Properties

Label 2100.2.x.a.1693.2
Level $2100$
Weight $2$
Character 2100.1693
Analytic conductor $16.769$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1693.2
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2100.1693
Dual form 2100.2.x.a.1357.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{3} +(2.63896 + 0.189469i) q^{7} -1.00000i q^{9} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{3} +(2.63896 + 0.189469i) q^{7} -1.00000i q^{9} -3.00000 q^{11} +(1.41421 - 1.41421i) q^{13} +(-4.24264 - 4.24264i) q^{17} +3.46410 q^{19} +(-2.00000 + 1.73205i) q^{21} +(-3.67423 - 3.67423i) q^{23} +(0.707107 + 0.707107i) q^{27} -9.00000i q^{29} +6.92820i q^{31} +(2.12132 - 2.12132i) q^{33} +(-3.67423 + 3.67423i) q^{37} +2.00000i q^{39} +(-8.57321 - 8.57321i) q^{43} +(-4.24264 - 4.24264i) q^{47} +(6.92820 + 1.00000i) q^{49} +6.00000 q^{51} +(-2.44949 + 2.44949i) q^{57} +10.3923 q^{59} -6.92820i q^{61} +(0.189469 - 2.63896i) q^{63} +(-8.57321 + 8.57321i) q^{67} +5.19615 q^{69} +3.00000 q^{71} +(-1.41421 + 1.41421i) q^{73} +(-7.91688 - 0.568406i) q^{77} -1.00000i q^{79} -1.00000 q^{81} +(8.48528 - 8.48528i) q^{83} +(6.36396 + 6.36396i) q^{87} -10.3923 q^{89} +(4.00000 - 3.46410i) q^{91} +(-4.89898 - 4.89898i) q^{93} +(-2.82843 - 2.82843i) q^{97} +3.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{11} - 16 q^{21} + 48 q^{51} + 24 q^{71} - 8 q^{81} + 32 q^{91}+O(q^{100}) \) 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
\(7\) 2.63896 + 0.189469i 0.997433 + 0.0716124i
\(8\) 0 0
\(9\) 1.00000i 0.333333i
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) 0 0
\(13\) 1.41421 1.41421i 0.392232 0.392232i −0.483250 0.875482i \(-0.660544\pi\)
0.875482 + 0.483250i \(0.160544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.24264 4.24264i −1.02899 1.02899i −0.999567 0.0294245i \(-0.990633\pi\)
−0.0294245 0.999567i \(-0.509367\pi\)
\(18\) 0 0
\(19\) 3.46410 0.794719 0.397360 0.917663i \(-0.369927\pi\)
0.397360 + 0.917663i \(0.369927\pi\)
\(20\) 0 0
\(21\) −2.00000 + 1.73205i −0.436436 + 0.377964i
\(22\) 0 0
\(23\) −3.67423 3.67423i −0.766131 0.766131i 0.211292 0.977423i \(-0.432233\pi\)
−0.977423 + 0.211292i \(0.932233\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.707107 + 0.707107i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 9.00000i 1.67126i −0.549294 0.835629i \(-0.685103\pi\)
0.549294 0.835629i \(-0.314897\pi\)
\(30\) 0 0
\(31\) 6.92820i 1.24434i 0.782881 + 0.622171i \(0.213749\pi\)
−0.782881 + 0.622171i \(0.786251\pi\)
\(32\) 0 0
\(33\) 2.12132 2.12132i 0.369274 0.369274i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −3.67423 + 3.67423i −0.604040 + 0.604040i −0.941382 0.337342i \(-0.890472\pi\)
0.337342 + 0.941382i \(0.390472\pi\)
\(38\) 0 0
\(39\) 2.00000i 0.320256i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −8.57321 8.57321i −1.30740 1.30740i −0.923283 0.384120i \(-0.874505\pi\)
−0.384120 0.923283i \(-0.625495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.24264 4.24264i −0.618853 0.618853i 0.326384 0.945237i \(-0.394170\pi\)
−0.945237 + 0.326384i \(0.894170\pi\)
\(48\) 0 0
\(49\) 6.92820 + 1.00000i 0.989743 + 0.142857i
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −2.44949 + 2.44949i −0.324443 + 0.324443i
\(58\) 0 0
\(59\) 10.3923 1.35296 0.676481 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) 6.92820i 0.887066i −0.896258 0.443533i \(-0.853725\pi\)
0.896258 0.443533i \(-0.146275\pi\)
\(62\) 0 0
\(63\) 0.189469 2.63896i 0.0238708 0.332478i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −8.57321 + 8.57321i −1.04738 + 1.04738i −0.0485648 + 0.998820i \(0.515465\pi\)
−0.998820 + 0.0485648i \(0.984535\pi\)
\(68\) 0 0
\(69\) 5.19615 0.625543
\(70\) 0 0
