Properties

Label 2100.2.x.c.1693.4
Level $2100$
Weight $2$
Character 2100.1693
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(1357,2100)]
 
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, 0, 1, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1357");
 
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.x (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: 16.0.478584585616890104119296.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 31x^{12} + 336x^{8} - 19375x^{4} + 390625 \) 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 1693.4
Root \(1.97578 - 1.04705i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1693
Dual form 2100.2.x.c.1357.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(2.49342 - 0.884806i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(2.49342 - 0.884806i) q^{7} -1.00000i q^{9} +3.27492 q^{11} +(-3.02282 + 3.02282i) q^{13} +(1.41421 + 1.41421i) q^{17} +2.15068 q^{19} +(-1.13746 + 2.38876i) q^{21} +(-0.296014 - 0.296014i) q^{23} +(0.707107 + 0.707107i) q^{27} +0.725083i q^{29} +1.31342i q^{31} +(-2.31572 + 2.31572i) q^{33} +(1.56145 - 1.56145i) q^{37} -4.27492i q^{39} -5.25370i q^{41} +(0.632717 + 0.632717i) q^{43} +(1.41421 + 1.41421i) q^{47} +(5.43424 - 4.41238i) q^{49} -2.00000 q^{51} +(3.71492 + 3.71492i) q^{53} +(-1.52076 + 1.52076i) q^{57} +8.71780 q^{59} -1.31342i q^{61} +(-0.884806 - 2.49342i) q^{63} +(-3.08221 + 3.08221i) q^{67} +0.418627 q^{69} +9.27492 q^{71} +(-7.07107 + 7.07107i) q^{73} +(8.16573 - 2.89767i) q^{77} -11.8248i q^{79} -1.00000 q^{81} +(-11.7025 + 11.7025i) q^{83} +(-0.512711 - 0.512711i) q^{87} +18.2728 q^{89} +(-4.86254 + 10.2118i) q^{91} +(-0.928731 - 0.928731i) q^{93} +(7.26546 + 7.26546i) q^{97} -3.27492i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{11} + 12 q^{21} - 32 q^{51} + 88 q^{71} - 16 q^{81} - 108 q^{91}+O(q^{100}) \) 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
\(3\) −0.707107 + 0.707107i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.49342 0.884806i 0.942422 0.334425i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) 3.27492 0.987425 0.493712 0.869625i \(-0.335640\pi\)
0.493712 + 0.869625i \(0.335640\pi\)
\(12\) 0 0
\(13\) −3.02282 + 3.02282i −0.838380 + 0.838380i −0.988646 0.150265i \(-0.951987\pi\)
0.150265 + 0.988646i \(0.451987\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421 + 1.41421i 0.342997 + 0.342997i 0.857493 0.514496i \(-0.172021\pi\)
−0.514496 + 0.857493i \(0.672021\pi\)
\(18\) 0 0
\(19\) 2.15068 0.493399 0.246700 0.969092i \(-0.420654\pi\)
0.246700 + 0.969092i \(0.420654\pi\)
\(20\) 0 0
\(21\) −1.13746 + 2.38876i −0.248214 + 0.521271i
\(22\) 0 0
\(23\) −0.296014 0.296014i −0.0617231 0.0617231i 0.675571 0.737295i \(-0.263897\pi\)
−0.737295 + 0.675571i \(0.763897\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 0.725083i 0.134644i 0.997731 + 0.0673222i \(0.0214456\pi\)
−0.997731 + 0.0673222i \(0.978554\pi\)
\(30\) 0 0
\(31\) 1.31342i 0.235898i 0.993020 + 0.117949i \(0.0376319\pi\)
−0.993020 + 0.117949i \(0.962368\pi\)
\(32\) 0 0
\(33\) −2.31572 + 2.31572i −0.403114 + 0.403114i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.56145 1.56145i 0.256701 0.256701i −0.567010 0.823711i \(-0.691900\pi\)
0.823711 + 0.567010i \(0.191900\pi\)
\(38\) 0 0
\(39\) 4.27492i 0.684535i
\(40\) 0 0
\(41\) 5.25370i 0.820490i −0.911975 0.410245i \(-0.865443\pi\)
0.911975 0.410245i \(-0.134557\pi\)
\(42\) 0 0
\(43\) 0.632717 + 0.632717i 0.0964885 + 0.0964885i 0.753703 0.657215i \(-0.228266\pi\)
−0.657215 + 0.753703i \(0.728266\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.41421 + 1.41421i 0.206284 + 0.206284i 0.802686 0.596402i \(-0.203403\pi\)
−0.596402 + 0.802686i \(0.703403\pi\)
\(48\) 0 0
\(49\) 5.43424 4.41238i 0.776320 0.630339i
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) 0 0
\(53\) 3.71492 + 3.71492i 0.510284 + 0.510284i 0.914613 0.404329i \(-0.132495\pi\)
−0.404329 + 0.914613i \(0.632495\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.52076 + 1.52076i −0.201429 + 0.201429i
\(58\) 0 0
\(59\) 8.71780 1.13496 0.567480 0.823387i \(-0.307918\pi\)
0.567480 + 0.823387i \(0.307918\pi\)
\(60\) 0 0
\(61\) 1.31342i 0.168167i −0.996459 0.0840834i \(-0.973204\pi\)
0.996459 0.0840834i \(-0.0267962\pi\)
\(62\) 0 0
\(63\) −0.884806 2.49342i −0.111475 0.314141i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.08221 + 3.08221i −0.376552 + 0.376552i −0.869856 0.493305i \(-0.835789\pi\)
0.493305 + 0.869856i \(0.335789\pi\)
\(68\) 0 0
\(69\) 0.418627 0.0503967
\(70\) 0 0
\(71\) 9.27492 1.10073 0.550365 0.834924i \(-0.314489\pi\)
0.550365 + 0.834924i \(0.314489\pi\)
\(72\) 0 0
\(73\) −7.07107 + 7.07107i −0.827606 + 0.827606i −0.987185 0.159579i \(-0.948986\pi\)
