Properties

Label 2100.2.ce.a.493.1
Level $2100$
Weight $2$
Character 2100.493
Analytic conductor $16.769$
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] = [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 493.1
Root \(-0.258819 + 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2100.493
Dual form 2100.2.ce.a.1657.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.258819 - 0.965926i) q^{3} +(-1.15539 + 2.38014i) q^{7} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\) \(q+(-0.258819 - 0.965926i) q^{3} +(-1.15539 + 2.38014i) q^{7} +(-0.866025 + 0.500000i) q^{9} +(0.366025 - 0.633975i) q^{11} +(0.707107 - 0.707107i) q^{13} +(-3.15660 + 0.845807i) q^{17} +(-0.767949 - 1.33013i) q^{19} +(2.59808 + 0.500000i) q^{21} +(0.138701 - 0.517638i) q^{23} +(0.707107 + 0.707107i) q^{27} -6.92820i q^{29} +(1.73205 + 1.00000i) q^{31} +(-0.707107 - 0.189469i) q^{33} +(-0.776457 - 0.208051i) q^{37} +(-0.866025 - 0.500000i) q^{39} -6.19615i q^{41} +(-5.27792 - 5.27792i) q^{43} +(-0.0507680 + 0.189469i) q^{47} +(-4.33013 - 5.50000i) q^{49} +(1.63397 + 2.83013i) q^{51} +(-5.08845 + 1.36345i) q^{53} +(-1.08604 + 1.08604i) q^{57} +(6.36603 - 11.0263i) q^{59} +(5.42820 - 3.13397i) q^{61} +(-0.189469 - 2.63896i) q^{63} +(1.81173 + 6.76148i) q^{67} -0.535898 q^{69} -8.00000 q^{71} +(-1.39563 - 5.20857i) q^{73} +(1.08604 + 1.60368i) q^{77} +(4.96410 - 2.86603i) q^{79} +(0.500000 - 0.866025i) q^{81} +(0.896575 - 0.896575i) q^{83} +(-6.69213 + 1.79315i) q^{87} +(-5.36603 - 9.29423i) q^{89} +(0.866025 + 2.50000i) q^{91} +(0.517638 - 1.93185i) q^{93} +(-7.39924 - 7.39924i) q^{97} +0.732051i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) 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
\(11\) 0.366025 0.633975i 0.110361 0.191151i −0.805555 0.592521i \(-0.798133\pi\)
0.915916 + 0.401371i \(0.131466\pi\)
\(12\) 0 0
\(13\) 0.707107 0.707107i 0.196116 0.196116i −0.602217 0.798333i \(-0.705716\pi\)
0.798333 + 0.602217i \(0.205716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.15660 + 0.845807i −0.765587 + 0.205138i −0.620421 0.784269i \(-0.713038\pi\)
−0.145166 + 0.989407i \(0.546372\pi\)
\(18\) 0 0
\(19\) −0.767949 1.33013i −0.176180 0.305152i 0.764389 0.644755i \(-0.223041\pi\)
−0.940569 + 0.339603i \(0.889707\pi\)
\(20\) 0 0
\(21\) 2.59808 + 0.500000i 0.566947 + 0.109109i
\(22\) 0 0
\(23\) 0.138701 0.517638i 0.0289211 0.107935i −0.949956 0.312382i \(-0.898873\pi\)
0.978878 + 0.204447i \(0.0655397\pi\)
\(24\) 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 \(-0.777578\pi\)
0.765641 0.643268i \(-0.222422\pi\)
\(30\) 0 0
\(31\) 1.73205 + 1.00000i 0.311086 + 0.179605i 0.647412 0.762140i \(-0.275851\pi\)
−0.336327 + 0.941745i \(0.609185\pi\)
\(32\) 0 0
\(33\) −0.707107 0.189469i −0.123091 0.0329823i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.776457 0.208051i −0.127649 0.0342034i 0.194429 0.980917i \(-0.437715\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(38\) 0 0
\(39\) −0.866025 0.500000i −0.138675 0.0800641i
\(40\) 0 0
\(41\) 6.19615i 0.967676i −0.875157 0.483838i \(-0.839242\pi\)
0.875157 0.483838i \(-0.160758\pi\)
\(42\) 0 0
\(43\) −5.27792 5.27792i −0.804875 0.804875i 0.178978 0.983853i \(-0.442721\pi\)
−0.983853 + 0.178978i \(0.942721\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.0507680 + 0.189469i −0.00740527 + 0.0276368i −0.969530 0.244974i \(-0.921221\pi\)
0.962124 + 0.272611i \(0.0878872\pi\)
\(48\) 0 0
\(49\) −4.33013 5.50000i −0.618590 0.785714i
\(50\) 0 0
\(51\) 1.63397 + 2.83013i 0.228802 + 0.396297i
\(52\) 0 0
\(53\) −5.08845 + 1.36345i −0.698952 + 0.187284i −0.590761 0.806846i \(-0.701172\pi\)
−0.108191 + 0.994130i \(0.534506\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.08604 + 1.08604i −0.143850 + 0.143850i
\(58\) 0 0
\(59\) 6.36603 11.0263i 0.828786 1.43550i −0.0702053 0.997533i \(-0.522365\pi\)
0.898991 0.437967i \(-0.144301\pi\)
\(60\) 0 0
\(61\) 5.42820 3.13397i 0.695010 0.401264i −0.110476 0.993879i \(-0.535238\pi\)
0.805486 + 0.592614i \(0.201904\pi\)
\(62\) 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
\(69\) −0.535898 −0.0645146
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) −1.39563 5.20857i −0.163346 0.609617i −0.998245 0.0592135i \(-0.981141\pi\)
0.834899 0.550403i \(-0.185526\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.08604 + 1.60368i 0.123766 + 0.182757i
\(78\) 0 0
\(79\) 4.96410 2.86603i 0.558505 0.322453i −0.194040 0.980994i \(-0.562159\pi\)
0.752545 + 0.658541i \(0.228826\pi\)
\(80\) 0 0
\(81\) 0.500000 0.866025i 0.0555556 0.0962250i
\(82\) 0 0
\(83\) 0.896575 0.896575i 0.0984119 0.0984119i −0.656187 0.754599i \(-0.727832\pi\)
0.754599 + 0.656187i \(0.227832\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −6.69213 + 1.79315i −0.717472 + 0.192246i
\(88\) 0 0
