Properties

Label 2100.2.s.a.1793.3
Level $2100$
Weight $2$
Character 2100.1793
Analytic conductor $16.769$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1457,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1457");
 
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.s (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 24x^{12} + 424x^{8} - 159x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1793.3
Root \(0.660512 + 0.0465948i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1793
Dual form 2100.2.s.a.1457.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.541157 + 1.64534i) q^{3} +(0.707107 - 0.707107i) q^{7} +(-2.41430 - 1.78078i) q^{9} +O(q^{10})\) \(q+(-0.541157 + 1.64534i) q^{3} +(0.707107 - 0.707107i) q^{7} +(-2.41430 - 1.78078i) q^{9} +3.09218i q^{11} +(0.310029 + 0.310029i) q^{13} +(3.41433 + 3.41433i) q^{17} -1.12311i q^{19} +(0.780776 + 1.54609i) q^{21} +(-1.22783 + 1.22783i) q^{23} +(4.23650 - 3.00867i) q^{27} -1.35576 q^{29} -8.24621 q^{31} +(-5.08769 - 1.67335i) q^{33} +(2.20837 - 2.20837i) q^{37} +(-0.677878 + 0.342329i) q^{39} +6.18435i q^{41} +(7.24517 + 7.24517i) q^{43} +(2.18650 + 2.18650i) q^{47} -1.00000i q^{49} +(-7.46543 + 3.77005i) q^{51} +(1.22783 - 1.22783i) q^{53} +(1.84789 + 0.607777i) q^{57} -9.65719 q^{59} -5.12311 q^{61} +(-2.96637 + 0.447967i) q^{63} +(-7.86522 + 7.86522i) q^{67} +(-1.35576 - 2.68466i) q^{69} +14.1051i q^{71} +(4.24264 + 4.24264i) q^{73} +(2.18650 + 2.18650i) q^{77} +5.56155i q^{79} +(2.65767 + 8.59865i) q^{81} +(-1.22783 + 1.22783i) q^{83} +(0.733677 - 2.23068i) q^{87} -12.3687 q^{89} +0.438447 q^{91} +(4.46250 - 13.5678i) q^{93} +(2.51840 - 2.51840i) q^{97} +(5.50647 - 7.46543i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{21} - 4 q^{51} - 16 q^{61} + 92 q^{81} + 40 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −0.541157 + 1.64534i −0.312437 + 0.949938i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0.707107 0.707107i 0.267261 0.267261i
\(8\) 0 0
\(9\) −2.41430 1.78078i −0.804766 0.593592i
\(10\) 0 0
\(11\) 3.09218i 0.932326i 0.884699 + 0.466163i \(0.154364\pi\)
−0.884699 + 0.466163i \(0.845636\pi\)
\(12\) 0 0
\(13\) 0.310029 + 0.310029i 0.0859866 + 0.0859866i 0.748792 0.662805i \(-0.230634\pi\)
−0.662805 + 0.748792i \(0.730634\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.41433 + 3.41433i 0.828097 + 0.828097i 0.987253 0.159156i \(-0.0508773\pi\)
−0.159156 + 0.987253i \(0.550877\pi\)
\(18\) 0 0
\(19\) 1.12311i 0.257658i −0.991667 0.128829i \(-0.958878\pi\)
0.991667 0.128829i \(-0.0411218\pi\)
\(20\) 0 0
\(21\) 0.780776 + 1.54609i 0.170379 + 0.337384i
\(22\) 0 0
\(23\) −1.22783 + 1.22783i −0.256021 + 0.256021i −0.823434 0.567413i \(-0.807945\pi\)
0.567413 + 0.823434i \(0.307945\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 4.23650 3.00867i 0.815315 0.579018i
\(28\) 0 0
\(29\) −1.35576 −0.251758 −0.125879 0.992046i \(-0.540175\pi\)
−0.125879 + 0.992046i \(0.540175\pi\)
\(30\) 0 0
\(31\) −8.24621 −1.48106 −0.740532 0.672022i \(-0.765426\pi\)
−0.740532 + 0.672022i \(0.765426\pi\)
\(32\) 0 0
\(33\) −5.08769 1.67335i −0.885652 0.291293i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.20837 2.20837i 0.363054 0.363054i −0.501882 0.864936i \(-0.667359\pi\)
0.864936 + 0.501882i \(0.167359\pi\)
\(38\) 0 0
\(39\) −0.677878 + 0.342329i −0.108547 + 0.0548165i
\(40\) 0 0
\(41\) 6.18435i 0.965834i 0.875666 + 0.482917i \(0.160423\pi\)
−0.875666 + 0.482917i \(0.839577\pi\)
\(42\) 0 0
\(43\) 7.24517 + 7.24517i 1.10488 + 1.10488i 0.993813 + 0.111064i \(0.0354259\pi\)
0.111064 + 0.993813i \(0.464574\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.18650 + 2.18650i 0.318934 + 0.318934i 0.848357 0.529424i \(-0.177592\pi\)
−0.529424 + 0.848357i \(0.677592\pi\)
\(48\) 0 0
\(49\) 1.00000i 0.142857i
\(50\) 0 0
\(51\) −7.46543 + 3.77005i −1.04537 + 0.527913i
\(52\) 0 0
\(53\) 1.22783 1.22783i 0.168656 0.168656i −0.617732 0.786388i \(-0.711948\pi\)
0.786388 + 0.617732i \(0.211948\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.84789 + 0.607777i 0.244759 + 0.0805020i
\(58\) 0 0
\(59\) −9.65719 −1.25726 −0.628630 0.777705i \(-0.716384\pi\)
−0.628630 + 0.777705i \(0.716384\pi\)
\(60\) 0 0
\(61\) −5.12311 −0.655946 −0.327973 0.944687i \(-0.606366\pi\)
−0.327973 + 0.944687i \(0.606366\pi\)
\(62\) 0 0
\(63\) −2.96637 + 0.447967i −0.373727 + 0.0564386i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −7.86522 + 7.86522i −0.960890 + 0.960890i −0.999263 0.0383735i \(-0.987782\pi\)
0.0383735 + 0.999263i \(0.487782\pi\)
\(68\) 0 0
\(69\) −1.35576 2.68466i −0.163214 0.323195i
\(70\) 0 0
\(71\) 14.1051i 1.67397i 0.547226 + 0.836985i \(0.315684\pi\)
−0.547226 + 0.836985i \(0.684316\pi\)
\(72\) 0 0
\(73\) 4.24264 + 4.24264i 0.496564 + 0.496564i 0.910366 0.413803i \(-0.135800\pi\)