0.159579 + 0.987185i \(0.448986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.16573 2.89767i 0.930571 0.330220i
\(78\) 0 0
\(79\) 11.8248i 1.33039i −0.746670 0.665194i \(-0.768349\pi\)
0.746670 0.665194i \(-0.231651\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) −11.7025 + 11.7025i −1.28452 + 1.28452i −0.346447 + 0.938070i \(0.612612\pi\)
−0.938070 + 0.346447i \(0.887388\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −0.512711 0.512711i −0.0549684 0.0549684i
\(88\) 0 0
\(89\) 18.2728 1.93692 0.968459 0.249173i \(-0.0801587\pi\)
0.968459 + 0.249173i \(0.0801587\pi\)
\(90\) 0 0
\(91\) −4.86254 + 10.2118i −0.509733 + 1.07048i
\(92\) 0 0
\(93\) −0.928731 0.928731i −0.0963049 0.0963049i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.26546 + 7.26546i 0.737696 + 0.737696i 0.972132 0.234436i \(-0.0753242\pi\)
−0.234436 + 0.972132i \(0.575324\pi\)
\(98\) 0 0
\(99\) 3.27492i 0.329142i
\(100\) 0 0
\(101\) 7.76546i 0.772692i −0.922354 0.386346i \(-0.873737\pi\)
0.922354 0.386346i \(-0.126263\pi\)
\(102\) 0 0
\(103\) −4.63143 + 4.63143i −0.456349 + 0.456349i −0.897455 0.441106i \(-0.854586\pi\)
0.441106 + 0.897455i \(0.354586\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.30695 4.30695i 0.416369 0.416369i −0.467581 0.883950i \(-0.654874\pi\)
0.883950 + 0.467581i \(0.154874\pi\)
\(108\) 0 0
\(109\) 1.00000i 0.0957826i −0.998853 0.0478913i \(-0.984750\pi\)
0.998853 0.0478913i \(-0.0152501\pi\)
\(110\) 0 0
\(111\) 2.20822i 0.209595i
\(112\) 0 0
\(113\) 11.3594 + 11.3594i 1.06860 + 1.06860i 0.997467 + 0.0711366i \(0.0226626\pi\)
0.0711366 + 0.997467i \(0.477337\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.02282 + 3.02282i 0.279460 + 0.279460i
\(118\) 0 0
\(119\) 4.77753 + 2.27492i 0.437955 + 0.208541i
\(120\) 0 0
\(121\) −0.274917 −0.0249925
\(122\) 0 0
\(123\) 3.71492 + 3.71492i 0.334963 + 0.334963i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −9.50195 + 9.50195i −0.843161 + 0.843161i −0.989269 0.146107i \(-0.953325\pi\)
0.146107 + 0.989269i \(0.453325\pi\)
\(128\) 0 0
\(129\) −0.894797 −0.0787825
\(130\) 0 0
\(131\) 19.9474i 1.74281i 0.490566 + 0.871404i \(0.336790\pi\)
−0.490566 + 0.871404i \(0.663210\pi\)
\(132\) 0 0
\(133\) 5.36253 1.90293i 0.464990 0.165005i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.3703 15.3703i 1.31318 1.31318i 0.394117 0.919060i \(-0.371051\pi\)
0.919060 0.394117i \(-0.128949\pi\)
\(138\) 0 0
\(139\) 3.46410 0.293821 0.146911 0.989150i \(-0.453067\pi\)
0.146911 + 0.989150i \(0.453067\pi\)
\(140\) 0 0
\(141\) −2.00000 −0.168430
\(142\) 0 0
\(143\) −9.89949 + 9.89949i −0.827837 + 0.827837i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −0.722565 + 6.96261i −0.0595962 + 0.574266i
\(148\) 0 0
\(149\) 23.2749i 1.90676i 0.301779 + 0.953378i \(0.402419\pi\)
−0.301779 + 0.953378i \(0.597581\pi\)
\(150\) 0 0
\(151\) −11.0000 −0.895167 −0.447584 0.894242i \(-0.647715\pi\)
−0.447584 + 0.894242i \(0.647715\pi\)
\(152\) 0 0
\(153\) 1.41421 1.41421i 0.114332 0.114332i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.65426 + 7.65426i 0.610876 + 0.610876i 0.943174 0.332298i \(-0.107824\pi\)
−0.332298 + 0.943174i \(0.607824\pi\)
\(158\) 0 0
\(159\) −5.25370 −0.416645
\(160\) 0 0
\(161\) −1.00000 0.476171i −0.0788110 0.0375275i
\(162\) 0 0
\(163\) 15.7070 + 15.7070i 1.23027 + 1.23027i 0.963859 + 0.266412i \(0.0858381\pi\)
0.266412 + 0.963859i \(0.414162\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.45986 7.45986i −0.577261 0.577261i 0.356886 0.934148i \(-0.383838\pi\)
−0.934148 + 0.356886i \(0.883838\pi\)
\(168\) 0 0
\(169\) 5.27492i 0.405763i
\(170\) 0 0
\(171\) 2.15068i 0.164466i
\(172\) 0 0
\(173\) 13.5055 13.5055i 1.02680 1.02680i 0.0271738 0.999631i \(-0.491349\pi\)
0.999631 0.0271738i \(-0.00865077\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −6.16441 + 6.16441i −0.463346 + 0.463346i
\(178\) 0 0
\(179\) 7.45017i 0.556852i 0.960458 + 0.278426i \(0.0898126\pi\)
−0.960458 + 0.278426i \(0.910187\pi\)
\(180\) 0 0
\(181\) 3.10302i 0.230646i 0.993328 + 0.115323i \(0.0367902\pi\)
−0.993328 + 0.115323i \(0.963210\pi\)
\(182\) 0 0
\(183\) 0.928731 + 0.928731i 0.0686538 + 0.0686538i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.63143 + 4.63143i 0.338684 + 0.338684i
\(188\) 0 0
\(189\) 2.38876 + 1.13746i 0.173757 + 0.0827379i
\(190\) 0 0
\(191\) 12.5498 0.908074 0.454037 0.890983i \(-0.349983\pi\)
0.454037 + 0.890983i \(0.349983\pi\)
\(192\) 0 0
\(193\) −13.5536 13.5536i −0.975608 0.975608i 0.0241020 0.999710i \(-0.492327\pi\)
−0.999710 + 0.0241020i \(0.992327\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 12.0328 12.0328i 0.857303 0.857303i −0.133717 0.991020i \(-0.542691\pi\)
0.991020 + 0.133717i \(0.0426913\pi\)
\(198\) 0 0
\(199\) −15.1698 −1.07536 −0.537680 0.843149i \(-0.680699\pi\)