\(89\) −5.36603 9.29423i −0.568798 0.985186i −0.996685 0.0813544i \(-0.974075\pi\)
0.427888 0.903832i \(-0.359258\pi\)
\(90\) 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
\(97\) −7.39924 7.39924i −0.751279 0.751279i 0.223439 0.974718i \(-0.428272\pi\)
−0.974718 + 0.223439i \(0.928272\pi\)
\(98\) 0 0
\(99\) 0.732051i 0.0735739i
\(100\) 0 0
\(101\) −2.83013 1.63397i −0.281608 0.162587i 0.352543 0.935796i \(-0.385317\pi\)
−0.634151 + 0.773209i \(0.718650\pi\)
\(102\) 0 0
\(103\) −12.2289 3.27671i −1.20495 0.322864i −0.400169 0.916441i \(-0.631049\pi\)
−0.804777 + 0.593577i \(0.797715\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 10.8840 + 2.91636i 1.05220 + 0.281935i 0.743159 0.669115i \(-0.233327\pi\)
0.309037 + 0.951050i \(0.399993\pi\)
\(108\) 0 0
\(109\) 0.0621778 + 0.0358984i 0.00595556 + 0.00343844i 0.502975 0.864301i \(-0.332239\pi\)
−0.497019 + 0.867740i \(0.665572\pi\)
\(110\) 0 0
\(111\) 0.803848i 0.0762978i
\(112\) 0 0
\(113\) −10.9348 10.9348i −1.02866 1.02866i −0.999577 0.0290796i \(-0.990742\pi\)
−0.0290796 0.999577i \(-0.509258\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.258819 + 0.965926i −0.0239278 + 0.0892999i
\(118\) 0 0
\(119\) 1.63397 8.49038i 0.149786 0.778312i
\(120\) 0 0
\(121\) 5.23205 + 9.06218i 0.475641 + 0.823834i
\(122\) 0 0
\(123\) −5.98502 + 1.60368i −0.539651 + 0.144599i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −9.84873 + 9.84873i −0.873933 + 0.873933i −0.992898 0.118965i \(-0.962042\pi\)
0.118965 + 0.992898i \(0.462042\pi\)
\(128\) 0 0
\(129\) −3.73205 + 6.46410i −0.328589 + 0.569132i
\(130\) 0 0
\(131\) −2.66025 + 1.53590i −0.232427 + 0.134192i −0.611691 0.791096i \(-0.709511\pi\)
0.379264 + 0.925289i \(0.376177\pi\)
\(132\) 0 0
\(133\) 4.05317 0.291005i 0.351455 0.0252333i
\(134\) 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
\(139\) −10.8564 −0.920828 −0.460414 0.887704i \(-0.652299\pi\)
−0.460414 + 0.887704i \(0.652299\pi\)
\(140\) 0 0
\(141\) 0.196152 0.0165190
\(142\) 0 0
\(143\) −0.189469 0.707107i −0.0158442 0.0591312i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −4.19187 + 5.60609i −0.345740 + 0.462382i
\(148\) 0 0
\(149\) 13.0981 7.56218i 1.07304 0.619518i 0.144027 0.989574i \(-0.453995\pi\)
0.929009 + 0.370056i \(0.120662\pi\)
\(150\) 0 0
\(151\) −3.33013 + 5.76795i −0.271002 + 0.469389i −0.969119 0.246595i \(-0.920688\pi\)
0.698117 + 0.715984i \(0.254022\pi\)
\(152\) 0 0
\(153\) 2.31079 2.31079i 0.186816 0.186816i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.3502 + 3.84512i −1.14527 + 0.306874i −0.781068 0.624446i \(-0.785325\pi\)
−0.364202 + 0.931320i \(0.618658\pi\)
\(158\) 0 0
\(159\) 2.63397 + 4.56218i 0.208888 + 0.361804i
\(160\) 0 0
\(161\) 1.07180 + 0.928203i 0.0844694 + 0.0731527i
\(162\) 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
\(167\) 12.4877 + 12.4877i 0.966326 + 0.966326i 0.999451 0.0331251i \(-0.0105460\pi\)
−0.0331251 + 0.999451i \(0.510546\pi\)
\(168\) 0 0
\(169\) 12.0000i 0.923077i
\(170\) 0 0
\(171\) 1.33013 + 0.767949i 0.101717 + 0.0587265i
\(172\) 0 0
\(173\) −18.8009 5.03768i −1.42940 0.383008i −0.540594 0.841283i \(-0.681801\pi\)
−0.888810 + 0.458276i \(0.848467\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −12.2982 3.29530i −0.924391 0.247690i
\(178\) 0 0
\(179\) 2.19615 + 1.26795i 0.164148 + 0.0947710i 0.579824 0.814742i \(-0.303121\pi\)
−0.415675 + 0.909513i \(0.636455\pi\)
\(180\) 0 0
\(181\) 14.3923i 1.06977i −0.844924 0.534886i \(-0.820355\pi\)
0.844924 0.534886i \(-0.179645\pi\)
\(182\) 0 0
\(183\) −4.43211 4.43211i −0.327631 0.327631i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −0.619174 + 2.31079i −0.0452785 + 0.168982i
\(188\) 0 0
\(189\) −2.50000 + 0.866025i −0.181848 + 0.0629941i
\(190\) 0 0
\(191\) 6.90192 + 11.9545i 0.499406 + 0.864996i 1.00000 0.000686128i \(-0.000218401\pi\)
−0.500594 + 0.865682i \(0.666885\pi\)
\(192\) 0 0
\(193\) −20.5940 + 5.51815i −1.48239 + 0.397205i −0.907160 0.420786i \(-0.861754\pi\)
−0.575231 + 0.817991i \(0.695088\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.3843 13.3843i 0.953589 0.953589i −0.0453807 0.998970i \(-0.514450\pi\)
0.998970 + 0.0453807i \(0.0144501\pi\)
\(198\) 0 0
\(199\) −0.767949 + 1.33013i −0.0544385 + 0.0942902i −0.891960 0.452113i \(-0.850670\pi\)
0.837522 + 0.546404i \(0.184004\pi\)
\(200\) 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
\(207\) 0.138701 + 0.517638i 0.00964037 + 0.0359783i
\(208\) 0 0
\(209\) −1.12436 −0.0777733
\(210\) 0 0
\(211\) −12.2679 −0.844560 −0.422280 0.906465i \(-0.638770\pi\)
−0.422280 + 0.906465i \(0.638770\pi\)
\(212\) 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
\(219\) −4.66987 + 2.69615i −0.315561 + 0.182189i
\(220\) 0 0
\(221\) −1.63397 + 2.83013i −0.109913 + 0.190375i
\(222\) 0 0
\(223\) 13.1948 13.1948i 0.883589 0.883589i −0.110309 0.993897i \(-0.535184\pi\)