−0.413803 + 0.910366i \(0.635800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.18650 + 2.18650i 0.249175 + 0.249175i
\(78\) 0 0
\(79\) 5.56155i 0.625724i 0.949799 + 0.312862i \(0.101288\pi\)
−0.949799 + 0.312862i \(0.898712\pi\)
\(80\) 0 0
\(81\) 2.65767 + 8.59865i 0.295297 + 0.955406i
\(82\) 0 0
\(83\) −1.22783 + 1.22783i −0.134772 + 0.134772i −0.771275 0.636502i \(-0.780381\pi\)
0.636502 + 0.771275i \(0.280381\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0.733677 2.23068i 0.0786584 0.239154i
\(88\) 0 0
\(89\) −12.3687 −1.31108 −0.655540 0.755160i \(-0.727559\pi\)
−0.655540 + 0.755160i \(0.727559\pi\)
\(90\) 0 0
\(91\) 0.438447 0.0459618
\(92\) 0 0
\(93\) 4.46250 13.5678i 0.462739 1.40692i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.51840 2.51840i 0.255705 0.255705i −0.567600 0.823305i \(-0.692128\pi\)
0.823305 + 0.567600i \(0.192128\pi\)
\(98\) 0 0
\(99\) 5.50647 7.46543i 0.553421 0.750304i
\(100\) 0 0
\(101\) 3.47284i 0.345561i −0.984960 0.172780i \(-0.944725\pi\)
0.984960 0.172780i \(-0.0552751\pi\)
\(102\) 0 0
\(103\) −2.69250 2.69250i −0.265299 0.265299i 0.561903 0.827203i \(-0.310069\pi\)
−0.827203 + 0.561903i \(0.810069\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −14.3468 14.3468i −1.38696 1.38696i −0.831630 0.555330i \(-0.812592\pi\)
−0.555330 0.831630i \(-0.687408\pi\)
\(108\) 0 0
\(109\) 12.9309i 1.23855i −0.785173 0.619276i \(-0.787426\pi\)
0.785173 0.619276i \(-0.212574\pi\)
\(110\) 0 0
\(111\) 2.43845 + 4.82860i 0.231447 + 0.458310i
\(112\) 0 0
\(113\) 1.22783 1.22783i 0.115505 0.115505i −0.646992 0.762497i \(-0.723973\pi\)
0.762497 + 0.646992i \(0.223973\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.196410 1.30059i −0.0181581 0.120240i
\(118\) 0 0
\(119\) 4.82860 0.442637
\(120\) 0 0
\(121\) 1.43845 0.130768
\(122\) 0 0
\(123\) −10.1754 3.34671i −0.917482 0.301762i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −6.27691 + 6.27691i −0.556986 + 0.556986i −0.928448 0.371462i \(-0.878857\pi\)
0.371462 + 0.928448i \(0.378857\pi\)
\(128\) 0 0
\(129\) −15.8415 + 8.00000i −1.39477 + 0.704361i
\(130\) 0 0
\(131\) 18.5531i 1.62099i 0.585747 + 0.810494i \(0.300801\pi\)
−0.585747 + 0.810494i \(0.699199\pi\)
\(132\) 0 0
\(133\) −0.794156 0.794156i −0.0688620 0.0688620i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 3.14516 + 3.14516i 0.268709 + 0.268709i 0.828580 0.559871i \(-0.189149\pi\)
−0.559871 + 0.828580i \(0.689149\pi\)
\(138\) 0 0
\(139\) 10.0000i 0.848189i −0.905618 0.424094i \(-0.860592\pi\)
0.905618 0.424094i \(-0.139408\pi\)
\(140\) 0 0
\(141\) −4.78078 + 2.41430i −0.402614 + 0.203321i
\(142\) 0 0
\(143\) −0.958664 + 0.958664i −0.0801675 + 0.0801675i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.64534 + 0.541157i 0.135705 + 0.0446339i
\(148\) 0 0
\(149\) −15.8415 −1.29779 −0.648895 0.760878i \(-0.724769\pi\)
−0.648895 + 0.760878i \(0.724769\pi\)
\(150\) 0 0
\(151\) −5.56155 −0.452593 −0.226296 0.974058i \(-0.572662\pi\)
−0.226296 + 0.974058i \(0.572662\pi\)
\(152\) 0 0
\(153\) −2.16305 14.3234i −0.174873 1.15798i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.4878 + 11.4878i −0.916827 + 0.916827i −0.996797 0.0799705i \(-0.974517\pi\)
0.0799705 + 0.996797i \(0.474517\pi\)
\(158\) 0 0
\(159\) 1.35576 + 2.68466i 0.107518 + 0.212907i
\(160\) 0 0
\(161\) 1.73642i 0.136849i
\(162\) 0 0
\(163\) 10.6937 + 10.6937i 0.837591 + 0.837591i 0.988541 0.150950i \(-0.0482333\pi\)
−0.150950 + 0.988541i \(0.548233\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −13.3882 13.3882i −1.03601 1.03601i −0.999327 0.0366800i \(-0.988322\pi\)
−0.0366800 0.999327i \(-0.511678\pi\)
\(168\) 0 0
\(169\) 12.8078i 0.985213i
\(170\) 0 0
\(171\) −2.00000 + 2.71151i −0.152944 + 0.207354i
\(172\) 0 0
\(173\) −0.958664 + 0.958664i −0.0728859 + 0.0728859i −0.742610 0.669724i \(-0.766412\pi\)
0.669724 + 0.742610i \(0.266412\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.22606 15.8894i 0.392815 1.19432i
\(178\) 0 0
\(179\) 17.5780 1.31384 0.656919 0.753961i \(-0.271859\pi\)
0.656919 + 0.753961i \(0.271859\pi\)
\(180\) 0 0
\(181\) −12.2462 −0.910254 −0.455127 0.890427i \(-0.650406\pi\)
−0.455127 + 0.890427i \(0.650406\pi\)
\(182\) 0 0
\(183\) 2.77240 8.42926i 0.204942 0.623109i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −10.5577 + 10.5577i −0.772057 + 0.772057i
\(188\) 0 0
\(189\) 0.868210 5.12311i 0.0631530 0.372651i
\(190\) 0 0
\(191\) 6.56502i 0.475028i −0.971384 0.237514i \(-0.923667\pi\)
0.971384 0.237514i \(-0.0763325\pi\)
\(192\) 0 0
\(193\) 8.48528 + 8.48528i 0.610784 + 0.610784i 0.943150 0.332366i \(-0.107847\pi\)
−0.332366 + 0.943150i \(0.607847\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −9.97383 9.97383i −0.710606 0.710606i 0.256056 0.966662i \(-0.417577\pi\)
−0.966662 + 0.256056i \(0.917577\pi\)
\(198\) 0 0
\(199\) 22.4924i 1.59445i 0.603685 + 0.797223i \(0.293698\pi\)
−0.603685 + 0.797223i \(0.706302\pi\)