\(71\) 3.00000 0.356034 0.178017 0.984027i \(-0.443032\pi\)
0.178017 + 0.984027i \(0.443032\pi\)
\(72\) 0 0
\(73\) −1.41421 + 1.41421i −0.165521 + 0.165521i −0.785007 0.619486i \(-0.787341\pi\)
0.619486 + 0.785007i \(0.287341\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.91688 0.568406i −0.902212 0.0647759i
\(78\) 0 0
\(79\) 1.00000i 0.112509i −0.998416 0.0562544i \(-0.982084\pi\)
0.998416 0.0562544i \(-0.0179158\pi\)
\(80\) 0 0
\(81\) −1.00000 −0.111111
\(82\) 0 0
\(83\) 8.48528 8.48528i 0.931381 0.931381i −0.0664117 0.997792i \(-0.521155\pi\)
0.997792 + 0.0664117i \(0.0211551\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.36396 + 6.36396i 0.682288 + 0.682288i
\(88\) 0 0
\(89\) −10.3923 −1.10158 −0.550791 0.834643i \(-0.685674\pi\)
−0.550791 + 0.834643i \(0.685674\pi\)
\(90\) 0 0
\(91\) 4.00000 3.46410i 0.419314 0.363137i
\(92\) 0 0
\(93\) −4.89898 4.89898i −0.508001 0.508001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.82843 2.82843i −0.287183 0.287183i 0.548782 0.835965i \(-0.315092\pi\)
−0.835965 + 0.548782i \(0.815092\pi\)
\(98\) 0 0
\(99\) 3.00000i 0.301511i
\(100\) 0 0
\(101\) 10.3923i 1.03407i 0.855963 + 0.517036i \(0.172965\pi\)
−0.855963 + 0.517036i \(0.827035\pi\)
\(102\) 0 0
\(103\) 9.89949 9.89949i 0.975426 0.975426i −0.0242790 0.999705i \(-0.507729\pi\)
0.999705 + 0.0242790i \(0.00772901\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.34847 7.34847i 0.710403 0.710403i −0.256216 0.966620i \(-0.582476\pi\)
0.966620 + 0.256216i \(0.0824759\pi\)
\(108\) 0 0
\(109\) 17.0000i 1.62830i −0.580651 0.814152i \(-0.697202\pi\)
0.580651 0.814152i \(-0.302798\pi\)
\(110\) 0 0
\(111\) 5.19615i 0.493197i
\(112\) 0 0
\(113\) −3.67423 3.67423i −0.345643 0.345643i 0.512841 0.858484i \(-0.328593\pi\)
−0.858484 + 0.512841i \(0.828593\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.41421 1.41421i −0.130744 0.130744i
\(118\) 0 0
\(119\) −10.3923 12.0000i −0.952661 1.10004i
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.67423 3.67423i 0.326036 0.326036i −0.525041 0.851077i \(-0.675950\pi\)
0.851077 + 0.525041i \(0.175950\pi\)
\(128\) 0 0
\(129\) 12.1244 1.06749
\(130\) 0 0
\(131\) 10.3923i 0.907980i 0.891007 + 0.453990i \(0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(132\) 0 0
\(133\) 9.14162 + 0.656339i 0.792679 + 0.0569118i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(138\) 0 0
\(139\) 10.3923 0.881464 0.440732 0.897639i \(-0.354719\pi\)
0.440732 + 0.897639i \(0.354719\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) −4.24264 + 4.24264i −0.354787 + 0.354787i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −5.60609 + 4.19187i −0.462382 + 0.345740i
\(148\) 0 0
\(149\) 15.0000i 1.22885i −0.788976 0.614424i \(-0.789388\pi\)
0.788976 0.614424i \(-0.210612\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 0 0
\(153\) −4.24264 + 4.24264i −0.342997 + 0.342997i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −11.3137 11.3137i −0.902932 0.902932i 0.0927566 0.995689i \(-0.470432\pi\)
−0.995689 + 0.0927566i \(0.970432\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.00000 10.3923i −0.709299 0.819028i
\(162\) 0 0
\(163\) −2.44949 2.44949i −0.191859 0.191859i 0.604640 0.796499i \(-0.293317\pi\)
−0.796499 + 0.604640i \(0.793317\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −4.24264 4.24264i −0.328305 0.328305i 0.523636 0.851942i \(-0.324575\pi\)
−0.851942 + 0.523636i \(0.824575\pi\)
\(168\) 0 0
\(169\) 9.00000i 0.692308i
\(170\) 0 0
\(171\) 3.46410i 0.264906i
\(172\) 0 0
\(173\) 12.7279 12.7279i 0.967686 0.967686i −0.0318080 0.999494i \(-0.510127\pi\)
0.999494 + 0.0318080i \(0.0101265\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −7.34847 + 7.34847i −0.552345 + 0.552345i
\(178\) 0 0