0.993897 + 0.110309i \(0.0351840\pi\)
\(224\) 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
\(229\) 4.13397 + 7.16025i 0.273181 + 0.473163i 0.969675 0.244400i \(-0.0785910\pi\)
−0.696494 + 0.717563i \(0.745258\pi\)
\(230\) 0 0
\(231\) 1.26795 1.46410i 0.0834249 0.0963308i
\(232\) 0 0
\(233\) −1.88108 + 7.02030i −0.123234 + 0.459915i −0.999771 0.0214199i \(-0.993181\pi\)
0.876537 + 0.481335i \(0.159848\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.05317 4.05317i −0.263282 0.263282i
\(238\) 0 0
\(239\) 11.3205i 0.732263i −0.930563 0.366131i \(-0.880682\pi\)
0.930563 0.366131i \(-0.119318\pi\)
\(240\) 0 0
\(241\) 0.696152 + 0.401924i 0.0448431 + 0.0258902i 0.522254 0.852790i \(-0.325091\pi\)
−0.477411 + 0.878680i \(0.658425\pi\)
\(242\) 0 0
\(243\) −0.965926 0.258819i −0.0619642 0.0166032i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.48356 0.397520i −0.0943969 0.0252936i
\(248\) 0 0
\(249\) −1.09808 0.633975i −0.0695878 0.0401765i
\(250\) 0 0
\(251\) 20.5885i 1.29953i −0.760134 0.649766i \(-0.774867\pi\)
0.760134 0.649766i \(-0.225133\pi\)
\(252\) 0 0
\(253\) −0.277401 0.277401i −0.0174401 0.0174401i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.08656 22.7153i 0.379669 1.41694i −0.466732 0.884399i \(-0.654569\pi\)
0.846401 0.532546i \(-0.178765\pi\)
\(258\) 0 0
\(259\) 1.39230 1.60770i 0.0865136 0.0998973i
\(260\) 0 0
\(261\) 3.46410 + 6.00000i 0.214423 + 0.371391i
\(262\) 0 0
\(263\) 22.1469 5.93426i 1.36564 0.365922i 0.499755 0.866167i \(-0.333423\pi\)
0.865884 + 0.500245i \(0.166757\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −7.58871 + 7.58871i −0.464421 + 0.464421i
\(268\) 0 0
\(269\) 11.2942 19.5622i 0.688621 1.19273i −0.283663 0.958924i \(-0.591550\pi\)
0.972284 0.233803i \(-0.0751171\pi\)
\(270\) 0 0
\(271\) −0.928203 + 0.535898i −0.0563843 + 0.0325535i −0.527927 0.849290i \(-0.677031\pi\)
0.471543 + 0.881843i \(0.343697\pi\)
\(272\) 0 0
\(273\) 2.19067 1.48356i 0.132585 0.0897894i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −2.79624 10.4357i −0.168010 0.627021i −0.997637 0.0687037i \(-0.978114\pi\)
0.829627 0.558318i \(-0.188553\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) −33.1244 −1.97603 −0.988017 0.154347i \(-0.950673\pi\)
−0.988017 + 0.154347i \(0.950673\pi\)
\(282\) 0 0
\(283\) 6.38254 + 23.8200i 0.379403 + 1.41595i 0.846804 + 0.531905i \(0.178524\pi\)
−0.467401 + 0.884045i \(0.654810\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 14.7477 + 7.15900i 0.870530 + 0.422582i
\(288\) 0 0
\(289\) −5.47372 + 3.16025i −0.321984 + 0.185897i
\(290\) 0 0
\(291\) −5.23205 + 9.06218i −0.306708 + 0.531234i
\(292\) 0 0
\(293\) −3.10583 + 3.10583i −0.181444 + 0.181444i −0.791985 0.610541i \(-0.790952\pi\)
0.610541 + 0.791985i \(0.290952\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0.707107 0.189469i 0.0410305 0.0109941i
\(298\) 0 0
\(299\) −0.267949 0.464102i −0.0154959 0.0268397i
\(300\) 0 0
\(301\) 18.6603 6.46410i 1.07556 0.372585i
\(302\) 0 0
\(303\) −0.845807 + 3.15660i −0.0485904 + 0.181342i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 6.69213 + 6.69213i 0.381940 + 0.381940i 0.871801 0.489861i \(-0.162952\pi\)
−0.489861 + 0.871801i \(0.662952\pi\)
\(308\) 0 0
\(309\) 12.6603i 0.720217i
\(310\) 0 0
\(311\) −5.19615 3.00000i −0.294647 0.170114i 0.345389 0.938460i \(-0.387747\pi\)
−0.640036 + 0.768345i \(0.721080\pi\)
\(312\) 0 0
\(313\) 9.14162 + 2.44949i 0.516715 + 0.138453i 0.507747 0.861506i \(-0.330479\pi\)
0.00896828 + 0.999960i \(0.497145\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 17.7656 + 4.76028i 0.997816 + 0.267364i 0.720530 0.693424i \(-0.243899\pi\)
0.277286 + 0.960788i \(0.410565\pi\)
\(318\) 0 0
\(319\) −4.39230 2.53590i −0.245922 0.141983i
\(320\) 0 0
\(321\) 11.2679i 0.628916i
\(322\) 0 0
\(323\) 3.54914 + 3.54914i 0.197479 + 0.197479i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.0185824 0.0693504i 0.00102761 0.00383508i
\(328\) 0 0
\(329\) −0.392305 0.339746i −0.0216285 0.0187308i
\(330\) 0 0
\(331\) 15.3301 + 26.5526i 0.842620 + 1.45946i 0.887672 + 0.460476i \(0.152321\pi\)
−0.0450522 + 0.998985i \(0.514345\pi\)
\(332\) 0 0
\(333\) 0.776457 0.208051i 0.0425496 0.0114011i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.14162 9.14162i 0.497976 0.497976i −0.412832 0.910807i \(-0.635460\pi\)
0.910807 + 0.412832i \(0.135460\pi\)
\(338\) 0 0
\(339\) −7.73205 + 13.3923i −0.419947 + 0.727370i
\(340\) 0 0
\(341\) 1.26795 0.732051i 0.0686633 0.0396428i
\(342\) 0 0
\(343\) 18.0938 3.95164i 0.976972 0.213368i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.46309 9.19239i −0.132226 0.493473i 0.867768 0.496969i \(-0.165554\pi\)
−0.999994 + 0.00349604i \(0.998887\pi\)
\(348\) 0 0
\(349\) 0.928203 0.0496856 0.0248428 0.999691i \(-0.492091\pi\)