\(200\) 0 0
\(201\) −8.68466 17.1973i −0.612569 1.21300i
\(202\) 0 0
\(203\) −0.958664 + 0.958664i −0.0672850 + 0.0672850i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 5.15086 0.777860i 0.358009 0.0540650i
\(208\) 0 0
\(209\) 3.47284 0.240221
\(210\) 0 0
\(211\) 7.31534 0.503609 0.251804 0.967778i \(-0.418976\pi\)
0.251804 + 0.967778i \(0.418976\pi\)
\(212\) 0 0
\(213\) −23.2077 7.63309i −1.59017 0.523011i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −5.83095 + 5.83095i −0.395831 + 0.395831i
\(218\) 0 0
\(219\) −9.27653 + 4.68466i −0.626850 + 0.316560i
\(220\) 0 0
\(221\) 2.11708i 0.142411i
\(222\) 0 0
\(223\) 14.0062 + 14.0062i 0.937925 + 0.937925i 0.998183 0.0602580i \(-0.0191924\pi\)
−0.0602580 + 0.998183i \(0.519192\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.3055 + 15.3055i 1.01586 + 1.01586i 0.999872 + 0.0159889i \(0.00508965\pi\)
0.0159889 + 0.999872i \(0.494910\pi\)
\(228\) 0 0
\(229\) 8.24621i 0.544925i 0.962166 + 0.272462i \(0.0878381\pi\)
−0.962166 + 0.272462i \(0.912162\pi\)
\(230\) 0 0
\(231\) −4.78078 + 2.41430i −0.314552 + 0.158849i
\(232\) 0 0
\(233\) 11.8912 11.8912i 0.779016 0.779016i −0.200648 0.979663i \(-0.564305\pi\)
0.979663 + 0.200648i \(0.0643047\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −9.15065 3.00967i −0.594399 0.195499i
\(238\) 0 0
\(239\) 18.9337 1.22472 0.612360 0.790579i \(-0.290220\pi\)
0.612360 + 0.790579i \(0.290220\pi\)
\(240\) 0 0
\(241\) 0.246211 0.0158599 0.00792993 0.999969i \(-0.497476\pi\)
0.00792993 + 0.999969i \(0.497476\pi\)
\(242\) 0 0
\(243\) −15.5859 0.280444i −0.999838 0.0179905i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0.348195 0.348195i 0.0221551 0.0221551i
\(248\) 0 0
\(249\) −1.35576 2.68466i −0.0859175 0.170133i
\(250\) 0 0
\(251\) 9.65719i 0.609557i 0.952423 + 0.304778i \(0.0985824\pi\)
−0.952423 + 0.304778i \(0.901418\pi\)
\(252\) 0 0
\(253\) −3.79668 3.79668i −0.238695 0.238695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.9477 + 19.9477i 1.24430 + 1.24430i 0.958199 + 0.286101i \(0.0923594\pi\)
0.286101 + 0.958199i \(0.407641\pi\)
\(258\) 0 0
\(259\) 3.12311i 0.194060i
\(260\) 0 0
\(261\) 3.27320 + 2.41430i 0.202606 + 0.149441i
\(262\) 0 0
\(263\) −5.60083 + 5.60083i −0.345362 + 0.345362i −0.858379 0.513017i \(-0.828528\pi\)
0.513017 + 0.858379i \(0.328528\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.69341 20.3507i 0.409630 1.24545i
\(268\) 0 0
\(269\) −25.4987 −1.55469 −0.777343 0.629077i \(-0.783433\pi\)
−0.777343 + 0.629077i \(0.783433\pi\)
\(270\) 0 0
\(271\) 17.1231 1.04015 0.520077 0.854119i \(-0.325903\pi\)
0.520077 + 0.854119i \(0.325903\pi\)
\(272\) 0 0
\(273\) −0.237269 + 0.721395i −0.0143602 + 0.0436608i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.6937 10.6937i 0.642519 0.642519i −0.308655 0.951174i \(-0.599879\pi\)
0.951174 + 0.308655i \(0.0998788\pi\)
\(278\) 0 0
\(279\) 19.9088 + 14.6847i 1.19191 + 0.879148i
\(280\) 0 0
\(281\) 7.54011i 0.449805i −0.974381 0.224903i \(-0.927794\pi\)
0.974381 0.224903i \(-0.0722064\pi\)
\(282\) 0 0
\(283\) −3.31255 3.31255i −0.196911 0.196911i 0.601763 0.798674i \(-0.294465\pi\)
−0.798674 + 0.601763i \(0.794465\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.37300 + 4.37300i 0.258130 + 0.258130i
\(288\) 0 0
\(289\) 6.31534i 0.371491i
\(290\) 0 0
\(291\) 2.78078 + 5.50647i 0.163012 + 0.322795i
\(292\) 0 0
\(293\) 12.1603 12.1603i 0.710414 0.710414i −0.256208 0.966622i \(-0.582473\pi\)
0.966622 + 0.256208i \(0.0824731\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 9.30332 + 13.1000i 0.539834 + 0.760139i
\(298\) 0 0
\(299\) −0.761329 −0.0440288
\(300\) 0 0
\(301\) 10.2462 0.590582
\(302\) 0 0
\(303\) 5.71401 + 1.87935i 0.328261 + 0.107966i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −3.58442 + 3.58442i −0.204573 + 0.204573i −0.801956 0.597383i \(-0.796207\pi\)
0.597383 + 0.801956i \(0.296207\pi\)
\(308\) 0 0
\(309\) 5.88714 2.97301i 0.334908 0.169129i
\(310\) 0 0
\(311\) 28.2102i 1.59966i −0.600229 0.799828i \(-0.704924\pi\)
0.600229 0.799828i \(-0.295076\pi\)
\(312\) 0 0
\(313\) 23.9057 + 23.9057i 1.35123 + 1.35123i 0.884288 + 0.466942i \(0.154644\pi\)
0.466942 + 0.884288i \(0.345356\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.6372 20.6372i −1.15910 1.15910i −0.984670 0.174428i \(-0.944192\pi\)
−0.174428 0.984670i \(-0.555808\pi\)
\(318\) 0 0
\(319\) 4.19224i 0.234720i
\(320\) 0 0
\(321\) 31.3693 15.8415i 1.75086 0.884189i
\(322\) 0 0
\(323\) 3.83466 3.83466i 0.213366 0.213366i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 21.2757 + 6.99763i 1.17655 + 0.386970i
\(328\) 0 0
\(329\) 3.09218 0.170477
\(330\) 0 0
\(331\) −6.24621 −0.343323 −0.171661 0.985156i \(-0.554914\pi\)
−0.171661 + 0.985156i \(0.554914\pi\)
\(332\) 0 0
\(333\) −9.26427 + 1.39905i −0.507679 + 0.0766675i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.21342 8.21342i 0.447413 0.447413i −0.447080 0.894494i \(-0.647536\pi\)