−0.537680 + 0.843149i \(0.680699\pi\)
\(200\) 0 0
\(201\) 4.35890i 0.307453i
\(202\) 0 0
\(203\) 0.641557 + 1.80793i 0.0450285 + 0.126892i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.296014 + 0.296014i −0.0205744 + 0.0205744i
\(208\) 0 0
\(209\) 7.04329 0.487195
\(210\) 0 0
\(211\) 1.72508 0.118760 0.0593798 0.998235i \(-0.481088\pi\)
0.0593798 + 0.998235i \(0.481088\pi\)
\(212\) 0 0
\(213\) −6.55836 + 6.55836i −0.449371 + 0.449371i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.16213 + 3.27491i 0.0788902 + 0.222315i
\(218\) 0 0
\(219\) 10.0000i 0.675737i
\(220\) 0 0
\(221\) −8.54983 −0.575124
\(222\) 0 0
\(223\) 3.41161 3.41161i 0.228459 0.228459i −0.583590 0.812049i \(-0.698352\pi\)
0.812049 + 0.583590i \(0.198352\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −13.8943 13.8943i −0.922197 0.922197i 0.0749876 0.997184i \(-0.476108\pi\)
−0.997184 + 0.0749876i \(0.976108\pi\)
\(228\) 0 0
\(229\) 12.5430 0.828864 0.414432 0.910080i \(-0.363980\pi\)
0.414432 + 0.910080i \(0.363980\pi\)
\(230\) 0 0
\(231\) −3.72508 + 7.82300i −0.245092 + 0.514716i
\(232\) 0 0
\(233\) −6.46043 6.46043i −0.423237 0.423237i 0.463080 0.886317i \(-0.346744\pi\)
−0.886317 + 0.463080i \(0.846744\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 8.36136 + 8.36136i 0.543129 + 0.543129i
\(238\) 0 0
\(239\) 9.09967i 0.588609i 0.955712 + 0.294304i \(0.0950879\pi\)
−0.955712 + 0.294304i \(0.904912\pi\)
\(240\) 0 0
\(241\) 28.1890i 1.81581i −0.419174 0.907906i \(-0.637680\pi\)
0.419174 0.907906i \(-0.362320\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.50112 + 6.50112i −0.413656 + 0.413656i
\(248\) 0 0
\(249\) 16.5498i 1.04880i
\(250\) 0 0
\(251\) 6.09095i 0.384457i −0.981350 0.192229i \(-0.938428\pi\)
0.981350 0.192229i \(-0.0615715\pi\)
\(252\) 0 0
\(253\) −0.969421 0.969421i −0.0609470 0.0609470i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.60601 3.60601i −0.224937 0.224937i 0.585637 0.810574i \(-0.300844\pi\)
−0.810574 + 0.585637i \(0.800844\pi\)
\(258\) 0 0
\(259\) 2.51176 5.27492i 0.156073 0.327767i
\(260\) 0 0
\(261\) 0.725083 0.0448815
\(262\) 0 0
\(263\) −7.05246 7.05246i −0.434873 0.434873i 0.455409 0.890282i \(-0.349493\pi\)
−0.890282 + 0.455409i \(0.849493\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −12.9209 + 12.9209i −0.790744 + 0.790744i
\(268\) 0 0
\(269\) −18.1578 −1.10710 −0.553549 0.832817i \(-0.686727\pi\)
−0.553549 + 0.832817i \(0.686727\pi\)
\(270\) 0 0
\(271\) 22.5742i 1.37129i 0.727938 + 0.685643i \(0.240479\pi\)
−0.727938 + 0.685643i \(0.759521\pi\)
\(272\) 0 0
\(273\) −3.78247 10.6591i −0.228926 0.645121i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.97025 3.97025i 0.238549 0.238549i −0.577700 0.816249i \(-0.696050\pi\)
0.816249 + 0.577700i \(0.196050\pi\)
\(278\) 0 0
\(279\) 1.31342 0.0786326
\(280\) 0 0
\(281\) 12.7251 0.759115 0.379557 0.925168i \(-0.376076\pi\)
0.379557 + 0.925168i \(0.376076\pi\)
\(282\) 0 0
\(283\) −18.9680 + 18.9680i −1.12753 + 1.12753i −0.136951 + 0.990578i \(0.543730\pi\)
−0.990578 + 0.136951i \(0.956270\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.64850 13.0996i −0.274392 0.773248i
\(288\) 0 0
\(289\) 13.0000i 0.764706i
\(290\) 0 0
\(291\) −10.2749 −0.602326
\(292\) 0 0
\(293\) 10.6771 10.6771i 0.623762 0.623762i −0.322730 0.946491i \(-0.604600\pi\)
0.946491 + 0.322730i \(0.104600\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.31572 + 2.31572i 0.134371 + 0.134371i
\(298\) 0 0
\(299\) 1.78959 0.103495
\(300\) 0 0
\(301\) 2.13746 + 1.01779i 0.123201 + 0.0586647i
\(302\) 0 0
\(303\) 5.49101 + 5.49101i 0.315450 + 0.315450i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −11.1193 11.1193i −0.634613 0.634613i 0.314609 0.949221i \(-0.398127\pi\)
−0.949221 + 0.314609i \(0.898127\pi\)
\(308\) 0 0
\(309\) 6.54983i 0.372607i
\(310\) 0 0
\(311\) 33.0816i 1.87589i −0.346790 0.937943i \(-0.612729\pi\)
0.346790 0.937943i \(-0.387271\pi\)
\(312\) 0 0
\(313\) 19.6046 19.6046i 1.10812 1.10812i 0.114719 0.993398i \(-0.463403\pi\)
0.993398 0.114719i \(-0.0365968\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.64448 + 7.64448i −0.429357 + 0.429357i −0.888409 0.459052i \(-0.848189\pi\)
0.459052 + 0.888409i \(0.348189\pi\)
\(318\) 0 0
\(319\) 2.37459i 0.132951i
\(320\) 0 0
\(321\) 6.09095i 0.339964i
\(322\) 0 0
\(323\) 3.04152 + 3.04152i 0.169235 + 0.169235i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.707107 + 0.707107i 0.0391031 + 0.0391031i
\(328\) 0 0
\(329\) 4.77753 + 2.27492i 0.263394 + 0.125420i
\(330\) 0 0
\(331\) 10.3746 0.570239 0.285119 0.958492i \(-0.407967\pi\)
0.285119 + 0.958492i \(0.407967\pi\)
\(332\) 0 0
\(333\) −1.56145 1.56145i −0.0855668 0.0855668i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −12.5842 + 12.5842i −0.685502 + 0.685502i −0.961235 0.275732i \(-0.911080\pi\)