0.0248428 + 0.999691i \(0.492091\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 1.46498 + 5.46739i 0.0779731 + 0.291000i 0.993891 0.110366i \(-0.0352023\pi\)
−0.915918 + 0.401366i \(0.868536\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −8.62398 + 0.619174i −0.456430 + 0.0327702i
\(358\) 0 0
\(359\) 8.70577 5.02628i 0.459473 0.265277i −0.252350 0.967636i \(-0.581203\pi\)
0.711823 + 0.702359i \(0.247870\pi\)
\(360\) 0 0
\(361\) 8.32051 14.4115i 0.437921 0.758502i
\(362\) 0 0
\(363\) 7.39924 7.39924i 0.388359 0.388359i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.517638 + 0.138701i −0.0270205 + 0.00724012i −0.272304 0.962211i \(-0.587786\pi\)
0.245283 + 0.969451i \(0.421119\pi\)
\(368\) 0 0
\(369\) 3.09808 + 5.36603i 0.161279 + 0.279344i
\(370\) 0 0
\(371\) 2.63397 13.6865i 0.136749 0.710569i
\(372\) 0 0
\(373\) −5.96644 + 22.2671i −0.308931 + 1.15294i 0.620578 + 0.784145i \(0.286898\pi\)
−0.929509 + 0.368800i \(0.879769\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.89898 4.89898i −0.252310 0.252310i
\(378\) 0 0
\(379\) 35.4449i 1.82068i 0.413861 + 0.910340i \(0.364180\pi\)
−0.413861 + 0.910340i \(0.635820\pi\)
\(380\) 0 0
\(381\) 12.0622 + 6.96410i 0.617964 + 0.356782i
\(382\) 0 0
\(383\) 24.8367 + 6.65497i 1.26909 + 0.340053i 0.829684 0.558233i \(-0.188521\pi\)
0.439411 + 0.898286i \(0.355187\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 7.20977 + 1.93185i 0.366493 + 0.0982015i
\(388\) 0 0
\(389\) 24.4186 + 14.0981i 1.23807 + 0.714801i 0.968700 0.248235i \(-0.0798507\pi\)
0.269372 + 0.963036i \(0.413184\pi\)
\(390\) 0 0
\(391\) 1.75129i 0.0885665i
\(392\) 0 0
\(393\) 2.17209 + 2.17209i 0.109567 + 0.109567i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.48477 13.0053i 0.174895 0.652718i −0.821674 0.569958i \(-0.806960\pi\)
0.996569 0.0827608i \(-0.0263737\pi\)
\(398\) 0 0
\(399\) −1.33013 3.83975i −0.0665896 0.192228i
\(400\) 0 0
\(401\) −2.29423 3.97372i −0.114568 0.198438i 0.803039 0.595927i \(-0.203215\pi\)
−0.917607 + 0.397489i \(0.869882\pi\)
\(402\) 0 0
\(403\) 1.93185 0.517638i 0.0962324 0.0257854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.416102 + 0.416102i −0.0206254 + 0.0206254i
\(408\) 0 0
\(409\) −9.79423 + 16.9641i −0.484293 + 0.838821i −0.999837 0.0180424i \(-0.994257\pi\)
0.515544 + 0.856863i \(0.327590\pi\)
\(410\) 0 0
\(411\) −5.19615 + 3.00000i −0.256307 + 0.147979i
\(412\) 0 0
\(413\) 18.8888 + 27.8917i 0.929458 + 1.37246i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 2.80984 + 10.4865i 0.137599 + 0.513525i
\(418\) 0 0
\(419\) −38.0526 −1.85899 −0.929495 0.368836i \(-0.879756\pi\)
−0.929495 + 0.368836i \(0.879756\pi\)
\(420\) 0 0
\(421\) −10.6077 −0.516987 −0.258494 0.966013i \(-0.583226\pi\)
−0.258494 + 0.966013i \(0.583226\pi\)
\(422\) 0 0
\(423\) −0.0507680 0.189469i −0.00246842 0.00921228i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.18758 + 16.5409i 0.0574710 + 0.800468i
\(428\) 0 0
\(429\) −0.633975 + 0.366025i −0.0306086 + 0.0176719i
\(430\) 0 0
\(431\) −12.4641 + 21.5885i −0.600375 + 1.03988i 0.392390 + 0.919799i \(0.371649\pi\)
−0.992764 + 0.120080i \(0.961685\pi\)
\(432\) 0 0
\(433\) 26.4911 26.4911i 1.27308 1.27308i 0.328620 0.944462i \(-0.393417\pi\)
0.944462 0.328620i \(-0.106583\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.795040 + 0.213030i −0.0380319 + 0.0101906i
\(438\) 0 0
\(439\) 13.8923 + 24.0622i 0.663044 + 1.14843i 0.979812 + 0.199922i \(0.0640689\pi\)
−0.316768 + 0.948503i \(0.602598\pi\)
\(440\) 0 0
\(441\) 6.50000 + 2.59808i 0.309524 + 0.123718i
\(442\) 0 0
\(443\) −10.1397 + 37.8420i −0.481753 + 1.79793i 0.112503 + 0.993651i \(0.464113\pi\)
−0.594256 + 0.804276i \(0.702553\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −10.6945 10.6945i −0.505834 0.505834i
\(448\) 0 0
\(449\) 27.6603i 1.30537i 0.757630 + 0.652684i \(0.226357\pi\)
−0.757630 + 0.652684i \(0.773643\pi\)
\(450\) 0 0
\(451\) −3.92820 2.26795i −0.184972 0.106794i
\(452\) 0 0
\(453\) 6.43331 + 1.72380i 0.302263 + 0.0809912i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.84752 2.10274i −0.367092 0.0983619i 0.0705575 0.997508i \(-0.477522\pi\)
−0.437649 + 0.899146i \(0.644189\pi\)
\(458\) 0 0
\(459\) −2.83013 1.63397i −0.132099 0.0762674i
\(460\) 0 0
\(461\) 37.7128i 1.75646i −0.478238 0.878230i \(-0.658724\pi\)
0.478238 0.878230i \(-0.341276\pi\)
\(462\) 0 0
\(463\) −12.1967 12.1967i −0.566828 0.566828i 0.364411 0.931238i \(-0.381270\pi\)
−0.931238 + 0.364411i \(0.881270\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.0371647 + 0.138701i −0.00171978 + 0.00641830i −0.966780 0.255608i \(-0.917724\pi\)
0.965061 + 0.262027i \(0.0843909\pi\)
\(468\) 0 0
\(469\) −18.1865 3.50000i −0.839776 0.161615i
\(470\) 0 0
\(471\) 7.42820 + 12.8660i 0.342274 + 0.592835i
\(472\) 0 0
\(473\) −5.27792 + 1.41421i −0.242679 + 0.0650256i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.72500 3.72500i 0.170556 0.170556i