0.894494 + 0.447080i \(0.147536\pi\)
\(338\) 0 0
\(339\) 1.35576 + 2.68466i 0.0736346 + 0.145811i
\(340\) 0 0
\(341\) 25.4987i 1.38083i
\(342\) 0 0
\(343\) −0.707107 0.707107i −0.0381802 0.0381802i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 23.6312 + 23.6312i 1.26859 + 1.26859i 0.946818 + 0.321769i \(0.104277\pi\)
0.321769 + 0.946818i \(0.395723\pi\)
\(348\) 0 0
\(349\) 19.3693i 1.03682i −0.855133 0.518408i \(-0.826525\pi\)
0.855133 0.518408i \(-0.173475\pi\)
\(350\) 0 0
\(351\) 2.24621 + 0.380664i 0.119894 + 0.0203184i
\(352\) 0 0
\(353\) −16.5333 + 16.5333i −0.879980 + 0.879980i −0.993532 0.113552i \(-0.963777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −2.61303 + 7.94469i −0.138296 + 0.420478i
\(358\) 0 0
\(359\) 23.0010 1.21395 0.606973 0.794723i \(-0.292384\pi\)
0.606973 + 0.794723i \(0.292384\pi\)
\(360\) 0 0
\(361\) 17.7386 0.933612
\(362\) 0 0
\(363\) −0.778426 + 2.36674i −0.0408568 + 0.124221i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8.34935 + 8.34935i −0.435833 + 0.435833i −0.890607 0.454774i \(-0.849720\pi\)
0.454774 + 0.890607i \(0.349720\pi\)
\(368\) 0 0
\(369\) 11.0129 14.9309i 0.573311 0.777270i
\(370\) 0 0
\(371\) 1.73642i 0.0901504i
\(372\) 0 0
\(373\) 23.3238 + 23.3238i 1.20766 + 1.20766i 0.971783 + 0.235878i \(0.0757967\pi\)
0.235878 + 0.971783i \(0.424203\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −0.420324 0.420324i −0.0216478 0.0216478i
\(378\) 0 0
\(379\) 16.4924i 0.847159i 0.905859 + 0.423579i \(0.139227\pi\)
−0.905859 + 0.423579i \(0.860773\pi\)
\(380\) 0 0
\(381\) −6.93087 13.7245i −0.355079 0.703125i
\(382\) 0 0
\(383\) −18.7198 + 18.7198i −0.956538 + 0.956538i −0.999094 0.0425558i \(-0.986450\pi\)
0.0425558 + 0.999094i \(0.486450\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.58997 30.3940i −0.233321 1.54501i
\(388\) 0 0
\(389\) −1.35576 −0.0687396 −0.0343698 0.999409i \(-0.510942\pi\)
−0.0343698 + 0.999409i \(0.510942\pi\)
\(390\) 0 0
\(391\) −8.38447 −0.424021
\(392\) 0 0
\(393\) −30.5261 10.0401i −1.53984 0.506457i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.52345 + 8.52345i −0.427780 + 0.427780i −0.887871 0.460092i \(-0.847816\pi\)
0.460092 + 0.887871i \(0.347816\pi\)
\(398\) 0 0
\(399\) 1.73642 0.876894i 0.0869297 0.0438996i
\(400\) 0 0
\(401\) 19.9088i 0.994199i −0.867694 0.497099i \(-0.834398\pi\)
0.867694 0.497099i \(-0.165602\pi\)
\(402\) 0 0
\(403\) −2.55656 2.55656i −0.127352 0.127352i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.82867 + 6.82867i 0.338484 + 0.338484i
\(408\) 0 0
\(409\) 5.12311i 0.253321i −0.991946 0.126661i \(-0.959574\pi\)
0.991946 0.126661i \(-0.0404259\pi\)
\(410\) 0 0
\(411\) −6.87689 + 3.47284i −0.339212 + 0.171303i
\(412\) 0 0
\(413\) −6.82867 + 6.82867i −0.336017 + 0.336017i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 16.4534 + 5.41157i 0.805727 + 0.265006i
\(418\) 0 0
\(419\) 9.65719 0.471785 0.235892 0.971779i \(-0.424199\pi\)
0.235892 + 0.971779i \(0.424199\pi\)
\(420\) 0 0
\(421\) 23.1771 1.12958 0.564791 0.825234i \(-0.308957\pi\)
0.564791 + 0.825234i \(0.308957\pi\)
\(422\) 0 0
\(423\) −1.38519 9.17252i −0.0673505 0.445983i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.62258 + 3.62258i −0.175309 + 0.175309i
\(428\) 0 0
\(429\) −1.05854 2.09612i −0.0511069 0.101202i
\(430\) 0 0
\(431\) 18.9337i 0.912005i −0.889978 0.456003i \(-0.849281\pi\)
0.889978 0.456003i \(-0.150719\pi\)
\(432\) 0 0
\(433\) 14.3162 + 14.3162i 0.687994 + 0.687994i 0.961788 0.273794i \(-0.0882787\pi\)
−0.273794 + 0.961788i \(0.588279\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.37899 + 1.37899i 0.0659659 + 0.0659659i
\(438\) 0 0
\(439\) 22.4924i 1.07350i 0.843740 + 0.536752i \(0.180349\pi\)
−0.843740 + 0.536752i \(0.819651\pi\)
\(440\) 0 0
\(441\) −1.78078 + 2.41430i −0.0847989 + 0.114967i
\(442\) 0 0
\(443\) 28.0042 28.0042i 1.33052 1.33052i 0.425612 0.904906i \(-0.360059\pi\)
0.904906 0.425612i \(-0.139941\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 8.57276 26.0648i 0.405478 1.23282i
\(448\) 0 0
\(449\) 4.82860 0.227876 0.113938 0.993488i \(-0.463654\pi\)
0.113938 + 0.993488i \(0.463654\pi\)
\(450\) 0 0
\(451\) −19.1231 −0.900472
\(452\) 0 0
\(453\) 3.00967 9.15065i 0.141407 0.429935i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 26.7723 26.7723i 1.25236 1.25236i 0.297694 0.954661i \(-0.403783\pi\)
0.954661 0.297694i \(-0.0962174\pi\)
\(458\) 0 0
\(459\) 24.7374 + 4.19224i 1.15464 + 0.195677i
\(460\) 0 0
\(461\) 24.7374i 1.15214i −0.817402 0.576068i \(-0.804586\pi\)
0.817402 0.576068i \(-0.195414\pi\)
\(462\) 0 0
\(463\) 1.58831 + 1.58831i 0.0738151 + 0.0738151i 0.743050 0.669235i \(-0.233378\pi\)
−0.669235 + 0.743050i \(0.733378\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.64816 + 1.64816i 0.0762677 + 0.0762677i 0.744212 0.667944i \(-0.232825\pi\)