\(179\) 12.0000i 0.896922i −0.893802 0.448461i \(-0.851972\pi\)
0.893802 0.448461i \(-0.148028\pi\)
\(180\) 0 0
\(181\) 3.46410i 0.257485i −0.991678 0.128742i \(-0.958906\pi\)
0.991678 0.128742i \(-0.0410940\pi\)
\(182\) 0 0
\(183\) 4.89898 + 4.89898i 0.362143 + 0.362143i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 12.7279 + 12.7279i 0.930758 + 0.930758i
\(188\) 0 0
\(189\) 1.73205 + 2.00000i 0.125988 + 0.145479i
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −13.4722 13.4722i −0.969750 0.969750i 0.0298060 0.999556i \(-0.490511\pi\)
−0.999556 + 0.0298060i \(0.990511\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.0227 11.0227i 0.785335 0.785335i −0.195390 0.980726i \(-0.562597\pi\)
0.980726 + 0.195390i \(0.0625974\pi\)
\(198\) 0 0
\(199\) 6.92820 0.491127 0.245564 0.969380i \(-0.421027\pi\)
0.245564 + 0.969380i \(0.421027\pi\)
\(200\) 0 0
\(201\) 12.1244i 0.855186i
\(202\) 0 0
\(203\) 1.70522 23.7506i 0.119683 1.66697i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −3.67423 + 3.67423i −0.255377 + 0.255377i
\(208\) 0 0
\(209\) −10.3923 −0.718851
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) −2.12132 + 2.12132i −0.145350 + 0.145350i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.31268 + 18.2832i −0.0891104 + 1.24115i
\(218\) 0 0
\(219\) 2.00000i 0.135147i
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) −9.89949 + 9.89949i −0.662919 + 0.662919i −0.956067 0.293148i \(-0.905297\pi\)
0.293148 + 0.956067i \(0.405297\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.24264 4.24264i −0.281594 0.281594i 0.552151 0.833744i \(-0.313807\pi\)
−0.833744 + 0.552151i \(0.813807\pi\)
\(228\) 0 0
\(229\) 6.92820 0.457829 0.228914 0.973447i \(-0.426482\pi\)
0.228914 + 0.973447i \(0.426482\pi\)
\(230\) 0 0
\(231\) 6.00000 5.19615i 0.394771 0.341882i
\(232\) 0 0
\(233\) 18.3712 + 18.3712i 1.20354 + 1.20354i 0.973084 + 0.230452i \(0.0740204\pi\)
0.230452 + 0.973084i \(0.425980\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.707107 + 0.707107i 0.0459315 + 0.0459315i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 17.3205i 1.11571i 0.829938 + 0.557856i \(0.188376\pi\)
−0.829938 + 0.557856i \(0.811624\pi\)
\(242\) 0 0
\(243\) 0.707107 0.707107i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.89898 4.89898i 0.311715 0.311715i
\(248\) 0 0
\(249\) 12.0000i 0.760469i
\(250\) 0 0
\(251\) 31.1769i 1.96787i 0.178529 + 0.983935i \(0.442866\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 0 0
\(253\) 11.0227 + 11.0227i 0.692991 + 0.692991i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.48528 8.48528i −0.529297 0.529297i 0.391066 0.920363i \(-0.372107\pi\)
−0.920363 + 0.391066i \(0.872107\pi\)
\(258\) 0 0
\(259\) −10.3923 + 9.00000i −0.645746 + 0.559233i
\(260\) 0 0
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 11.0227 + 11.0227i 0.679689 + 0.679689i 0.959930 0.280241i \(-0.0904143\pi\)
−0.280241 + 0.959930i \(0.590414\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.34847 7.34847i 0.449719 0.449719i
\(268\) 0 0
\(269\) −20.7846 −1.26726 −0.633630 0.773636i \(-0.718436\pi\)
−0.633630 + 0.773636i \(0.718436\pi\)
\(270\) 0 0
\(271\) 31.1769i 1.89386i −0.321436 0.946931i \(-0.604165\pi\)
0.321436 0.946931i \(-0.395835\pi\)
\(272\) 0 0
\(273\) −0.378937 + 5.27792i −0.0229343 + 0.319434i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −9.79796 + 9.79796i −0.588702 + 0.588702i −0.937280 0.348578i \(-0.886665\pi\)
0.348578 + 0.937280i \(0.386665\pi\)
\(278\) 0 0
\(279\) 6.92820 0.414781
\(280\) 0 0
\(281\) 3.00000 0.178965 0.0894825 0.995988i \(-0.471479\pi\)
0.0894825 + 0.995988i \(0.471479\pi\)
\(282\) 0 0
\(283\) −11.3137 + 11.3137i −0.672530 + 0.672530i −0.958299 0.285769i \(-0.907751\pi\)