\(478\) 0 0
\(479\) 16.2224 28.0981i 0.741222 1.28383i −0.210717 0.977547i \(-0.567580\pi\)
0.951939 0.306287i \(-0.0990867\pi\)
\(480\) 0 0
\(481\) −0.696152 + 0.401924i −0.0317418 + 0.0183261i
\(482\) 0 0
\(483\) 0.619174 1.27551i 0.0281734 0.0580378i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −2.41233 9.00292i −0.109313 0.407961i 0.889486 0.456963i \(-0.151063\pi\)
−0.998799 + 0.0490015i \(0.984396\pi\)
\(488\) 0 0
\(489\) −3.00000 −0.135665
\(490\) 0 0
\(491\) −6.39230 −0.288481 −0.144240 0.989543i \(-0.546074\pi\)
−0.144240 + 0.989543i \(0.546074\pi\)
\(492\) 0 0
\(493\) 5.85993 + 21.8695i 0.263918 + 0.984955i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.24316 19.0411i 0.414612 0.854111i
\(498\) 0 0
\(499\) −7.96410 + 4.59808i −0.356522 + 0.205838i −0.667554 0.744561i \(-0.732659\pi\)
0.311032 + 0.950399i \(0.399325\pi\)
\(500\) 0 0
\(501\) 8.83013 15.2942i 0.394501 0.683296i
\(502\) 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
\(509\) 7.26795 + 12.5885i 0.322146 + 0.557974i 0.980931 0.194358i \(-0.0622624\pi\)
−0.658784 + 0.752332i \(0.728929\pi\)
\(510\) 0 0
\(511\) 14.0096 + 2.69615i 0.619749 + 0.119271i
\(512\) 0 0
\(513\) 0.397520 1.48356i 0.0175509 0.0655009i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0.101536 + 0.101536i 0.00446555 + 0.00446555i
\(518\) 0 0
\(519\) 19.4641i 0.854379i
\(520\) 0 0
\(521\) −2.32051 1.33975i −0.101663 0.0586953i 0.448306 0.893880i \(-0.352027\pi\)
−0.549970 + 0.835185i \(0.685361\pi\)
\(522\) 0 0
\(523\) 14.7985 + 3.96524i 0.647092 + 0.173388i 0.567414 0.823433i \(-0.307944\pi\)
0.0796783 + 0.996821i \(0.474611\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.31319 1.69161i −0.275007 0.0736879i
\(528\) 0 0
\(529\) 19.6699 + 11.3564i 0.855212 + 0.493757i
\(530\) 0 0
\(531\) 12.7321i 0.552524i
\(532\) 0 0
\(533\) −4.38134 4.38134i −0.189777 0.189777i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0.656339 2.44949i 0.0283231 0.105703i
\(538\) 0 0
\(539\) −5.07180 + 0.732051i −0.218458 + 0.0315317i
\(540\) 0 0
\(541\) 1.83975 + 3.18653i 0.0790969 + 0.137000i 0.902861 0.429934i \(-0.141463\pi\)
−0.823764 + 0.566933i \(0.808130\pi\)
\(542\) 0 0
\(543\) −13.9019 + 3.72500i −0.596588 + 0.159855i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 9.04008 9.04008i 0.386526 0.386526i −0.486920 0.873446i \(-0.661880\pi\)
0.873446 + 0.486920i \(0.161880\pi\)
\(548\) 0 0
\(549\) −3.13397 + 5.42820i −0.133755 + 0.231670i
\(550\) 0 0
\(551\) −9.21539 + 5.32051i −0.392589 + 0.226661i
\(552\) 0 0
\(553\) 1.08604 + 15.1266i 0.0461833 + 0.643250i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.24264 15.8338i −0.179766 0.670898i −0.995691 0.0927384i \(-0.970438\pi\)
0.815924 0.578159i \(-0.196229\pi\)
\(558\) 0 0
\(559\) −7.46410 −0.315698
\(560\) 0 0
\(561\) 2.39230 0.101003
\(562\) 0 0
\(563\) 1.42782 + 5.32868i 0.0601753 + 0.224577i 0.989465 0.144776i \(-0.0462460\pi\)
−0.929289 + 0.369353i \(0.879579\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.48356 + 2.19067i 0.0623038 + 0.0919995i
\(568\) 0 0
\(569\) 6.41858 3.70577i 0.269081 0.155354i −0.359389 0.933188i \(-0.617015\pi\)
0.628470 + 0.777834i \(0.283682\pi\)
\(570\) 0 0
\(571\) −19.4545 + 33.6962i −0.814145 + 1.41014i 0.0957956 + 0.995401i \(0.469460\pi\)
−0.909940 + 0.414739i \(0.863873\pi\)
\(572\) 0 0
\(573\) 9.76079 9.76079i 0.407763 0.407763i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 9.65926 2.58819i 0.402120 0.107748i −0.0520899 0.998642i \(-0.516588\pi\)
0.454210 + 0.890895i \(0.349922\pi\)
\(578\) 0 0
\(579\) 10.6603 + 18.4641i 0.443025 + 0.767342i
\(580\) 0 0
\(581\) 1.09808 + 3.16987i 0.0455559 + 0.131508i
\(582\) 0 0
\(583\) −0.998111 + 3.72500i −0.0413376 + 0.154274i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.1817 + 18.1817i 0.750439 + 0.750439i 0.974561 0.224122i \(-0.0719514\pi\)
−0.224122 + 0.974561i \(0.571951\pi\)
\(588\) 0 0
\(589\) 3.07180i 0.126571i
\(590\) 0 0
\(591\) −16.3923 9.46410i −0.674289 0.389301i
\(592\) 0 0
\(593\) −26.2509 7.03390i −1.07799 0.288848i −0.324221 0.945982i \(-0.605102\pi\)
−0.753774 + 0.657134i \(0.771769\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.48356 + 0.397520i 0.0607182 + 0.0162694i
\(598\) 0 0
\(599\) 20.9545 + 12.0981i 0.856177 + 0.494314i 0.862730 0.505665i \(-0.168753\pi\)
−0.00655324 + 0.999979i \(0.502086\pi\)
\(600\) 0 0
\(601\) 32.5167i 1.32638i −0.748450 0.663191i \(-0.769202\pi\)
0.748450 0.663191i \(-0.230798\pi\)
\(602\) 0 0
\(603\) −4.94975 4.94975i −0.201569 0.201569i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0.688524 2.56961i 0.0279463 0.104297i −0.950544 0.310590i \(-0.899473\pi\)
0.978490 + 0.206293i \(0.0661400\pi\)
\(608\) 0 0
\(609\) 3.46410 18.0000i 0.140372 0.729397i
\(610\) 0 0