−0.667944 + 0.744212i \(0.732825\pi\)
\(468\) 0 0
\(469\) 11.1231i 0.513617i
\(470\) 0 0
\(471\) −12.6847 25.1181i −0.584478 1.15738i
\(472\) 0 0
\(473\) −22.4033 + 22.4033i −1.03011 + 1.03011i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.15086 + 0.777860i −0.235842 + 0.0356158i
\(478\) 0 0
\(479\) 30.9218 1.41285 0.706426 0.707787i \(-0.250306\pi\)
0.706426 + 0.707787i \(0.250306\pi\)
\(480\) 0 0
\(481\) 1.36932 0.0624355
\(482\) 0 0
\(483\) −2.85700 0.939676i −0.129998 0.0427567i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −25.1840 + 25.1840i −1.14120 + 1.14120i −0.152963 + 0.988232i \(0.548882\pi\)
−0.988232 + 0.152963i \(0.951118\pi\)
\(488\) 0 0
\(489\) −23.3817 + 11.8078i −1.05735 + 0.533966i
\(490\) 0 0
\(491\) 9.27653i 0.418644i 0.977847 + 0.209322i \(0.0671257\pi\)
−0.977847 + 0.209322i \(0.932874\pi\)
\(492\) 0 0
\(493\) −4.62900 4.62900i −0.208480 0.208480i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.97383 + 9.97383i 0.447387 + 0.447387i
\(498\) 0 0
\(499\) 14.0540i 0.629142i 0.949234 + 0.314571i \(0.101861\pi\)
−0.949234 + 0.314571i \(0.898139\pi\)
\(500\) 0 0
\(501\) 29.2732 14.7830i 1.30783 0.660456i
\(502\) 0 0
\(503\) 11.4708 11.4708i 0.511459 0.511459i −0.403514 0.914973i \(-0.632211\pi\)
0.914973 + 0.403514i \(0.132211\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 21.0731 + 6.93101i 0.935891 + 0.307817i
\(508\) 0 0
\(509\) 13.1300 0.581978 0.290989 0.956726i \(-0.406016\pi\)
0.290989 + 0.956726i \(0.406016\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 0 0
\(513\) −3.37905 4.75804i −0.149189 0.210072i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −6.76104 + 6.76104i −0.297350 + 0.297350i
\(518\) 0 0
\(519\) −1.05854 2.09612i −0.0464648 0.0920094i
\(520\) 0 0
\(521\) 8.89586i 0.389735i −0.980830 0.194867i \(-0.937572\pi\)
0.980830 0.194867i \(-0.0624277\pi\)
\(522\) 0 0
\(523\) −17.3188 17.3188i −0.757296 0.757296i 0.218533 0.975830i \(-0.429873\pi\)
−0.975830 + 0.218533i \(0.929873\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −28.1553 28.1553i −1.22646 1.22646i
\(528\) 0 0
\(529\) 19.9848i 0.868906i
\(530\) 0 0
\(531\) 23.3153 + 17.1973i 1.01180 + 0.746299i
\(532\) 0 0
\(533\) −1.91733 + 1.91733i −0.0830487 + 0.0830487i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −9.51244 + 28.9218i −0.410492 + 1.24807i
\(538\) 0 0
\(539\) 3.09218 0.133189
\(540\) 0 0
\(541\) 34.6847 1.49121 0.745605 0.666388i \(-0.232161\pi\)
0.745605 + 0.666388i \(0.232161\pi\)
\(542\) 0 0
\(543\) 6.62712 20.1492i 0.284397 0.864685i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.89697 6.89697i 0.294893 0.294893i −0.544117 0.839010i \(-0.683135\pi\)
0.839010 + 0.544117i \(0.183135\pi\)
\(548\) 0 0
\(549\) 12.3687 + 9.12311i 0.527883 + 0.389365i
\(550\) 0 0
\(551\) 1.52266i 0.0648674i
\(552\) 0 0
\(553\) 3.93261 + 3.93261i 0.167232 + 0.167232i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −10.5122 10.5122i −0.445415 0.445415i 0.448412 0.893827i \(-0.351990\pi\)
−0.893827 + 0.448412i \(0.851990\pi\)
\(558\) 0 0
\(559\) 4.49242i 0.190009i
\(560\) 0 0
\(561\) −11.6577 23.0844i −0.492187 0.974626i
\(562\) 0 0
\(563\) −9.97383 + 9.97383i −0.420347 + 0.420347i −0.885323 0.464976i \(-0.846063\pi\)
0.464976 + 0.885323i \(0.346063\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.95942 + 4.20091i 0.334264 + 0.176422i
\(568\) 0 0
\(569\) −12.3687 −0.518523 −0.259262 0.965807i \(-0.583479\pi\)
−0.259262 + 0.965807i \(0.583479\pi\)
\(570\) 0 0
\(571\) −26.7386 −1.11898 −0.559488 0.828838i \(-0.689002\pi\)
−0.559488 + 0.828838i \(0.689002\pi\)
\(572\) 0 0
\(573\) 10.8017 + 3.55270i 0.451247 + 0.148416i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −5.07496 + 5.07496i −0.211273 + 0.211273i −0.804808 0.593535i \(-0.797732\pi\)
0.593535 + 0.804808i \(0.297732\pi\)
\(578\) 0 0
\(579\) −18.5531 + 9.36932i −0.771039 + 0.389376i
\(580\) 0 0
\(581\) 1.73642i 0.0720388i
\(582\) 0 0
\(583\) 3.79668 + 3.79668i 0.157242 + 0.157242i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −18.7198 18.7198i −0.772650 0.772650i 0.205919 0.978569i \(-0.433982\pi\)
−0.978569 + 0.205919i \(0.933982\pi\)
\(588\) 0 0
\(589\) 9.26137i 0.381608i
\(590\) 0 0
\(591\) 21.8078 11.0129i 0.897052 0.453012i
\(592\) 0 0
\(593\) −7.78733 + 7.78733i −0.319787 + 0.319787i −0.848685 0.528898i \(-0.822605\pi\)
0.528898 + 0.848685i \(0.322605\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −37.0077 12.1719i −1.51463 0.498164i
\(598\) 0 0
\(599\) 11.9880 0.489818 0.244909 0.969546i \(-0.421242\pi\)
0.244909 + 0.969546i \(0.421242\pi\)
\(600\) 0 0
\(601\) −20.6307 −0.841543 −0.420772 0.907167i \(-0.638241\pi\)
−0.420772 + 0.907167i \(0.638241\pi\)
\(602\) 0 0
\(603\) 32.9952 4.98279i 1.34367 0.202915i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −16.4864 + 16.4864i −0.669164 + 0.669164i −0.957522 0.288359i \(-0.906890\pi\)