0.275732 + 0.961235i \(0.411080\pi\)
\(338\) 0 0
\(339\) −16.0646 −0.872511
\(340\) 0 0
\(341\) 4.30136i 0.232931i
\(342\) 0 0
\(343\) 9.64572 15.8101i 0.520820 0.853667i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.50195 9.50195i 0.510091 0.510091i −0.404463 0.914554i \(-0.632542\pi\)
0.914554 + 0.404463i \(0.132542\pi\)
\(348\) 0 0
\(349\) −18.2728 −0.978123 −0.489062 0.872249i \(-0.662661\pi\)
−0.489062 + 0.872249i \(0.662661\pi\)
\(350\) 0 0
\(351\) −4.27492 −0.228178
\(352\) 0 0
\(353\) −5.26806 + 5.26806i −0.280391 + 0.280391i −0.833265 0.552874i \(-0.813531\pi\)
0.552874 + 0.833265i \(0.313531\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −4.98683 + 1.76961i −0.263931 + 0.0936578i
\(358\) 0 0
\(359\) 26.3746i 1.39200i 0.718043 + 0.695999i \(0.245038\pi\)
−0.718043 + 0.695999i \(0.754962\pi\)
\(360\) 0 0
\(361\) −14.3746 −0.756557
\(362\) 0 0
\(363\) 0.194396 0.194396i 0.0102031 0.0102031i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −18.9680 18.9680i −0.990120 0.990120i 0.00983201 0.999952i \(-0.496870\pi\)
−0.999952 + 0.00983201i \(0.996870\pi\)
\(368\) 0 0
\(369\) −5.25370 −0.273497
\(370\) 0 0
\(371\) 12.5498 + 5.97586i 0.651555 + 0.310251i
\(372\) 0 0
\(373\) −12.9615 12.9615i −0.671123 0.671123i 0.286852 0.957975i \(-0.407391\pi\)
−0.957975 + 0.286852i \(0.907391\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.19180 2.19180i −0.112883 0.112883i
\(378\) 0 0
\(379\) 15.0000i 0.770498i 0.922813 + 0.385249i \(0.125884\pi\)
−0.922813 + 0.385249i \(0.874116\pi\)
\(380\) 0 0
\(381\) 13.4378i 0.688438i
\(382\) 0 0
\(383\) 15.1676 15.1676i 0.775026 0.775026i −0.203954 0.978980i \(-0.565379\pi\)
0.978980 + 0.203954i \(0.0653794\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.632717 0.632717i 0.0321628 0.0321628i
\(388\) 0 0
\(389\) 18.3746i 0.931628i 0.884883 + 0.465814i \(0.154239\pi\)
−0.884883 + 0.465814i \(0.845761\pi\)
\(390\) 0 0
\(391\) 0.837253i 0.0423417i
\(392\) 0 0
\(393\) −14.1049 14.1049i −0.711499 0.711499i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.43704 + 4.43704i 0.222688 + 0.222688i 0.809630 0.586941i \(-0.199668\pi\)
−0.586941 + 0.809630i \(0.699668\pi\)
\(398\) 0 0
\(399\) −2.44631 + 5.13746i −0.122469 + 0.257195i
\(400\) 0 0
\(401\) −1.27492 −0.0636663 −0.0318332 0.999493i \(-0.510135\pi\)
−0.0318332 + 0.999493i \(0.510135\pi\)
\(402\) 0 0
\(403\) −3.97025 3.97025i −0.197772 0.197772i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 5.11361 5.11361i 0.253472 0.253472i
\(408\) 0 0
\(409\) −31.7682 −1.57084 −0.785418 0.618966i \(-0.787552\pi\)
−0.785418 + 0.618966i \(0.787552\pi\)
\(410\) 0 0
\(411\) 21.7370i 1.07220i
\(412\) 0 0
\(413\) 21.7371 7.71356i 1.06961 0.379559i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.44949 + 2.44949i −0.119952 + 0.119952i
\(418\) 0 0
\(419\) −12.1819 −0.595125 −0.297562 0.954702i \(-0.596174\pi\)
−0.297562 + 0.954702i \(0.596174\pi\)
\(420\) 0 0
\(421\) 0.725083 0.0353384 0.0176692 0.999844i \(-0.494375\pi\)
0.0176692 + 0.999844i \(0.494375\pi\)
\(422\) 0 0
\(423\) 1.41421 1.41421i 0.0687614 0.0687614i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.16213 3.27491i −0.0562392 0.158484i
\(428\) 0 0
\(429\) 14.0000i 0.675926i
\(430\) 0 0
\(431\) 25.6495 1.23549 0.617747 0.786377i \(-0.288046\pi\)
0.617747 + 0.786377i \(0.288046\pi\)
\(432\) 0 0
\(433\) 4.43704 4.43704i 0.213230 0.213230i −0.592408 0.805638i \(-0.701823\pi\)
0.805638 + 0.592408i \(0.201823\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.636630 0.636630i −0.0304542 0.0304542i
\(438\) 0 0
\(439\) −33.4427 −1.59613 −0.798066 0.602570i \(-0.794143\pi\)
−0.798066 + 0.602570i \(0.794143\pi\)
\(440\) 0 0
\(441\) −4.41238 5.43424i −0.210113 0.258773i
\(442\) 0 0
\(443\) −19.0039 19.0039i −0.902902 0.902902i 0.0927842 0.995686i \(-0.470423\pi\)
−0.995686 + 0.0927842i \(0.970423\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −16.4579 16.4579i −0.778430 0.778430i
\(448\) 0 0
\(449\) 7.82475i 0.369273i 0.982807 + 0.184636i \(0.0591108\pi\)
−0.982807 + 0.184636i \(0.940889\pi\)
\(450\) 0 0
\(451\) 17.2054i 0.810172i
\(452\) 0 0
\(453\) 7.77817 7.77817i 0.365451 0.365451i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.86840 5.86840i 0.274512 0.274512i −0.556401 0.830914i \(-0.687818\pi\)
0.830914 + 0.556401i \(0.187818\pi\)
\(458\) 0 0
\(459\) 2.00000i 0.0933520i
\(460\) 0 0
\(461\) 21.6219i 1.00703i −0.863986 0.503515i \(-0.832040\pi\)
0.863986 0.503515i \(-0.167960\pi\)
\(462\) 0 0
\(463\) −14.7783 14.7783i −0.686807 0.686807i 0.274718 0.961525i \(-0.411415\pi\)
−0.961525 + 0.274718i \(0.911415\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −0.388792 0.388792i −0.0179911 0.0179911i 0.698054 0.716045i \(-0.254050\pi\)