\(611\) 0.0980762 + 0.169873i 0.00396774 + 0.00687233i
\(612\) 0 0
\(613\) 0.138701 0.0371647i 0.00560207 0.00150107i −0.256017 0.966672i \(-0.582410\pi\)
0.261619 + 0.965171i \(0.415744\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 15.4548 15.4548i 0.622187 0.622187i −0.323903 0.946090i \(-0.604995\pi\)
0.946090 + 0.323903i \(0.104995\pi\)
\(618\) 0 0
\(619\) −8.80385 + 15.2487i −0.353857 + 0.612897i −0.986922 0.161201i \(-0.948463\pi\)
0.633065 + 0.774099i \(0.281797\pi\)
\(620\) 0 0
\(621\) 0.464102 0.267949i 0.0186238 0.0107524i
\(622\) 0 0
\(623\) 28.3214 2.03339i 1.13467 0.0814660i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0.291005 + 1.08604i 0.0116216 + 0.0433724i
\(628\) 0 0
\(629\) 2.62693 0.104743
\(630\) 0 0
\(631\) −37.1962 −1.48076 −0.740378 0.672191i \(-0.765353\pi\)
−0.740378 + 0.672191i \(0.765353\pi\)
\(632\) 0 0
\(633\) 3.17518 + 11.8499i 0.126202 + 0.470992i
\(634\) 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
\(641\) −8.58846 + 14.8756i −0.339224 + 0.587553i −0.984287 0.176577i \(-0.943498\pi\)
0.645063 + 0.764129i \(0.276831\pi\)
\(642\) 0 0
\(643\) −8.09274 + 8.09274i −0.319147 + 0.319147i −0.848439 0.529293i \(-0.822457\pi\)
0.529293 + 0.848439i \(0.322457\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.19239 + 2.46309i −0.361390 + 0.0968342i −0.434945 0.900457i \(-0.643232\pi\)
0.0735550 + 0.997291i \(0.476566\pi\)
\(648\) 0 0
\(649\) −4.66025 8.07180i −0.182931 0.316846i
\(650\) 0 0
\(651\) 4.00000 + 3.46410i 0.156772 + 0.135769i
\(652\) 0 0
\(653\) −7.91688 + 29.5462i −0.309811 + 1.15623i 0.618913 + 0.785460i \(0.287573\pi\)
−0.928724 + 0.370772i \(0.879093\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 3.81294 + 3.81294i 0.148757 + 0.148757i
\(658\) 0 0
\(659\) 11.6603i 0.454219i 0.973869 + 0.227110i \(0.0729275\pi\)
−0.973869 + 0.227110i \(0.927072\pi\)
\(660\) 0 0
\(661\) 25.6244 + 14.7942i 0.996672 + 0.575429i 0.907262 0.420566i \(-0.138169\pi\)
0.0894100 + 0.995995i \(0.471502\pi\)
\(662\) 0 0
\(663\) 3.15660 + 0.845807i 0.122592 + 0.0328484i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.58630 0.960947i −0.138862 0.0372080i
\(668\) 0 0
\(669\) −16.1603 9.33013i −0.624792 0.360724i
\(670\) 0 0
\(671\) 4.58846i 0.177135i
\(672\) 0 0
\(673\) −5.84632 5.84632i −0.225359 0.225359i 0.585392 0.810751i \(-0.300941\pi\)
−0.810751 + 0.585392i \(0.800941\pi\)
\(674\) 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
\(679\) 26.1603 9.06218i 1.00394 0.347774i
\(680\) 0 0
\(681\) −6.92820 12.0000i −0.265489 0.459841i
\(682\) 0 0
\(683\) 14.2301 3.81294i 0.544498 0.145898i 0.0239233 0.999714i \(-0.492384\pi\)
0.520575 + 0.853816i \(0.325718\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 5.84632 5.84632i 0.223051 0.223051i
\(688\) 0 0
\(689\) −2.63397 + 4.56218i −0.100346 + 0.173805i
\(690\) 0 0
\(691\) −30.5263 + 17.6244i −1.16127 + 0.670462i −0.951609 0.307311i \(-0.900571\pi\)
−0.209665 + 0.977773i \(0.567237\pi\)
\(692\) 0 0
\(693\) −1.74238 0.845807i −0.0661877 0.0321296i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.24075 + 19.5588i 0.198508 + 0.740841i
\(698\) 0 0
\(699\) 7.26795 0.274899
\(700\) 0 0
\(701\) −35.6603 −1.34687 −0.673435 0.739247i \(-0.735182\pi\)
−0.673435 + 0.739247i \(0.735182\pi\)
\(702\) 0 0
\(703\) 0.319545 + 1.19256i 0.0120519 + 0.0449782i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.15900 4.84821i 0.269242 0.182336i
\(708\) 0 0
\(709\) 28.4545 16.4282i 1.06863 0.616974i 0.140824 0.990035i \(-0.455025\pi\)
0.927807 + 0.373061i \(0.121692\pi\)
\(710\) 0 0
\(711\) −2.86603 + 4.96410i −0.107484 + 0.186168i
\(712\) 0 0
\(713\) 0.757875 0.757875i 0.0283826 0.0283826i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.9348 + 2.92996i −0.408367 + 0.109421i
\(718\) 0 0
\(719\) −17.7583 30.7583i −0.662274 1.14709i −0.980017 0.198915i \(-0.936258\pi\)
0.317743 0.948177i \(-0.397075\pi\)
\(720\) 0 0
\(721\) 21.9282 25.3205i 0.816649 0.942985i
\(722\) 0 0
\(723\) 0.208051 0.776457i 0.00773750 0.0288768i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −7.91688 7.91688i −0.293621 0.293621i 0.544888 0.838509i \(-0.316572\pi\)
−0.838509 + 0.544888i \(0.816572\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 21.1244 + 12.1962i 0.781313 + 0.451091i
\(732\) 0 0
\(733\) −9.21097 2.46807i −0.340215 0.0911603i 0.0846664 0.996409i \(-0.473018\pi\)
−0.424881 + 0.905249i \(0.639684\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.94975 + 1.32628i 0.182326 + 0.0488542i
\(738\) 0 0
\(739\) −31.9641 18.4545i −1.17582 0.678859i −0.220775 0.975325i \(-0.570859\pi\)
−0.955044 + 0.296466i \(0.904192\pi\)
\(740\) 0 0
\(741\) 1.53590i 0.0564226i
\(742\) 0 0
\(743\) −29.7728 29.7728i −1.09226 1.09226i −0.995287 0.0969715i \(-0.969084\pi\)
−0.0969715 0.995287i \(-0.530916\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −0.328169 + 1.22474i −0.0120071 + 0.0448111i