0.288359 + 0.957522i \(0.406890\pi\)
\(608\) 0 0
\(609\) −1.05854 2.09612i −0.0428943 0.0849390i
\(610\) 0 0
\(611\) 1.35576i 0.0548480i
\(612\) 0 0
\(613\) −4.06854 4.06854i −0.164327 0.164327i 0.620154 0.784480i \(-0.287070\pi\)
−0.784480 + 0.620154i \(0.787070\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −27.4658 27.4658i −1.10573 1.10573i −0.993705 0.112027i \(-0.964266\pi\)
−0.112027 0.993705i \(-0.535734\pi\)
\(618\) 0 0
\(619\) 18.4924i 0.743273i −0.928378 0.371637i \(-0.878797\pi\)
0.928378 0.371637i \(-0.121203\pi\)
\(620\) 0 0
\(621\) −1.50758 + 8.89586i −0.0604970 + 0.356979i
\(622\) 0 0
\(623\) −8.74599 + 8.74599i −0.350401 + 0.350401i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −1.87935 + 5.71401i −0.0750541 + 0.228196i
\(628\) 0 0
\(629\) 15.0802 0.601288
\(630\) 0 0
\(631\) 26.0540 1.03719 0.518596 0.855019i \(-0.326455\pi\)
0.518596 + 0.855019i \(0.326455\pi\)
\(632\) 0 0
\(633\) −3.95875 + 12.0362i −0.157346 + 0.478397i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.310029 0.310029i 0.0122838 0.0122838i
\(638\) 0 0
\(639\) 25.1181 34.0540i 0.993656 1.34715i
\(640\) 0 0
\(641\) 12.3687i 0.488534i 0.969708 + 0.244267i \(0.0785474\pi\)
−0.969708 + 0.244267i \(0.921453\pi\)
\(642\) 0 0
\(643\) −15.8664 15.8664i −0.625709 0.625709i 0.321277 0.946985i \(-0.395888\pi\)
−0.946985 + 0.321277i \(0.895888\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.9215 + 29.9215i 1.17634 + 1.17634i 0.980671 + 0.195664i \(0.0626862\pi\)
0.195664 + 0.980671i \(0.437314\pi\)
\(648\) 0 0
\(649\) 29.8617i 1.17218i
\(650\) 0 0
\(651\) −6.43845 12.7494i −0.252343 0.499687i
\(652\) 0 0
\(653\) 34.2945 34.2945i 1.34205 1.34205i 0.448027 0.894020i \(-0.352127\pi\)
0.894020 0.448027i \(-0.147873\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −2.68780 17.7982i −0.104861 0.694374i
\(658\) 0 0
\(659\) −16.2222 −0.631928 −0.315964 0.948771i \(-0.602328\pi\)
−0.315964 + 0.948771i \(0.602328\pi\)
\(660\) 0 0
\(661\) 19.3693 0.753379 0.376690 0.926340i \(-0.377062\pi\)
0.376690 + 0.926340i \(0.377062\pi\)
\(662\) 0 0
\(663\) −3.48333 1.14568i −0.135281 0.0444943i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.66464 1.66464i 0.0644553 0.0644553i
\(668\) 0 0
\(669\) −30.6245 + 15.4654i −1.18401 + 0.597928i
\(670\) 0 0
\(671\) 15.8415i 0.611556i
\(672\) 0 0
\(673\) 22.6274 + 22.6274i 0.872223 + 0.872223i 0.992714 0.120492i \(-0.0384471\pi\)
−0.120492 + 0.992714i \(0.538447\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.0777 + 14.0777i 0.541048 + 0.541048i 0.923836 0.382788i \(-0.125036\pi\)
−0.382788 + 0.923836i \(0.625036\pi\)
\(678\) 0 0
\(679\) 3.56155i 0.136680i
\(680\) 0 0
\(681\) −33.4654 + 16.9001i −1.28240 + 0.647613i
\(682\) 0 0
\(683\) 10.5122 10.5122i 0.402237 0.402237i −0.476784 0.879021i \(-0.658197\pi\)
0.879021 + 0.476784i \(0.158197\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −13.5678 4.46250i −0.517645 0.170255i
\(688\) 0 0
\(689\) 0.761329 0.0290043
\(690\) 0 0
\(691\) 12.6307 0.480494 0.240247 0.970712i \(-0.422772\pi\)
0.240247 + 0.970712i \(0.422772\pi\)
\(692\) 0 0
\(693\) −1.38519 9.17252i −0.0526192 0.348435i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −21.1154 + 21.1154i −0.799804 + 0.799804i
\(698\) 0 0
\(699\) 13.1300 + 26.0000i 0.496623 + 0.983410i
\(700\) 0 0
\(701\) 33.0388i 1.24786i −0.781480 0.623930i \(-0.785535\pi\)
0.781480 0.623930i \(-0.214465\pi\)
\(702\) 0 0
\(703\) −2.48023 2.48023i −0.0935437 0.0935437i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.45567 2.45567i −0.0923549 0.0923549i
\(708\) 0 0
\(709\) 17.8078i 0.668785i 0.942434 + 0.334392i \(0.108531\pi\)
−0.942434 + 0.334392i \(0.891469\pi\)
\(710\) 0 0
\(711\) 9.90388 13.4272i 0.371425 0.503561i
\(712\) 0 0
\(713\) 10.1250 10.1250i 0.379184 0.379184i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.2461 + 31.1524i −0.382648 + 1.16341i
\(718\) 0 0
\(719\) 9.65719 0.360153 0.180076 0.983653i \(-0.442366\pi\)
0.180076 + 0.983653i \(0.442366\pi\)
\(720\) 0 0
\(721\) −3.80776 −0.141809
\(722\) 0 0
\(723\) −0.133239 + 0.405102i −0.00495521 + 0.0150659i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 30.2208 30.2208i 1.12083 1.12083i 0.129209 0.991617i \(-0.458756\pi\)
0.991617 0.129209i \(-0.0412439\pi\)
\(728\) 0 0
\(729\) 8.89586 25.4924i 0.329476 0.944164i
\(730\) 0 0
\(731\) 49.4748i 1.82989i
\(732\) 0 0
\(733\) −26.1141 26.1141i −0.964545 0.964545i 0.0348475 0.999393i \(-0.488905\pi\)
−0.999393 + 0.0348475i \(0.988905\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −24.3207 24.3207i −0.895863 0.895863i
\(738\) 0 0
\(739\) 37.1771i 1.36758i −0.729678 0.683791i \(-0.760330\pi\)
0.729678 0.683791i \(-0.239670\pi\)
\(740\) 0 0
\(741\) 0.384472 + 0.761329i 0.0141239 + 0.0279681i
\(742\) 0 0