0.285769 + 0.958299i \(0.407751\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000i 1.11765i
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) −12.7279 + 12.7279i −0.743573 + 0.743573i −0.973264 0.229691i \(-0.926229\pi\)
0.229691 + 0.973264i \(0.426229\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.12132 2.12132i −0.123091 0.123091i
\(298\) 0 0
\(299\) −10.3923 −0.601003
\(300\) 0 0
\(301\) −21.0000 24.2487i −1.21042 1.39767i
\(302\) 0 0
\(303\) −7.34847 7.34847i −0.422159 0.422159i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.07107 + 7.07107i 0.403567 + 0.403567i 0.879488 0.475921i \(-0.157885\pi\)
−0.475921 + 0.879488i \(0.657885\pi\)
\(308\) 0 0
\(309\) 14.0000i 0.796432i
\(310\) 0 0
\(311\) 31.1769i 1.76788i 0.467600 + 0.883940i \(0.345119\pi\)
−0.467600 + 0.883940i \(0.654881\pi\)
\(312\) 0 0
\(313\) −9.89949 + 9.89949i −0.559553 + 0.559553i −0.929180 0.369627i \(-0.879485\pi\)
0.369627 + 0.929180i \(0.379485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.67423 3.67423i 0.206366 0.206366i −0.596355 0.802721i \(-0.703385\pi\)
0.802721 + 0.596355i \(0.203385\pi\)
\(318\) 0 0
\(319\) 27.0000i 1.51171i
\(320\) 0 0
\(321\) 10.3923i 0.580042i
\(322\) 0 0
\(323\) −14.6969 14.6969i −0.817760 0.817760i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 12.0208 + 12.0208i 0.664753 + 0.664753i
\(328\) 0 0
\(329\) −10.3923 12.0000i −0.572946 0.661581i
\(330\) 0 0
\(331\) 19.0000 1.04433 0.522167 0.852843i \(-0.325124\pi\)
0.522167 + 0.852843i \(0.325124\pi\)
\(332\) 0 0
\(333\) 3.67423 + 3.67423i 0.201347 + 0.201347i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −4.89898 + 4.89898i −0.266864 + 0.266864i −0.827835 0.560971i \(-0.810428\pi\)
0.560971 + 0.827835i \(0.310428\pi\)
\(338\) 0 0
\(339\) 5.19615 0.282216
\(340\) 0 0
\(341\) 20.7846i 1.12555i
\(342\) 0 0
\(343\) 18.0938 + 3.95164i 0.976972 + 0.213368i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.7196 25.7196i 1.38070 1.38070i 0.537332 0.843371i \(-0.319432\pi\)
0.843371 0.537332i \(-0.180568\pi\)
\(348\) 0 0
\(349\) 10.3923 0.556287 0.278144 0.960539i \(-0.410281\pi\)
0.278144 + 0.960539i \(0.410281\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) −25.4558 + 25.4558i −1.35488 + 1.35488i −0.474766 + 0.880112i \(0.657467\pi\)
−0.880112 + 0.474766i \(0.842533\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 15.8338 + 1.13681i 0.838011 + 0.0601665i
\(358\) 0 0
\(359\) 27.0000i 1.42501i 0.701669 + 0.712503i \(0.252438\pi\)
−0.701669 + 0.712503i \(0.747562\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) 0 0
\(363\) 1.41421 1.41421i 0.0742270 0.0742270i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 22.6274 + 22.6274i 1.18114 + 1.18114i 0.979449 + 0.201693i \(0.0646442\pi\)
0.201693 + 0.979449i \(0.435356\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −6.12372 6.12372i −0.317074 0.317074i 0.530568 0.847642i \(-0.321979\pi\)
−0.847642 + 0.530568i \(0.821979\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.7279 12.7279i −0.655521 0.655521i
\(378\) 0 0
\(379\) 1.00000i 0.0513665i −0.999670 0.0256833i \(-0.991824\pi\)
0.999670 0.0256833i \(-0.00817614\pi\)
\(380\) 0 0
\(381\) 5.19615i 0.266207i
\(382\) 0 0
\(383\) −4.24264 + 4.24264i −0.216789 + 0.216789i −0.807144 0.590355i \(-0.798988\pi\)
0.590355 + 0.807144i \(0.298988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.57321 + 8.57321i −0.435801 + 0.435801i
\(388\) 0 0
\(389\) 3.00000i 0.152106i 0.997104 + 0.0760530i \(0.0242318\pi\)
−0.997104 + 0.0760530i \(0.975768\pi\)
\(390\) 0 0
\(391\) 31.1769i 1.57668i
\(392\) 0 0
\(393\) −7.34847 7.34847i −0.370681 0.370681i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5.65685 5.65685i −0.283909 0.283909i 0.550757 0.834666i \(-0.314339\pi\)