\(748\) 0 0
\(749\) −19.5167 + 22.5359i −0.713123 + 0.823444i
\(750\) 0 0
\(751\) 12.0622 + 20.8923i 0.440155 + 0.762371i 0.997701 0.0677754i \(-0.0215901\pi\)
−0.557545 + 0.830146i \(0.688257\pi\)
\(752\) 0 0
\(753\) −19.8869 + 5.32868i −0.724720 + 0.194188i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 11.4016 11.4016i 0.414400 0.414400i −0.468868 0.883268i \(-0.655338\pi\)
0.883268 + 0.468868i \(0.155338\pi\)
\(758\) 0 0
\(759\) −0.196152 + 0.339746i −0.00711988 + 0.0123320i
\(760\) 0 0
\(761\) −5.49038 + 3.16987i −0.199026 + 0.114908i −0.596201 0.802835i \(-0.703324\pi\)
0.397175 + 0.917743i \(0.369991\pi\)
\(762\) 0 0
\(763\) −0.157283 + 0.106515i −0.00569403 + 0.00385611i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.29530 12.2982i −0.118986 0.444063i
\(768\) 0 0
\(769\) 13.3205 0.480350 0.240175 0.970730i \(-0.422795\pi\)
0.240175 + 0.970730i \(0.422795\pi\)
\(770\) 0 0
\(771\) −23.5167 −0.846932
\(772\) 0 0
\(773\) −11.5911 43.2586i −0.416903 1.55590i −0.780993 0.624540i \(-0.785286\pi\)
0.364089 0.931364i \(-0.381380\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −1.91327 0.928761i −0.0686382 0.0333191i
\(778\) 0 0
\(779\) −8.24167 + 4.75833i −0.295288 + 0.170485i
\(780\) 0 0
\(781\) −2.92820 + 5.07180i −0.104779 + 0.181483i
\(782\) 0 0
\(783\) 4.89898 4.89898i 0.175075 0.175075i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 30.5121 8.17569i 1.08764 0.291432i 0.329917 0.944010i \(-0.392979\pi\)
0.757722 + 0.652578i \(0.226312\pi\)
\(788\) 0 0
\(789\) −11.4641 19.8564i −0.408133 0.706907i
\(790\) 0 0
\(791\) 38.6603 13.3923i 1.37460 0.476176i
\(792\) 0 0
\(793\) 1.62226 6.05437i 0.0576083 0.214997i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 15.2789 + 15.2789i 0.541208 + 0.541208i 0.923883 0.382675i \(-0.124997\pi\)
−0.382675 + 0.923883i \(0.624997\pi\)
\(798\) 0 0
\(799\) 0.641016i 0.0226775i
\(800\) 0 0
\(801\) 9.29423 + 5.36603i 0.328395 + 0.189599i
\(802\) 0 0
\(803\) −3.81294 1.02167i −0.134556 0.0360541i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −21.8188 5.84632i −0.768057 0.205800i
\(808\) 0 0
\(809\) 38.5692 + 22.2679i 1.35602 + 0.782899i 0.989085 0.147347i \(-0.0470733\pi\)
0.366937 + 0.930246i \(0.380407\pi\)
\(810\) 0 0
\(811\) 54.8564i 1.92627i −0.269020 0.963134i \(-0.586700\pi\)
0.269020 0.963134i \(-0.413300\pi\)
\(812\) 0 0
\(813\) 0.757875 + 0.757875i 0.0265798 + 0.0265798i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −2.96713 + 11.0735i −0.103807 + 0.387412i
\(818\) 0 0
\(819\) −2.00000 1.73205i −0.0698857 0.0605228i
\(820\) 0 0
\(821\) 17.1699 + 29.7391i 0.599233 + 1.03790i 0.992935 + 0.118663i \(0.0378609\pi\)
−0.393702 + 0.919238i \(0.628806\pi\)
\(822\) 0 0
\(823\) −33.1511 + 8.88280i −1.15557 + 0.309635i −0.785197 0.619247i \(-0.787438\pi\)
−0.370377 + 0.928882i \(0.620771\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15.1774 + 15.1774i −0.527770 + 0.527770i −0.919907 0.392137i \(-0.871736\pi\)
0.392137 + 0.919907i \(0.371736\pi\)
\(828\) 0 0
\(829\) −6.72243 + 11.6436i −0.233480 + 0.404399i −0.958830 0.283981i \(-0.908345\pi\)
0.725350 + 0.688380i \(0.241678\pi\)
\(830\) 0 0
\(831\) −9.35641 + 5.40192i −0.324570 + 0.187391i
\(832\) 0 0
\(833\) 18.3204 + 13.6988i 0.634764 + 0.474636i
\(834\) 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
\(843\) 8.57321 + 31.9957i 0.295277 + 1.10199i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −27.6143 + 1.98262i −0.948840 + 0.0681236i
\(848\) 0 0
\(849\) 21.3564 12.3301i 0.732950 0.423169i
\(850\) 0 0
\(851\) −0.215390 + 0.373067i −0.00738348 + 0.0127886i
\(852\) 0 0
\(853\) 15.5563 15.5563i 0.532639 0.532639i −0.388718 0.921357i \(-0.627082\pi\)
0.921357 + 0.388718i \(0.127082\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −43.1199 + 11.5539i −1.47295 + 0.394675i −0.903941 0.427657i \(-0.859339\pi\)
−0.569007 + 0.822333i \(0.692672\pi\)
\(858\) 0 0
\(859\) 6.46410 + 11.1962i 0.220552 + 0.382008i 0.954976 0.296684i \(-0.0958807\pi\)
−0.734424 + 0.678691i \(0.762547\pi\)
\(860\) 0 0
\(861\) 3.09808 16.0981i 0.105582 0.548621i
\(862\) 0 0
\(863\) 3.47116 12.9546i 0.118160 0.440978i −0.881344 0.472475i \(-0.843361\pi\)
0.999504 + 0.0314967i \(0.0100274\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 4.46927 + 4.46927i 0.151785 + 0.151785i
\(868\) 0 0
\(869\) 4.19615i 0.142345i
\(870\) 0 0
\(871\) 6.06218 + 3.50000i 0.205409 + 0.118593i
\(872\) 0 0
\(873\) 10.1075 + 2.70831i 0.342089 + 0.0916624i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −33.3405 8.93357i −1.12583 0.301665i −0.352589 0.935778i \(-0.614699\pi\)
−0.773240 + 0.634113i \(0.781365\pi\)
\(878\) 0 0
\(879\) 3.80385 + 2.19615i 0.128301 + 0.0740744i
\(880\) 0 0
\(881\) 20.0000i 0.673817i 0.941537 + 0.336909i \(0.109381\pi\)
−0.941537 + 0.336909i \(0.890619\pi\)