\(743\) 32.3772 32.3772i 1.18780 1.18780i 0.210129 0.977674i \(-0.432612\pi\)
0.977674 0.210129i \(-0.0673884\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 5.15086 0.777860i 0.188460 0.0284604i
\(748\) 0 0
\(749\) −20.2895 −0.741361
\(750\) 0 0
\(751\) 39.4233 1.43858 0.719288 0.694712i \(-0.244468\pi\)
0.719288 + 0.694712i \(0.244468\pi\)
\(752\) 0 0
\(753\) −15.8894 5.22606i −0.579041 0.190448i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 4.06854 4.06854i 0.147874 0.147874i −0.629294 0.777168i \(-0.716656\pi\)
0.777168 + 0.629294i \(0.216656\pi\)
\(758\) 0 0
\(759\) 8.30144 4.19224i 0.301323 0.152169i
\(760\) 0 0
\(761\) 41.3403i 1.49858i −0.662240 0.749292i \(-0.730394\pi\)
0.662240 0.749292i \(-0.269606\pi\)
\(762\) 0 0
\(763\) −9.14351 9.14351i −0.331017 0.331017i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −2.99401 2.99401i −0.108107 0.108107i
\(768\) 0 0
\(769\) 17.1231i 0.617475i −0.951147 0.308737i \(-0.900094\pi\)
0.951147 0.308737i \(-0.0999065\pi\)
\(770\) 0 0
\(771\) −43.6155 + 22.0259i −1.57077 + 0.793243i
\(772\) 0 0
\(773\) −25.2793 + 25.2793i −0.909234 + 0.909234i −0.996210 0.0869764i \(-0.972280\pi\)
0.0869764 + 0.996210i \(0.472280\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5.13858 + 1.69009i 0.184345 + 0.0606317i
\(778\) 0 0
\(779\) 6.94568 0.248855
\(780\) 0 0
\(781\) −43.6155 −1.56069
\(782\) 0 0
\(783\) −5.74366 + 4.07902i −0.205262 + 0.145772i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −22.2196 + 22.2196i −0.792044 + 0.792044i −0.981826 0.189782i \(-0.939222\pi\)
0.189782 + 0.981826i \(0.439222\pi\)
\(788\) 0 0
\(789\) −6.18435 12.2462i −0.220169 0.435977i
\(790\) 0 0
\(791\) 1.73642i 0.0617400i
\(792\) 0 0
\(793\) −1.58831 1.58831i −0.0564026 0.0564026i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.6987 12.6987i −0.449810 0.449810i 0.445481 0.895291i \(-0.353033\pi\)
−0.895291 + 0.445481i \(0.853033\pi\)
\(798\) 0 0
\(799\) 14.9309i 0.528216i
\(800\) 0 0
\(801\) 29.8617 + 22.0259i 1.05511 + 0.778247i
\(802\) 0 0
\(803\) −13.1190 + 13.1190i −0.462959 + 0.462959i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 13.7988 41.9541i 0.485742 1.47686i
\(808\) 0 0
\(809\) −33.8002 −1.18835 −0.594175 0.804335i \(-0.702522\pi\)
−0.594175 + 0.804335i \(0.702522\pi\)
\(810\) 0 0
\(811\) −40.7386 −1.43053 −0.715263 0.698855i \(-0.753693\pi\)
−0.715263 + 0.698855i \(0.753693\pi\)
\(812\) 0 0
\(813\) −9.26629 + 28.1734i −0.324983 + 0.988083i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 8.13709 8.13709i 0.284681 0.284681i
\(818\) 0 0
\(819\) −1.05854 0.780776i −0.0369885 0.0272825i
\(820\) 0 0
\(821\) 45.4075i 1.58473i 0.610044 + 0.792367i \(0.291152\pi\)
−0.610044 + 0.792367i \(0.708848\pi\)
\(822\) 0 0
\(823\) −16.3505 16.3505i −0.569943 0.569943i 0.362169 0.932112i \(-0.382036\pi\)
−0.932112 + 0.362169i \(0.882036\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 21.1755 + 21.1755i 0.736344 + 0.736344i 0.971868 0.235524i \(-0.0756808\pi\)
−0.235524 + 0.971868i \(0.575681\pi\)
\(828\) 0 0
\(829\) 19.7538i 0.686077i −0.939321 0.343039i \(-0.888544\pi\)
0.939321 0.343039i \(-0.111456\pi\)
\(830\) 0 0
\(831\) 11.8078 + 23.3817i 0.409607 + 0.811101i
\(832\) 0 0
\(833\) 3.41433 3.41433i 0.118300 0.118300i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −34.9351 + 24.8101i −1.20753 + 0.857562i
\(838\) 0 0
\(839\) −44.8131 −1.54712 −0.773560 0.633723i \(-0.781526\pi\)
−0.773560 + 0.633723i \(0.781526\pi\)
\(840\) 0 0
\(841\) −27.1619 −0.936618
\(842\) 0 0
\(843\) 12.4061 + 4.08038i 0.427287 + 0.140536i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.01714 1.01714i 0.0349492 0.0349492i
\(848\) 0 0
\(849\) 7.24289 3.65767i 0.248575 0.125531i
\(850\) 0 0
\(851\) 5.42302i 0.185899i
\(852\) 0 0
\(853\) 14.6644 + 14.6644i 0.502100 + 0.502100i 0.912090 0.409990i \(-0.134468\pi\)
−0.409990 + 0.912090i \(0.634468\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 13.6573 + 13.6573i 0.466526 + 0.466526i 0.900787 0.434261i \(-0.142991\pi\)
−0.434261 + 0.900787i \(0.642991\pi\)
\(858\) 0 0
\(859\) 35.3693i 1.20679i 0.797444 + 0.603393i \(0.206185\pi\)
−0.797444 + 0.603393i \(0.793815\pi\)
\(860\) 0 0
\(861\) −9.56155 + 4.82860i −0.325857 + 0.164558i
\(862\) 0 0
\(863\) 5.06249 5.06249i 0.172329 0.172329i −0.615673 0.788002i \(-0.711116\pi\)
0.788002 + 0.615673i \(0.211116\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −10.3909 3.41759i −0.352893 0.116067i
\(868\) 0 0
\(869\) −17.1973 −0.583378
\(870\) 0 0
\(871\) −4.87689 −0.165247
\(872\) 0 0
\(873\) −10.5649 + 1.59546i −0.357567 + 0.0539981i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 11.6619 11.6619i 0.393795 0.393795i −0.482243 0.876037i \(-0.660178\pi\)
0.876037 + 0.482243i \(0.160178\pi\)
\(878\) 0 0
\(879\) 13.4272 + 26.5885i 0.452890 + 0.896809i
\(880\) 0 0
\(881\) 17.7917i 0.599419i 0.954031 + 0.299709i \(0.0968897\pi\)