−0.834666 + 0.550757i \(0.814339\pi\)
\(398\) 0 0
\(399\) −6.92820 + 6.00000i −0.346844 + 0.300376i
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 9.79796 + 9.79796i 0.488071 + 0.488071i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 11.0227 11.0227i 0.546375 0.546375i
\(408\) 0 0
\(409\) 38.1051 1.88418 0.942088 0.335365i \(-0.108860\pi\)
0.942088 + 0.335365i \(0.108860\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 27.4249 + 1.96902i 1.34949 + 0.0968890i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −7.34847 + 7.34847i −0.359856 + 0.359856i
\(418\) 0 0
\(419\) −20.7846 −1.01539 −0.507697 0.861536i \(-0.669503\pi\)
−0.507697 + 0.861536i \(0.669503\pi\)
\(420\) 0 0
\(421\) −17.0000 −0.828529 −0.414265 0.910156i \(-0.635961\pi\)
−0.414265 + 0.910156i \(0.635961\pi\)
\(422\) 0 0
\(423\) −4.24264 + 4.24264i −0.206284 + 0.206284i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.31268 18.2832i 0.0635249 0.884788i
\(428\) 0 0
\(429\) 6.00000i 0.289683i
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −5.65685 + 5.65685i −0.271851 + 0.271851i −0.829845 0.557994i \(-0.811571\pi\)
0.557994 + 0.829845i \(0.311571\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −12.7279 12.7279i −0.608859 0.608859i
\(438\) 0 0
\(439\) −38.1051 −1.81866 −0.909329 0.416078i \(-0.863404\pi\)
−0.909329 + 0.416078i \(0.863404\pi\)
\(440\) 0 0
\(441\) 1.00000 6.92820i 0.0476190 0.329914i
\(442\) 0 0
\(443\) 7.34847 + 7.34847i 0.349136 + 0.349136i 0.859788 0.510651i \(-0.170596\pi\)
−0.510651 + 0.859788i \(0.670596\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 10.6066 + 10.6066i 0.501675 + 0.501675i
\(448\) 0 0
\(449\) 27.0000i 1.27421i −0.770778 0.637104i \(-0.780132\pi\)
0.770778 0.637104i \(-0.219868\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −3.53553 + 3.53553i −0.166114 + 0.166114i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.67423 3.67423i 0.171873 0.171873i −0.615929 0.787802i \(-0.711219\pi\)
0.787802 + 0.615929i \(0.211219\pi\)
\(458\) 0 0
\(459\) 6.00000i 0.280056i
\(460\) 0 0
\(461\) 31.1769i 1.45205i −0.687666 0.726027i \(-0.741365\pi\)
0.687666 0.726027i \(-0.258635\pi\)
\(462\) 0 0
\(463\) 7.34847 + 7.34847i 0.341512 + 0.341512i 0.856936 0.515423i \(-0.172365\pi\)
−0.515423 + 0.856936i \(0.672365\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.48528 + 8.48528i 0.392652 + 0.392652i 0.875632 0.482980i \(-0.160445\pi\)
−0.482980 + 0.875632i \(0.660445\pi\)
\(468\) 0 0
\(469\) −24.2487 + 21.0000i −1.11970 + 0.969690i
\(470\) 0 0
\(471\) 16.0000 0.737241
\(472\) 0 0
\(473\) 25.7196 + 25.7196i 1.18259 + 1.18259i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 10.3923 0.474837 0.237418 0.971408i \(-0.423699\pi\)
0.237418 + 0.971408i \(0.423699\pi\)
\(480\) 0 0
\(481\) 10.3923i 0.473848i
\(482\) 0 0
\(483\) 13.7124 + 0.984508i 0.623937 + 0.0447967i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 15.9217 15.9217i 0.721480 0.721480i −0.247426 0.968907i \(-0.579585\pi\)
0.968907 + 0.247426i \(0.0795848\pi\)
\(488\) 0 0
\(489\) 3.46410 0.156652
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 0 0
\(493\) −38.1838 + 38.1838i −1.71971 + 1.71971i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.91688 + 0.568406i 0.355120 + 0.0254965i
\(498\) 0 0
\(499\) 20.0000i 0.895323i −0.894203 0.447661i \(-0.852257\pi\)
0.894203 0.447661i \(-0.147743\pi\)
\(500\) 0 0
\(501\) 6.00000 0.268060
\(502\) 0 0
\(503\) −25.4558 + 25.4558i −1.13502 + 1.13502i −0.145690 + 0.989330i \(0.546540\pi\)
−0.989330 + 0.145690i \(0.953460\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.36396 6.36396i −0.282633 0.282633i
\(508\) 0 0