−0.716045 + 0.698054i \(0.754050\pi\)
\(468\) 0 0
\(469\) −4.95807 + 10.4124i −0.228942 + 0.480799i
\(470\) 0 0
\(471\) −10.8248 −0.498778
\(472\) 0 0
\(473\) 2.07210 + 2.07210i 0.0952751 + 0.0952751i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 3.71492 3.71492i 0.170095 0.170095i
\(478\) 0 0
\(479\) −24.2487 −1.10795 −0.553976 0.832533i \(-0.686890\pi\)
−0.553976 + 0.832533i \(0.686890\pi\)
\(480\) 0 0
\(481\) 9.43996i 0.430425i
\(482\) 0 0
\(483\) 1.04381 0.370403i 0.0474950 0.0168539i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −15.4110 + 15.4110i −0.698341 + 0.698341i −0.964052 0.265712i \(-0.914393\pi\)
0.265712 + 0.964052i \(0.414393\pi\)
\(488\) 0 0
\(489\) −22.2131 −1.00451
\(490\) 0 0
\(491\) 28.9244 1.30534 0.652670 0.757642i \(-0.273649\pi\)
0.652670 + 0.757642i \(0.273649\pi\)
\(492\) 0 0
\(493\) −1.02542 + 1.02542i −0.0461827 + 0.0461827i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 23.1262 8.20650i 1.03735 0.368112i
\(498\) 0 0
\(499\) 9.17525i 0.410741i −0.978684 0.205370i \(-0.934160\pi\)
0.978684 0.205370i \(-0.0658399\pi\)
\(500\) 0 0
\(501\) 10.5498 0.471332
\(502\) 0 0
\(503\) 8.48528 8.48528i 0.378340 0.378340i −0.492163 0.870503i \(-0.663794\pi\)
0.870503 + 0.492163i \(0.163794\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 3.72993 + 3.72993i 0.165652 + 0.165652i
\(508\) 0 0
\(509\) 17.3205 0.767718 0.383859 0.923392i \(-0.374595\pi\)
0.383859 + 0.923392i \(0.374595\pi\)
\(510\) 0 0
\(511\) −11.3746 + 23.8876i −0.503182 + 1.05673i
\(512\) 0 0
\(513\) 1.52076 + 1.52076i 0.0671431 + 0.0671431i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4.63143 + 4.63143i 0.203690 + 0.203690i
\(518\) 0 0
\(519\) 19.0997i 0.838382i
\(520\) 0 0
\(521\) 32.0142i 1.40257i −0.712883 0.701283i \(-0.752611\pi\)
0.712883 0.701283i \(-0.247389\pi\)
\(522\) 0 0
\(523\) 20.1343 20.1343i 0.880413 0.880413i −0.113163 0.993576i \(-0.536098\pi\)
0.993576 + 0.113163i \(0.0360983\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.85746 + 1.85746i −0.0809123 + 0.0809123i
\(528\) 0 0
\(529\) 22.8248i 0.992381i
\(530\) 0 0
\(531\) 8.71780i 0.378320i
\(532\) 0 0
\(533\) 15.8810 + 15.8810i 0.687882 + 0.687882i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.26806 5.26806i −0.227334 0.227334i
\(538\) 0 0
\(539\) 17.7967 14.4502i 0.766557 0.622413i
\(540\) 0 0
\(541\) −33.1993 −1.42735 −0.713676 0.700476i \(-0.752971\pi\)
−0.713676 + 0.700476i \(0.752971\pi\)
\(542\) 0 0
\(543\) −2.19417 2.19417i −0.0941607 0.0941607i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.19499 + 5.19499i −0.222122 + 0.222122i −0.809391 0.587270i \(-0.800203\pi\)
0.587270 + 0.809391i \(0.300203\pi\)
\(548\) 0 0
\(549\) −1.31342 −0.0560556
\(550\) 0 0
\(551\) 1.55942i 0.0664335i
\(552\) 0 0
\(553\) −10.4626 29.4840i −0.444915 1.25379i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.9318 + 16.9318i −0.717423 + 0.717423i −0.968077 0.250654i \(-0.919354\pi\)
0.250654 + 0.968077i \(0.419354\pi\)
\(558\) 0 0
\(559\) −3.82518 −0.161788
\(560\) 0 0
\(561\) −6.54983 −0.276534
\(562\) 0 0
\(563\) −19.4102 + 19.4102i −0.818042 + 0.818042i −0.985824 0.167782i \(-0.946340\pi\)
0.167782 + 0.985824i \(0.446340\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.49342 + 0.884806i −0.104714 + 0.0371583i
\(568\) 0 0
\(569\) 41.8248i 1.75338i 0.481051 + 0.876692i \(0.340255\pi\)
−0.481051 + 0.876692i \(0.659745\pi\)
\(570\) 0 0
\(571\) 18.0997 0.757448 0.378724 0.925510i \(-0.376363\pi\)
0.378724 + 0.925510i \(0.376363\pi\)
\(572\) 0 0
\(573\) −8.87407 + 8.87407i −0.370720 + 0.370720i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.67968 8.67968i −0.361340 0.361340i 0.502966 0.864306i \(-0.332242\pi\)
−0.864306 + 0.502966i \(0.832242\pi\)
\(578\) 0 0
\(579\) 19.1676 0.796580
\(580\) 0 0
\(581\) −18.8248 + 39.5336i −0.780982 + 1.64013i
\(582\) 0 0
\(583\) 12.1661 + 12.1661i 0.503867 + 0.503867i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −4.63143 4.63143i −0.191160 0.191160i 0.605037 0.796197i \(-0.293158\pi\)
−0.796197 + 0.605037i \(0.793158\pi\)
\(588\) 0 0
\(589\) 2.82475i 0.116392i
\(590\) 0 0
\(591\) 17.0170i 0.699985i
\(592\) 0 0
\(593\) 14.1421 14.1421i 0.580748 0.580748i −0.354361 0.935109i \(-0.615302\pi\)
0.935109 + 0.354361i \(0.115302\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.7267 10.7267i 0.439014 0.439014i
\(598\) 0 0
\(599\) 22.3746i 0.914201i 0.889415 + 0.457100i \(0.151112\pi\)
−0.889415 + 0.457100i \(0.848888\pi\)
\(600\) 0 0
\(601\) 24.0027i 0.979091i −0.871978 0.489546i \(-0.837163\pi\)
0.871978 0.489546i \(-0.162837\pi\)
\(602\) 0 0
\(603\) 3.08221 + 3.08221i 0.125517 + 0.125517i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −5.40902 5.40902i −0.219545 0.219545i 0.588762 0.808307i \(-0.299616\pi\)