−0.954031 + 0.299709i \(0.903110\pi\)
\(882\) 0 0
\(883\) 12.0101 + 12.0101i 0.404172 + 0.404172i 0.879700 0.475528i \(-0.157743\pi\)
−0.475528 + 0.879700i \(0.657743\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.4295 + 12.4295i 0.417342 + 0.417342i 0.884286 0.466945i \(-0.154645\pi\)
−0.466945 + 0.884286i \(0.654645\pi\)
\(888\) 0 0
\(889\) 8.87689i 0.297721i
\(890\) 0 0
\(891\) −26.5885 + 8.21799i −0.890750 + 0.275313i
\(892\) 0 0
\(893\) 2.45567 2.45567i 0.0821758 0.0821758i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0.411998 1.25265i 0.0137562 0.0418246i
\(898\) 0 0
\(899\) 11.1798 0.372869
\(900\) 0 0
\(901\) 8.38447 0.279327
\(902\) 0 0
\(903\) −5.54481 + 16.8585i −0.184520 + 0.561016i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 13.5221 13.5221i 0.448993 0.448993i −0.446027 0.895020i \(-0.647161\pi\)
0.895020 + 0.446027i \(0.147161\pi\)
\(908\) 0 0
\(909\) −6.18435 + 8.38447i −0.205122 + 0.278095i
\(910\) 0 0
\(911\) 19.5281i 0.646996i −0.946229 0.323498i \(-0.895141\pi\)
0.946229 0.323498i \(-0.104859\pi\)
\(912\) 0 0
\(913\) −3.79668 3.79668i −0.125652 0.125652i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 13.1190 + 13.1190i 0.433227 + 0.433227i
\(918\) 0 0
\(919\) 1.17708i 0.0388283i −0.999812 0.0194142i \(-0.993820\pi\)
0.999812 0.0194142i \(-0.00618011\pi\)
\(920\) 0 0
\(921\) −3.95786 7.83732i −0.130416 0.258249i
\(922\) 0 0
\(923\) −4.37300 + 4.37300i −0.143939 + 0.143939i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.70575 + 11.2952i 0.0560243 + 0.370984i
\(928\) 0 0
\(929\) 34.3946 1.12845 0.564225 0.825621i \(-0.309175\pi\)
0.564225 + 0.825621i \(0.309175\pi\)
\(930\) 0 0
\(931\) −1.12311 −0.0368083
\(932\) 0 0
\(933\) 46.4155 + 15.2662i 1.51958 + 0.499792i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.2589 30.2589i 0.988517 0.988517i −0.0114182 0.999935i \(-0.503635\pi\)
0.999935 + 0.0114182i \(0.00363460\pi\)
\(938\) 0 0
\(939\) −52.2698 + 26.3963i −1.70576 + 0.861411i
\(940\) 0 0
\(941\) 50.9975i 1.66247i 0.555921 + 0.831235i \(0.312366\pi\)
−0.555921 + 0.831235i \(0.687634\pi\)
\(942\) 0 0
\(943\) −7.59336 7.59336i −0.247274 0.247274i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −25.0101 25.0101i −0.812721 0.812721i 0.172320 0.985041i \(-0.444874\pi\)
−0.985041 + 0.172320i \(0.944874\pi\)
\(948\) 0 0
\(949\) 2.63068i 0.0853956i
\(950\) 0 0
\(951\) 45.1231 22.7872i 1.46322 0.738926i
\(952\) 0 0
\(953\) −27.4658 + 27.4658i −0.889705 + 0.889705i −0.994494 0.104789i \(-0.966583\pi\)
0.104789 + 0.994494i \(0.466583\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 6.89766 + 2.26866i 0.222970 + 0.0733353i
\(958\) 0 0
\(959\) 4.44793 0.143631
\(960\) 0 0
\(961\) 37.0000 1.19355
\(962\) 0 0
\(963\) 9.08903 + 60.1860i 0.292890 + 1.93947i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −30.2208 + 30.2208i −0.971835 + 0.971835i −0.999614 0.0277794i \(-0.991156\pi\)
0.0277794 + 0.999614i \(0.491156\pi\)
\(968\) 0 0
\(969\) 4.23417 + 8.38447i 0.136021 + 0.269348i
\(970\) 0 0
\(971\) 50.2361i 1.61215i 0.591810 + 0.806077i \(0.298413\pi\)
−0.591810 + 0.806077i \(0.701587\pi\)
\(972\) 0 0
\(973\) −7.07107 7.07107i −0.226688 0.226688i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 41.6615 + 41.6615i 1.33287 + 1.33287i 0.902792 + 0.430077i \(0.141514\pi\)
0.430077 + 0.902792i \(0.358486\pi\)
\(978\) 0 0
\(979\) 38.2462i 1.22235i
\(980\) 0 0
\(981\) −23.0270 + 31.2190i −0.735195 + 0.996745i
\(982\) 0 0
\(983\) 4.10383 4.10383i 0.130892 0.130892i −0.638626 0.769517i \(-0.720497\pi\)
0.769517 + 0.638626i \(0.220497\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −1.67335 + 5.08769i −0.0532634 + 0.161943i
\(988\) 0 0
\(989\) −17.7917 −0.565744
\(990\) 0 0
\(991\) −21.7538 −0.691032 −0.345516 0.938413i \(-0.612296\pi\)
−0.345516 + 0.938413i \(0.612296\pi\)
\(992\) 0 0
\(993\) 3.38018 10.2772i 0.107267 0.326136i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 43.7047 43.7047i 1.38414 1.38414i 0.547025 0.837116i \(-0.315760\pi\)
0.837116 0.547025i \(-0.184240\pi\)
\(998\) 0 0
\(999\) 2.71151 16.0000i 0.0857884 0.506218i
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.s.a.1793.3 yes 16
3.2 odd 2 inner 2100.2.s.a.1793.7 yes 16
5.2 odd 4 inner 2100.2.s.a.1457.7 yes 16
5.3 odd 4 inner 2100.2.s.a.1457.2 16
5.4 even 2 inner 2100.2.s.a.1793.6 yes 16
15.2 even 4 inner 2100.2.s.a.1457.3 yes 16
15.8 even 4 inner 2100.2.s.a.1457.6 yes 16
15.14 odd 2 inner 2100.2.s.a.1793.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.s.a.1457.2 16 5.3 odd 4 inner
2100.2.s.a.1457.3 yes 16 15.2 even 4 inner
2100.2.s.a.1457.6 yes 16 15.8 even 4 inner
2100.2.s.a.1457.7 yes 16 5.2 odd 4 inner
2100.2.s.a.1793.2 yes 16 15.14 odd 2 inner
2100.2.s.a.1793.3 yes 16 1.1 even 1 trivial
2100.2.s.a.1793.6 yes 16 5.4 even 2 inner
2100.2.s.a.1793.7 yes 16 3.2 odd 2 inner