\(509\) −31.1769 −1.38189 −0.690946 0.722906i \(-0.742806\pi\)
−0.690946 + 0.722906i \(0.742806\pi\)
\(510\) 0 0
\(511\) −4.00000 + 3.46410i −0.176950 + 0.153243i
\(512\) 0 0
\(513\) 2.44949 + 2.44949i 0.108148 + 0.108148i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12.7279 + 12.7279i 0.559773 + 0.559773i
\(518\) 0 0
\(519\) 18.0000i 0.790112i
\(520\) 0 0
\(521\) 20.7846i 0.910590i 0.890341 + 0.455295i \(0.150466\pi\)
−0.890341 + 0.455295i \(0.849534\pi\)
\(522\) 0 0
\(523\) 31.1127 31.1127i 1.36046 1.36046i 0.487139 0.873325i \(-0.338041\pi\)
0.873325 0.487139i \(-0.161959\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 29.3939 29.3939i 1.28042 1.28042i
\(528\) 0 0
\(529\) 4.00000i 0.173913i
\(530\) 0 0
\(531\) 10.3923i 0.450988i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 8.48528 + 8.48528i 0.366167 + 0.366167i
\(538\) 0 0
\(539\) −20.7846 3.00000i −0.895257 0.129219i
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) 2.44949 + 2.44949i 0.105118 + 0.105118i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 11.0227 11.0227i 0.471297 0.471297i −0.431037 0.902334i \(-0.641852\pi\)
0.902334 + 0.431037i \(0.141852\pi\)
\(548\) 0 0
\(549\) −6.92820 −0.295689
\(550\) 0 0
\(551\) 31.1769i 1.32818i
\(552\) 0 0
\(553\) 0.189469 2.63896i 0.00805703 0.112220i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −25.7196 + 25.7196i −1.08978 + 1.08978i −0.0942253 + 0.995551i \(0.530037\pi\)
−0.995551 + 0.0942253i \(0.969963\pi\)
\(558\) 0 0
\(559\) −24.2487 −1.02561
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) 0 0
\(563\) −16.9706 + 16.9706i −0.715224 + 0.715224i −0.967623 0.252399i \(-0.918780\pi\)
0.252399 + 0.967623i \(0.418780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2.63896 0.189469i −0.110826 0.00795694i
\(568\) 0 0
\(569\) 39.0000i 1.63497i 0.575953 + 0.817483i \(0.304631\pi\)
−0.575953 + 0.817483i \(0.695369\pi\)
\(570\) 0 0
\(571\) −7.00000 −0.292941 −0.146470 0.989215i \(-0.546791\pi\)
−0.146470 + 0.989215i \(0.546791\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 7.07107 + 7.07107i 0.294372 + 0.294372i 0.838805 0.544432i \(-0.183255\pi\)
−0.544432 + 0.838805i \(0.683255\pi\)
\(578\) 0 0
\(579\) 19.0526 0.791797
\(580\) 0 0
\(581\) 24.0000 20.7846i 0.995688 0.862291i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.7279 12.7279i −0.525338 0.525338i 0.393841 0.919179i \(-0.371146\pi\)
−0.919179 + 0.393841i \(0.871146\pi\)
\(588\) 0 0
\(589\) 24.0000i 0.988903i
\(590\) 0 0
\(591\) 15.5885i 0.641223i
\(592\) 0 0
\(593\) 25.4558 25.4558i 1.04535 1.04535i 0.0464244 0.998922i \(-0.485217\pi\)
0.998922 0.0464244i \(-0.0147827\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.89898 + 4.89898i −0.200502 + 0.200502i
\(598\) 0 0
\(599\) 39.0000i 1.59350i 0.604311 + 0.796748i \(0.293448\pi\)
−0.604311 + 0.796748i \(0.706552\pi\)
\(600\) 0 0
\(601\) 27.7128i 1.13043i −0.824944 0.565215i \(-0.808793\pi\)
0.824944 0.565215i \(-0.191207\pi\)
\(602\) 0 0
\(603\) 8.57321 + 8.57321i 0.349128 + 0.349128i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 15.5563 + 15.5563i 0.631413 + 0.631413i 0.948422 0.317010i \(-0.102679\pi\)
−0.317010 + 0.948422i \(0.602679\pi\)
\(608\) 0 0
\(609\) 15.5885 + 18.0000i 0.631676 + 0.729397i
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 0 0
\(613\) 11.0227 + 11.0227i 0.445203 + 0.445203i 0.893756 0.448553i \(-0.148061\pi\)
−0.448553 + 0.893756i \(0.648061\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −25.7196 + 25.7196i −1.03543 + 1.03543i −0.0360851 + 0.999349i \(0.511489\pi\)
−0.999349 + 0.0360851i \(0.988511\pi\)
\(618\) 0 0
\(619\) 24.2487 0.974638 0.487319 0.873224i \(-0.337975\pi\)
0.487319 + 0.873224i \(0.337975\pi\)
\(620\) 0 0
\(621\) 5.19615i 0.208514i