−0.808307 + 0.588762i \(0.799616\pi\)
\(608\) 0 0
\(609\) −1.73205 0.824752i −0.0701862 0.0334206i
\(610\) 0 0
\(611\) −8.54983 −0.345889
\(612\) 0 0
\(613\) 29.7713 + 29.7713i 1.20245 + 1.20245i 0.973419 + 0.229031i \(0.0735558\pi\)
0.229031 + 0.973419i \(0.426444\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −5.27637 + 5.27637i −0.212419 + 0.212419i −0.805294 0.592875i \(-0.797993\pi\)
0.592875 + 0.805294i \(0.297993\pi\)
\(618\) 0 0
\(619\) −8.12654 −0.326633 −0.163317 0.986574i \(-0.552219\pi\)
−0.163317 + 0.986574i \(0.552219\pi\)
\(620\) 0 0
\(621\) 0.418627i 0.0167989i
\(622\) 0 0
\(623\) 45.5618 16.1679i 1.82539 0.647754i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.98036 + 4.98036i −0.198896 + 0.198896i
\(628\) 0 0
\(629\) 4.41644 0.176095
\(630\) 0 0
\(631\) −29.0000 −1.15447 −0.577236 0.816577i \(-0.695869\pi\)
−0.577236 + 0.816577i \(0.695869\pi\)
\(632\) 0 0
\(633\) −1.21982 + 1.21982i −0.0484834 + 0.0484834i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.08891 + 29.7646i −0.122387 + 1.17932i
\(638\) 0 0
\(639\) 9.27492i 0.366910i
\(640\) 0 0
\(641\) −13.2749 −0.524328 −0.262164 0.965023i \(-0.584436\pi\)
−0.262164 + 0.965023i \(0.584436\pi\)
\(642\) 0 0
\(643\) −21.3542 + 21.3542i −0.842126 + 0.842126i −0.989135 0.147009i \(-0.953035\pi\)
0.147009 + 0.989135i \(0.453035\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −17.7481 17.7481i −0.697752 0.697752i 0.266174 0.963925i \(-0.414241\pi\)
−0.963925 + 0.266174i \(0.914241\pi\)
\(648\) 0 0
\(649\) 28.5501 1.12069
\(650\) 0 0
\(651\) −3.13746 1.49397i −0.122967 0.0585531i
\(652\) 0 0
\(653\) −14.0235 14.0235i −0.548783 0.548783i 0.377306 0.926089i \(-0.376851\pi\)
−0.926089 + 0.377306i \(0.876851\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 7.07107 + 7.07107i 0.275869 + 0.275869i
\(658\) 0 0
\(659\) 33.0997i 1.28938i −0.764444 0.644690i \(-0.776986\pi\)
0.764444 0.644690i \(-0.223014\pi\)
\(660\) 0 0
\(661\) 33.8038i 1.31481i −0.753536 0.657407i \(-0.771653\pi\)
0.753536 0.657407i \(-0.228347\pi\)
\(662\) 0 0
\(663\) 6.04565 6.04565i 0.234793 0.234793i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.214635 0.214635i 0.00831068 0.00831068i
\(668\) 0 0
\(669\) 4.82475i 0.186536i
\(670\) 0 0
\(671\) 4.30136i 0.166052i
\(672\) 0 0
\(673\) −7.26709 7.26709i −0.280126 0.280126i 0.553033 0.833159i \(-0.313470\pi\)
−0.833159 + 0.553033i \(0.813470\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.26806 5.26806i −0.202468 0.202468i 0.598589 0.801057i \(-0.295728\pi\)
−0.801057 + 0.598589i \(0.795728\pi\)
\(678\) 0 0
\(679\) 24.5443 + 11.6873i 0.941925 + 0.448517i
\(680\) 0 0
\(681\) 19.6495 0.752971
\(682\) 0 0
\(683\) −13.7275 13.7275i −0.525269 0.525269i 0.393889 0.919158i \(-0.371129\pi\)
−0.919158 + 0.393889i \(0.871129\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −8.86923 + 8.86923i −0.338382 + 0.338382i
\(688\) 0 0
\(689\) −22.4591 −0.855624
\(690\) 0 0
\(691\) 47.7753i 1.81746i 0.417388 + 0.908728i \(0.362945\pi\)
−0.417388 + 0.908728i \(0.637055\pi\)
\(692\) 0 0
\(693\) −2.89767 8.16573i −0.110073 0.310190i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.42985 7.42985i 0.281426 0.281426i
\(698\) 0 0
\(699\) 9.13642 0.345571
\(700\) 0 0
\(701\) −45.2990 −1.71092 −0.855460 0.517869i \(-0.826725\pi\)
−0.855460 + 0.517869i \(0.826725\pi\)
\(702\) 0 0
\(703\) 3.35817 3.35817i 0.126656 0.126656i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.87092 19.3625i −0.258408 0.728202i
\(708\) 0 0
\(709\) 4.27492i 0.160548i −0.996773 0.0802739i \(-0.974420\pi\)
0.996773 0.0802739i \(-0.0255795\pi\)
\(710\) 0 0
\(711\) −11.8248 −0.443463
\(712\) 0 0
\(713\) 0.388792 0.388792i 0.0145604 0.0145604i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −6.43444 6.43444i −0.240298 0.240298i
\(718\) 0 0
\(719\) −32.2443 −1.20251 −0.601256 0.799057i \(-0.705333\pi\)
−0.601256 + 0.799057i \(0.705333\pi\)
\(720\) 0 0
\(721\) −7.45017 + 15.6460i −0.277459 + 0.582688i
\(722\) 0 0
\(723\) 19.9326 + 19.9326i 0.741302 + 0.741302i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −20.7710 20.7710i −0.770353 0.770353i 0.207815 0.978168i \(-0.433365\pi\)
−0.978168 + 0.207815i \(0.933365\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 1.78959i 0.0661905i
\(732\) 0 0
\(733\) 16.5818 16.5818i 0.612462 0.612462i −0.331125 0.943587i \(-0.607428\pi\)
0.943587 + 0.331125i \(0.107428\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −10.0940 + 10.0940i −0.371816 + 0.371816i
\(738\) 0 0
\(739\) 13.6254i 0.501219i 0.968088 + 0.250609i \(0.0806310\pi\)
−0.968088 + 0.250609i \(0.919369\pi\)
\(740\) 0 0
\(741\) 9.19397i 0.337749i
\(742\) 0 0
\(743\) 4.30695 + 4.30695i 0.158007 + 0.158007i 0.781683 0.623676i \(-0.214362\pi\)
