\) 6.19615i 0.210190i
\(870\) 0 0
\(871\) −6.06218 3.50000i −0.205409 0.118593i
\(872\) 0 0
\(873\) −3.41542 0.915158i −0.115594 0.0309734i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.1583 + 2.72191i 0.343022 + 0.0919124i 0.426217 0.904621i \(-0.359846\pi\)
−0.0831954 + 0.996533i \(0.526513\pi\)
\(878\) 0 0
\(879\) −14.1962 8.19615i −0.478824 0.276449i
\(880\) 0 0
\(881\) 20.0000i 0.673817i −0.941537 0.336909i \(-0.890619\pi\)
0.941537 0.336909i \(-0.109381\pi\)
\(882\) 0 0
\(883\) 0.226633 + 0.226633i 0.00762682 + 0.00762682i 0.710910 0.703283i \(-0.248283\pi\)
−0.703283 + 0.710910i \(0.748283\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −4.30701 + 16.0740i −0.144615 + 0.539712i 0.855157 + 0.518369i \(0.173461\pi\)
−0.999772 + 0.0213425i \(0.993206\pi\)
\(888\) 0 0
\(889\) −0.0621778 0.179492i −0.00208538 0.00601997i
\(890\) 0 0
\(891\) 1.36603 + 2.36603i 0.0457636 + 0.0792648i
\(892\) 0 0
\(893\) −83.3606 + 22.3364i −2.78956 + 0.747460i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.27792 + 5.27792i −0.176225 + 0.176225i
\(898\) 0 0
\(899\) 6.92820 12.0000i 0.231069 0.400222i
\(900\) 0 0
\(901\) −50.9090 + 29.3923i −1.69602 + 0.979200i
\(902\) 0 0
\(903\) 0.795040 + 1.17398i 0.0264573 + 0.0390675i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −4.22406 15.7644i −0.140258 0.523448i −0.999921 0.0125871i \(-0.995993\pi\)
0.859663 0.510861i \(-0.170673\pi\)
\(908\) 0 0
\(909\) −6.73205 −0.223288
\(910\) 0 0
\(911\) −9.71281 −0.321800 −0.160900 0.986971i \(-0.551440\pi\)
−0.160900 + 0.986971i \(0.551440\pi\)
\(912\) 0 0
\(913\) −3.34607 12.4877i −0.110739 0.413282i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.20736 + 44.6728i 0.105917 + 1.47523i
\(918\) 0 0
\(919\) 10.3923 6.00000i 0.342811 0.197922i −0.318704 0.947854i \(-0.603247\pi\)
0.661514 + 0.749933i \(0.269914\pi\)
\(920\) 0 0
\(921\) 1.26795 2.19615i 0.0417803 0.0723657i
\(922\) 0 0
\(923\) 5.65685 5.65685i 0.186198 0.186198i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.50146 1.20616i 0.147847 0.0396156i
\(928\) 0 0
\(929\) −6.09808 10.5622i −0.200071 0.346534i 0.748480 0.663158i \(-0.230784\pi\)
−0.948551 + 0.316624i \(0.897451\pi\)
\(930\) 0 0
\(931\) 21.9904 55.0167i 0.720706 1.80310i
\(932\) 0 0
\(933\) −1.55291 + 5.79555i −0.0508401 + 0.189738i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 11.8685 + 11.8685i 0.387727 + 0.387727i 0.873876 0.486149i \(-0.161599\pi\)
−0.486149 + 0.873876i \(0.661599\pi\)
\(938\) 0 0
\(939\) 2.53590i 0.0827559i
\(940\) 0 0
\(941\) −0.124356 0.0717968i −0.00405388 0.00234051i 0.497972 0.867193i \(-0.334078\pi\)
−0.502026 + 0.864853i \(0.667412\pi\)
\(942\) 0 0