\(622\) 0 0
\(623\) −27.4249 1.96902i −1.09875 0.0788870i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.34847 7.34847i 0.293470 0.293470i
\(628\) 0 0
\(629\) 31.1769 1.24310
\(630\) 0 0
\(631\) 35.0000 1.39333 0.696664 0.717398i \(-0.254667\pi\)
0.696664 + 0.717398i \(0.254667\pi\)
\(632\) 0 0
\(633\) −11.3137 + 11.3137i −0.449680 + 0.449680i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 11.2122 8.38375i 0.444242 0.332176i
\(638\) 0 0
\(639\) 3.00000i 0.118678i
\(640\) 0 0
\(641\) 9.00000 0.355479 0.177739 0.984078i \(-0.443122\pi\)
0.177739 + 0.984078i \(0.443122\pi\)
\(642\) 0 0
\(643\) 2.82843 2.82843i 0.111542 0.111542i −0.649133 0.760675i \(-0.724868\pi\)
0.760675 + 0.649133i \(0.224868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) −31.1769 −1.22380
\(650\) 0 0
\(651\) −12.0000 13.8564i −0.470317 0.543075i
\(652\) 0 0
\(653\) 29.3939 + 29.3939i 1.15027 + 1.15027i 0.986498 + 0.163773i \(0.0523666\pi\)
0.163773 + 0.986498i \(0.447633\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.41421 + 1.41421i 0.0551737 + 0.0551737i
\(658\) 0 0
\(659\) 24.0000i 0.934907i −0.884018 0.467454i \(-0.845171\pi\)
0.884018 0.467454i \(-0.154829\pi\)
\(660\) 0 0
\(661\) 31.1769i 1.21264i 0.795220 + 0.606321i \(0.207355\pi\)
−0.795220 + 0.606321i \(0.792645\pi\)
\(662\) 0 0
\(663\) 8.48528 8.48528i 0.329541 0.329541i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −33.0681 + 33.0681i −1.28040 + 1.28040i
\(668\) 0 0
\(669\) 14.0000i 0.541271i
\(670\) 0 0
\(671\) 20.7846i 0.802381i
\(672\) 0 0
\(673\) 14.6969 + 14.6969i 0.566525 + 0.566525i 0.931153 0.364628i \(-0.118804\pi\)
−0.364628 + 0.931153i \(0.618804\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 8.48528 + 8.48528i 0.326116 + 0.326116i 0.851107 0.524992i \(-0.175932\pi\)
−0.524992 + 0.851107i \(0.675932\pi\)
\(678\) 0 0
\(679\) −6.92820 8.00000i −0.265880 0.307012i
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) 0 0
\(683\) −25.7196 25.7196i −0.984135 0.984135i 0.0157413 0.999876i \(-0.494989\pi\)
−0.999876 + 0.0157413i \(0.994989\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4.89898 + 4.89898i −0.186908 + 0.186908i
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 20.7846i 0.790684i −0.918534 0.395342i \(-0.870626\pi\)
0.918534 0.395342i \(-0.129374\pi\)
\(692\) 0 0
\(693\) −0.568406 + 7.91688i −0.0215920 + 0.300737i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −25.9808 −0.982683
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −12.7279 + 12.7279i −0.480043 + 0.480043i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.96902 + 27.4249i −0.0740525 + 1.03142i
\(708\) 0 0
\(709\) 22.0000i 0.826227i −0.910679 0.413114i \(-0.864441\pi\)
0.910679 0.413114i \(-0.135559\pi\)
\(710\) 0 0
\(711\) −1.00000 −0.0375029
\(712\) 0 0
\(713\) 25.4558 25.4558i 0.953329 0.953329i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 28.0000 24.2487i 1.04277 0.903069i
\(722\) 0 0
\(723\) −12.2474 12.2474i −0.455488 0.455488i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −9.89949 9.89949i −0.367152 0.367152i 0.499286 0.866437i \(-0.333596\pi\)
−0.866437 + 0.499286i \(0.833596\pi\)
\(728\) 0 0
\(729\) 1.00000i 0.0370370i
\(730\) 0 0
\(731\) 72.7461i 2.69061i
\(732\) 0 0
\(733\) 2.82843 2.82843i 0.104470 0.104470i −0.652940 0.757410i \(-0.726464\pi\)
0.757410 + 0.652940i \(0.226464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.7196 25.7196i 0.947395 0.947395i
\(738\) 0 0
\(739\) 11.0000i 0.404642i −0.979319 0.202321i \(-0.935152\pi\)
0.979319 0.202321i \(-0.0648484\pi\)
\(740\) 0 0
\(741\) 6.92820i 0.254514i
\(742\) 0 0
\(743\) 7.34847 + 7.34847i 0.269589 + 0.269589i 0.828935 0.559345i \(-0.188947\pi\)