−0.623676 + 0.781683i \(0.714362\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.7025 + 11.7025i 0.428172 + 0.428172i
\(748\) 0 0
\(749\) 6.92820 14.5498i 0.253151 0.531639i
\(750\) 0 0
\(751\) 24.0000 0.875772 0.437886 0.899030i \(-0.355727\pi\)
0.437886 + 0.899030i \(0.355727\pi\)
\(752\) 0 0
\(753\) 4.30695 + 4.30695i 0.156954 + 0.156954i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.16524 9.16524i 0.333116 0.333116i −0.520652 0.853769i \(-0.674311\pi\)
0.853769 + 0.520652i \(0.174311\pi\)
\(758\) 0 0
\(759\) 1.37097 0.0497630
\(760\) 0 0
\(761\) 13.8564i 0.502294i −0.967949 0.251147i \(-0.919192\pi\)
0.967949 0.251147i \(-0.0808078\pi\)
\(762\) 0 0
\(763\) −0.884806 2.49342i −0.0320321 0.0902677i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −26.3524 + 26.3524i −0.951529 + 0.951529i
\(768\) 0 0
\(769\) −37.8591 −1.36523 −0.682617 0.730776i \(-0.739158\pi\)
−0.682617 + 0.730776i \(0.739158\pi\)
\(770\) 0 0
\(771\) 5.09967 0.183660
\(772\) 0 0
\(773\) −16.9706 + 16.9706i −0.610389 + 0.610389i −0.943047 0.332659i \(-0.892054\pi\)
0.332659 + 0.943047i \(0.392054\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.95385 + 5.50601i 0.0700939 + 0.197527i
\(778\) 0 0
\(779\) 11.2990i 0.404829i
\(780\) 0 0
\(781\) 30.3746 1.08689
\(782\) 0 0
\(783\) −0.512711 + 0.512711i −0.0183228 + 0.0183228i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 24.3770 + 24.3770i 0.868945 + 0.868945i 0.992356 0.123410i \(-0.0393831\pi\)
−0.123410 + 0.992356i \(0.539383\pi\)
\(788\) 0 0
\(789\) 9.97368 0.355072
\(790\) 0 0
\(791\) 38.3746 + 18.2728i 1.36444 + 0.649708i
\(792\) 0 0
\(793\) 3.97025 + 3.97025i 0.140988 + 0.140988i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 13.1167 + 13.1167i 0.464618 + 0.464618i 0.900166 0.435548i \(-0.143445\pi\)
−0.435548 + 0.900166i \(0.643445\pi\)
\(798\) 0 0
\(799\) 4.00000i 0.141510i
\(800\) 0 0
\(801\) 18.2728i 0.645639i
\(802\) 0 0
\(803\) −23.1572 + 23.1572i −0.817198 + 0.817198i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 12.8395 12.8395i 0.451971 0.451971i
\(808\) 0 0
\(809\) 6.72508i 0.236441i 0.992987 + 0.118221i \(0.0377190\pi\)
−0.992987 + 0.118221i \(0.962281\pi\)
\(810\) 0 0
\(811\) 19.4712i 0.683726i −0.939750 0.341863i \(-0.888942\pi\)
0.939750 0.341863i \(-0.111058\pi\)
\(812\) 0 0
\(813\) −15.9624 15.9624i −0.559825 0.559825i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.36077 + 1.36077i 0.0476073 + 0.0476073i
\(818\) 0 0
\(819\) 10.2118 + 4.86254i 0.356828 + 0.169911i
\(820\) 0 0
\(821\) −34.0000 −1.18661 −0.593304 0.804978i \(-0.702177\pi\)
−0.593304 + 0.804978i \(0.702177\pi\)
\(822\) 0 0
\(823\) 6.20510 + 6.20510i 0.216296 + 0.216296i 0.806936 0.590639i \(-0.201124\pi\)
−0.590639 + 0.806936i \(0.701124\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 37.2011 37.2011i 1.29361 1.29361i 0.361072 0.932538i \(-0.382411\pi\)
0.932538 0.361072i \(-0.117589\pi\)
\(828\) 0 0
\(829\) 35.7084 1.24021 0.620103 0.784521i \(-0.287091\pi\)
0.620103 + 0.784521i \(0.287091\pi\)
\(830\) 0 0
\(831\) 5.61478i 0.194775i
\(832\) 0 0
\(833\) 13.9252 + 1.44513i 0.482480 + 0.0500708i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.928731 + 0.928731i −0.0321016 + 0.0321016i
\(838\) 0 0
\(839\) −19.2252 −0.663727 −0.331864 0.943327i \(-0.607677\pi\)
−0.331864 + 0.943327i \(0.607677\pi\)
\(840\) 0 0
\(841\) 28.4743 0.981871
\(842\) 0 0
\(843\) −8.99799 + 8.99799i −0.309907 + 0.309907i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −0.685483 + 0.243248i −0.0235535 + 0.00835811i
\(848\) 0 0
\(849\) 26.8248i 0.920623i
\(850\) 0 0
\(851\) −0.924421 −0.0316887
\(852\) 0 0
\(853\) −24.2360 + 24.2360i −0.829826 + 0.829826i −0.987492 0.157667i \(-0.949603\pi\)
0.157667 + 0.987492i \(0.449603\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.26286 + 9.26286i 0.316413 + 0.316413i 0.847388 0.530974i \(-0.178174\pi\)
−0.530974 + 0.847388i \(0.678174\pi\)
\(858\) 0 0
\(859\) 51.3544 1.75219 0.876096 0.482138i \(-0.160139\pi\)
0.876096 + 0.482138i \(0.160139\pi\)
\(860\) 0 0
\(861\) 12.5498 + 5.97586i 0.427697 + 0.203657i
\(862\) 0 0
\(863\) −29.7713 29.7713i −1.01343 1.01343i −0.999909 0.0135172i \(-0.995697\pi\)
−0.0135172 0.999909i \(-0.504303\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.19239 + 9.19239i 0.312190 + 0.312190i
\(868\) 0 0
\(869\) 38.7251i 1.31366i
\(870\) 0 0
\(871\) 18.6339i 0.631387i
\(872\) 0 0
\(873\) 7.26546 7.26546i 0.245899 0.245899i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 23.0555 23.0555i 0.778530 0.778530i −0.201051 0.979581i \(-0.564436\pi\)
0.979581 + 0.201051i \(0.0644358\pi\)
\(878\) 0 0
\(879\) 15.0997i 0.509299i
\(880\) 0 0
\(881\) 45.1484i 1.52109i 0.649286 + 0.760544i \(0.275068\pi\)