\(943\) −30.2533 8.10634i −0.985183 0.263979i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 31.5288 + 8.44812i 1.02455 + 0.274527i 0.731696 0.681631i \(-0.238729\pi\)
0.292852 + 0.956158i \(0.405396\pi\)
\(948\) 0 0
\(949\) 13.3301 + 7.69615i 0.432714 + 0.249828i
\(950\) 0 0
\(951\) 2.39230i 0.0775758i
\(952\) 0 0
\(953\) −2.41233 2.41233i −0.0781429 0.0781429i 0.666955 0.745098i \(-0.267597\pi\)
−0.745098 + 0.666955i \(0.767597\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.89898 18.2832i 0.158362 0.591013i
\(958\) 0 0
\(959\) 15.5885 + 3.00000i 0.503378 + 0.0968751i
\(960\) 0 0
\(961\) −13.5000 23.3827i −0.435484 0.754280i
\(962\) 0 0
\(963\) −14.2301 + 3.81294i −0.458558 + 0.122870i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −2.50026 + 2.50026i −0.0804029 + 0.0804029i −0.746164 0.665762i \(-0.768107\pi\)
0.665762 + 0.746164i \(0.268107\pi\)
\(968\) 0 0
\(969\) −28.4904 + 49.3468i −0.915243 + 1.58525i
\(970\) 0 0
\(971\) −1.22243 + 0.705771i −0.0392297 + 0.0226493i −0.519487 0.854479i \(-0.673877\pi\)
0.480257 + 0.877128i \(0.340543\pi\)
\(972\) 0 0
\(973\) 19.4758 40.1206i 0.624365 1.28621i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 6.36396 + 23.7506i 0.203601 + 0.759850i 0.989871 + 0.141967i \(0.0453427\pi\)
−0.786270 + 0.617883i \(0.787991\pi\)
\(978\) 0 0
\(979\) −19.8564 −0.634614
\(980\) 0 0
\(981\) −13.9282 −0.444693
\(982\) 0 0
\(983\) 0.642736 + 2.39872i 0.0205001 + 0.0765073i 0.975418 0.220362i \(-0.0707240\pi\)
−0.954918 + 0.296870i \(0.904057\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 11.7806 24.2683i 0.374980 0.772467i
\(988\) 0 0
\(989\) −3.46410 + 2.00000i −0.110152 + 0.0635963i
\(990\) 0 0
\(991\) −10.6603 + 18.4641i −0.338634 + 0.586532i −0.984176 0.177193i \(-0.943298\pi\)
0.645542 + 0.763725i \(0.276632\pi\)
\(992\) 0 0
\(993\) −9.43262 + 9.43262i −0.299335 + 0.299335i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 33.6687 9.02150i 1.06630 0.285714i 0.317327 0.948316i \(-0.397215\pi\)
0.748971 + 0.662602i \(0.230548\pi\)
\(998\) 0 0
\(999\) 5.59808 + 9.69615i 0.177115 + 0.306773i
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 2100.2.ce.b.493.2 yes 8
5.2 odd 4 2100.2.ce.a.157.2 yes 8
5.3 odd 4 2100.2.ce.a.157.1 8
5.4 even 2 inner 2100.2.ce.b.493.1 yes 8
7.5 odd 6 2100.2.ce.a.1993.2 yes 8
35.12 even 12 inner 2100.2.ce.b.1657.2 yes 8
35.19 odd 6 2100.2.ce.a.1993.1 yes 8
35.33 even 12 inner 2100.2.ce.b.1657.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.ce.a.157.1 8 5.3 odd 4
2100.2.ce.a.157.2 yes 8 5.2 odd 4
2100.2.ce.a.1993.1 yes 8 35.19 odd 6
2100.2.ce.a.1993.2 yes 8 7.5 odd 6
2100.2.ce.b.493.1 yes 8 5.4 even 2 inner
2100.2.ce.b.493.2 yes 8 1.1 even 1 trivial
2100.2.ce.b.1657.1 yes 8 35.33 even 12 inner
2100.2.ce.b.1657.2 yes 8 35.12 even 12 inner