\(882\) 0 0
\(883\) −24.2683 24.2683i −0.816692 0.816692i 0.168935 0.985627i \(-0.445967\pi\)
−0.985627 + 0.168935i \(0.945967\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.6245 + 50.8473i −0.457466 + 1.70729i 0.223270 + 0.974757i \(0.428327\pi\)
−0.680736 + 0.732529i \(0.738340\pi\)
\(888\) 0 0
\(889\) −12.0622 34.8205i −0.404552 1.16784i
\(890\) 0 0
\(891\) −0.366025 0.633975i −0.0122623 0.0212389i
\(892\) 0 0
\(893\) 0.291005 0.0779745i 0.00973810 0.00260932i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −0.378937 + 0.378937i −0.0126524 + 0.0126524i
\(898\) 0 0
\(899\) 6.92820 12.0000i 0.231069 0.400222i
\(900\) 0 0
\(901\) 14.9090 8.60770i 0.496690 0.286764i
\(902\) 0 0
\(903\) −11.0735 16.3514i −0.368502 0.544140i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 4.74170 + 17.6963i 0.157445 + 0.587594i 0.998884 + 0.0472409i \(0.0150428\pi\)
−0.841438 + 0.540354i \(0.818291\pi\)
\(908\) 0 0
\(909\) 3.26795 0.108391
\(910\) 0 0
\(911\) 45.7128 1.51453 0.757267 0.653106i \(-0.226534\pi\)
0.757267 + 0.653106i \(0.226534\pi\)
\(912\) 0 0
\(913\) −0.240237 0.896575i −0.00795067 0.0296723i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.582009 8.10634i −0.0192196 0.267695i
\(918\) 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
\(921\) 4.73205 8.19615i 0.155926 0.270072i
\(922\) 0 0
\(923\) −5.65685 + 5.65685i −0.186198 + 0.186198i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 12.2289 3.27671i 0.401649 0.107621i
\(928\) 0 0
\(929\) 0.901924 + 1.56218i 0.0295912 + 0.0512534i 0.880442 0.474154i \(-0.157246\pi\)
−0.850851 + 0.525408i \(0.823913\pi\)
\(930\) 0 0
\(931\) −3.99038 + 9.98334i −0.130779 + 0.327191i
\(932\) 0 0
\(933\) −1.55291 + 5.79555i −0.0508401 + 0.189738i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 17.5254 + 17.5254i 0.572529 + 0.572529i 0.932834 0.360306i \(-0.117328\pi\)
−0.360306 + 0.932834i \(0.617328\pi\)
\(938\) 0 0
\(939\) 9.46410i 0.308849i
\(940\) 0 0
\(941\) 24.1244 + 13.9282i 0.786432 + 0.454046i 0.838705 0.544586i \(-0.183313\pi\)
−0.0522732 + 0.998633i \(0.516647\pi\)
\(942\) 0 0
\(943\) −3.20736 0.859411i −0.104446 0.0279863i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −35.3925 9.48339i −1.15010 0.308169i −0.367096 0.930183i \(-0.619648\pi\)
−0.783006 + 0.622014i \(0.786315\pi\)
\(948\) 0 0
\(949\) −4.66987 2.69615i −0.151590 0.0875208i
\(950\) 0 0
\(951\) 18.3923i 0.596411i
\(952\) 0 0
\(953\) −24.4577 24.4577i −0.792264 0.792264i 0.189598 0.981862i \(-0.439281\pi\)
−0.981862 + 0.189598i \(0.939281\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.31268 + 4.89898i −0.0424328 + 0.158362i
\(958\) 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
\(963\) −10.8840 + 2.91636i −0.350732 + 0.0939784i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −7.39924 + 7.39924i −0.237943 + 0.237943i −0.815998 0.578055i \(-0.803812\pi\)
0.578055 + 0.815998i \(0.303812\pi\)
\(968\) 0 0
\(969\) 2.50962 4.34679i 0.0806206 0.139639i
\(970\) 0 0
\(971\) 28.2224 16.2942i 0.905701 0.522907i 0.0266555 0.999645i \(-0.491514\pi\)
0.879045 + 0.476738i \(0.158181\pi\)
\(972\) 0 0
\(973\) 12.5434 25.8398i 0.402124 0.828385i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.70522 6.36396i −0.0545548 0.203601i 0.933269 0.359178i \(-0.116943\pi\)
−0.987824 + 0.155577i \(0.950276\pi\)
\(978\) 0 0
\(979\) −7.85641 −0.251092
\(980\) 0 0
\(981\) −0.0717968 −0.00229229
\(982\) 0 0
\(983\) −12.2982 45.8976i −0.392252 1.46391i −0.826410 0.563068i \(-0.809621\pi\)
0.434158 0.900837i \(-0.357046\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −0.226633 + 0.466870i −0.00721382 + 0.0148606i
\(988\) 0 0
\(989\) −3.46410 + 2.00000i −0.110152 + 0.0635963i
\(990\) 0 0
\(991\) 6.66025 11.5359i 0.211570 0.366450i −0.740636 0.671906i \(-0.765476\pi\)
0.952206 + 0.305456i \(0.0988091\pi\)
\(992\) 0 0
\(993\) 21.6801 21.6801i 0.687996 0.687996i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.90018 + 1.84890i −0.218531 + 0.0585552i −0.366423 0.930448i \(-0.619418\pi\)
0.147892 + 0.989003i \(0.452751\pi\)
\(998\) 0 0
\(999\) −0.401924 0.696152i −0.0127163 0.0220253i
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.a.493.1 8
5.2 odd 4 2100.2.ce.b.157.1 yes 8
5.3 odd 4 2100.2.ce.b.157.2 yes 8
5.4 even 2 inner 2100.2.ce.a.493.2 yes 8
7.5 odd 6 2100.2.ce.b.1993.1 yes 8
35.12 even 12 inner 2100.2.ce.a.1657.1 yes 8
35.19 odd 6 2100.2.ce.b.1993.2 yes 8
35.33 even 12 inner 2100.2.ce.a.1657.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.ce.a.493.1 8 1.1 even 1 trivial
2100.2.ce.a.493.2 yes 8 5.4 even 2 inner
2100.2.ce.a.1657.1 yes 8 35.12 even 12 inner
2100.2.ce.a.1657.2 yes 8 35.33 even 12 inner
2100.2.ce.b.157.1 yes 8 5.2 odd 4
2100.2.ce.b.157.2 yes 8 5.3 odd 4
2100.2.ce.b.1993.1 yes 8 7.5 odd 6
2100.2.ce.b.1993.2 yes 8 35.19 odd 6