−0.649286 + 0.760544i \(0.724932\pi\)
\(882\) 0 0
\(883\) 20.9834 + 20.9834i 0.706148 + 0.706148i 0.965723 0.259575i \(-0.0835824\pi\)
−0.259575 + 0.965723i \(0.583582\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.26806 + 5.26806i 0.176884 + 0.176884i 0.789996 0.613112i \(-0.210083\pi\)
−0.613112 + 0.789996i \(0.710083\pi\)
\(888\) 0 0
\(889\) −15.2849 + 32.0997i −0.512640 + 1.07659i
\(890\) 0 0
\(891\) −3.27492 −0.109714
\(892\) 0 0
\(893\) 3.04152 + 3.04152i 0.101780 + 0.101780i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −1.26543 + 1.26543i −0.0422516 + 0.0422516i
\(898\) 0 0
\(899\) −0.952341 −0.0317624
\(900\) 0 0
\(901\) 10.5074i 0.350052i
\(902\) 0 0
\(903\) −2.23110 + 0.791722i −0.0742464 + 0.0263469i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 22.2082 22.2082i 0.737410 0.737410i −0.234666 0.972076i \(-0.575400\pi\)
0.972076 + 0.234666i \(0.0753996\pi\)
\(908\) 0 0
\(909\) −7.76546 −0.257564
\(910\) 0 0
\(911\) −12.7251 −0.421601 −0.210800 0.977529i \(-0.567607\pi\)
−0.210800 + 0.977529i \(0.567607\pi\)
\(912\) 0 0
\(913\) −38.3247 + 38.3247i −1.26836 + 1.26836i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 17.6495 + 49.7370i 0.582839 + 1.64246i
\(918\) 0 0
\(919\) 36.2990i 1.19739i 0.800976 + 0.598697i \(0.204315\pi\)
−0.800976 + 0.598697i \(0.795685\pi\)
\(920\) 0 0
\(921\) 15.7251 0.518159
\(922\) 0 0
\(923\) −28.0364 + 28.0364i −0.922830 + 0.922830i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 4.63143 + 4.63143i 0.152116 + 0.152116i
\(928\) 0 0
\(929\) 39.7796 1.30513 0.652564 0.757734i \(-0.273694\pi\)
0.652564 + 0.757734i \(0.273694\pi\)
\(930\) 0 0
\(931\) 11.6873 9.48960i 0.383036 0.311009i
\(932\) 0 0
\(933\) 23.3922 + 23.3922i 0.765827 + 0.765827i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 24.6248 + 24.6248i 0.804458 + 0.804458i 0.983789 0.179331i \(-0.0573933\pi\)
−0.179331 + 0.983789i \(0.557393\pi\)
\(938\) 0 0
\(939\) 27.7251i 0.904774i
\(940\) 0 0
\(941\) 7.65037i 0.249395i −0.992195 0.124697i \(-0.960204\pi\)
0.992195 0.124697i \(-0.0397960\pi\)
\(942\) 0 0
\(943\) −1.55517 + 1.55517i −0.0506432 + 0.0506432i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.71658 + 9.71658i −0.315746 + 0.315746i −0.847131 0.531384i \(-0.821672\pi\)
0.531384 + 0.847131i \(0.321672\pi\)
\(948\) 0 0
\(949\) 42.7492i 1.38770i
\(950\) 0 0
\(951\) 10.8109i 0.350568i
\(952\) 0 0
\(953\) 9.42057 + 9.42057i 0.305162 + 0.305162i 0.843029 0.537867i \(-0.180770\pi\)
−0.537867 + 0.843029i \(0.680770\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −1.67909 1.67909i −0.0542771 0.0542771i
\(958\) 0 0
\(959\) 24.7249 51.9244i 0.798408 1.67673i
\(960\) 0 0
\(961\) 29.2749 0.944352
\(962\) 0 0
\(963\) −4.30695 4.30695i −0.138790 0.138790i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.5763 24.5763i 0.790320 0.790320i −0.191226 0.981546i \(-0.561246\pi\)
0.981546 + 0.191226i \(0.0612464\pi\)
\(968\) 0 0
\(969\) −4.30136 −0.138179
\(970\) 0 0
\(971\) 25.9232i 0.831916i 0.909384 + 0.415958i \(0.136554\pi\)
−0.909384 + 0.415958i \(0.863446\pi\)
\(972\) 0 0
\(973\) 8.63744 3.06506i 0.276904 0.0982612i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.0328 12.0328i 0.384964 0.384964i −0.487923 0.872887i \(-0.662245\pi\)
0.872887 + 0.487923i \(0.162245\pi\)
\(978\) 0 0
\(979\) 59.8421 1.91256
\(980\) 0 0
\(981\) −1.00000 −0.0319275
\(982\) 0 0
\(983\) −30.6170 + 30.6170i −0.976531 + 0.976531i −0.999731 0.0231995i \(-0.992615\pi\)
0.0231995 + 0.999731i \(0.492615\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −4.98683 + 1.76961i −0.158733 + 0.0563274i
\(988\) 0 0
\(989\) 0.374586i 0.0119111i
\(990\) 0 0
\(991\) −21.5498 −0.684553 −0.342277 0.939599i \(-0.611198\pi\)
−0.342277 + 0.939599i \(0.611198\pi\)
\(992\) 0 0
\(993\) −7.33594 + 7.33594i −0.232799 + 0.232799i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −32.0312 32.0312i −1.01444 1.01444i −0.999894 0.0145452i \(-0.995370\pi\)
−0.0145452 0.999894i \(-0.504630\pi\)
\(998\) 0 0
\(999\) 2.20822 0.0698650
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.x.c.1693.4 yes 16
5.2 odd 4 inner 2100.2.x.c.1357.6 yes 16
5.3 odd 4 inner 2100.2.x.c.1357.3 16
5.4 even 2 inner 2100.2.x.c.1693.5 yes 16
7.6 odd 2 inner 2100.2.x.c.1693.6 yes 16
35.13 even 4 inner 2100.2.x.c.1357.5 yes 16
35.27 even 4 inner 2100.2.x.c.1357.4 yes 16
35.34 odd 2 inner 2100.2.x.c.1693.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.x.c.1357.3 16 5.3 odd 4 inner
2100.2.x.c.1357.4 yes 16 35.27 even 4 inner
2100.2.x.c.1357.5 yes 16 35.13 even 4 inner
2100.2.x.c.1357.6 yes 16 5.2 odd 4 inner
2100.2.x.c.1693.3 yes 16 35.34 odd 2 inner
2100.2.x.c.1693.4 yes 16 1.1 even 1 trivial
2100.2.x.c.1693.5 yes 16 5.4 even 2 inner
2100.2.x.c.1693.6 yes 16 7.6 odd 2 inner