−0.559345 + 0.828935i \(0.688947\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8.48528 8.48528i −0.310460 0.310460i
\(748\) 0 0
\(749\) 20.7846 18.0000i 0.759453 0.657706i
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) −22.0454 22.0454i −0.803379 0.803379i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −1.22474 + 1.22474i −0.0445141 + 0.0445141i −0.729013 0.684499i \(-0.760021\pi\)
0.684499 + 0.729013i \(0.260021\pi\)
\(758\) 0 0
\(759\) −15.5885 −0.565825
\(760\) 0 0
\(761\) 41.5692i 1.50688i −0.657515 0.753442i \(-0.728392\pi\)
0.657515 0.753442i \(-0.271608\pi\)
\(762\) 0 0
\(763\) 3.22097 44.8623i 0.116607 1.62412i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.6969 14.6969i 0.530676 0.530676i
\(768\) 0 0
\(769\) 13.8564 0.499675 0.249837 0.968288i \(-0.419623\pi\)
0.249837 + 0.968288i \(0.419623\pi\)
\(770\) 0 0
\(771\) 12.0000 0.432169
\(772\) 0 0
\(773\) 16.9706 16.9706i 0.610389 0.610389i −0.332659 0.943047i \(-0.607946\pi\)
0.943047 + 0.332659i \(0.107946\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.984508 13.7124i 0.0353190 0.491931i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −9.00000 −0.322045
\(782\) 0 0
\(783\) 6.36396 6.36396i 0.227429 0.227429i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −26.8701 26.8701i −0.957814 0.957814i 0.0413314 0.999145i \(-0.486840\pi\)
−0.999145 + 0.0413314i \(0.986840\pi\)
\(788\) 0 0
\(789\) −15.5885 −0.554964
\(790\) 0 0
\(791\) −9.00000 10.3923i −0.320003 0.369508i
\(792\) 0 0
\(793\) −9.79796 9.79796i −0.347936 0.347936i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.2132 + 21.2132i 0.751410 + 0.751410i 0.974742 0.223332i \(-0.0716935\pi\)
−0.223332 + 0.974742i \(0.571693\pi\)
\(798\) 0 0
\(799\) 36.0000i 1.27359i
\(800\) 0 0
\(801\) 10.3923i 0.367194i
\(802\) 0 0
\(803\) 4.24264 4.24264i 0.149720 0.149720i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.6969 14.6969i 0.517357 0.517357i
\(808\) 0 0
\(809\) 51.0000i 1.79306i −0.442978 0.896532i \(-0.646078\pi\)
0.442978 0.896532i \(-0.353922\pi\)
\(810\) 0 0
\(811\) 27.7128i 0.973128i −0.873645 0.486564i \(-0.838250\pi\)
0.873645 0.486564i \(-0.161750\pi\)
\(812\) 0 0
\(813\) 22.0454 + 22.0454i 0.773166 + 0.773166i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −29.6985 29.6985i −1.03902 1.03902i
\(818\) 0 0
\(819\) −3.46410 4.00000i −0.121046 0.139771i
\(820\) 0 0
\(821\) 30.0000 1.04701 0.523504 0.852023i \(-0.324625\pi\)
0.523504 + 0.852023i \(0.324625\pi\)
\(822\) 0 0
\(823\) 1.22474 + 1.22474i 0.0426919 + 0.0426919i 0.728130 0.685439i \(-0.240389\pi\)
−0.685439 + 0.728130i \(0.740389\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −18.3712 + 18.3712i −0.638828 + 0.638828i −0.950266 0.311438i \(-0.899189\pi\)
0.311438 + 0.950266i \(0.399189\pi\)
\(828\) 0 0
\(829\) 10.3923 0.360940 0.180470 0.983581i \(-0.442238\pi\)
0.180470 + 0.983581i \(0.442238\pi\)
\(830\) 0 0
\(831\) 13.8564i 0.480673i
\(832\) 0 0
\(833\) −25.1512 33.6365i −0.871439 1.16544i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.89898 + 4.89898i −0.169334 + 0.169334i
\(838\) 0 0
\(839\) −10.3923 −0.358782 −0.179391 0.983778i \(-0.557413\pi\)
−0.179391 + 0.983778i \(0.557413\pi\)
\(840\) 0 0
\(841\) −52.0000 −1.79310
\(842\) 0 0
\(843\) −2.12132 + 2.12132i −0.0730622 + 0.0730622i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5.27792 0.378937i −0.181351 0.0130204i
\(848\) 0 0
\(849\) 16.0000i 0.549119i
\(850\) 0 0
\(851\) 27.0000 0.925548
\(852\) 0 0
\(853\) −2.82843 + 2.82843i −0.0968435 + 0.0968435i −0.753869 0.657025i \(-0.771815\pi\)
0.657025 + 0.753869i \(0.271815\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.48528 8.48528i −0.289852 0.289852i 0.547170 0.837022i \(-0.315705\pi\)
−0.837022 + 0.547170i \(0.815705\pi\)
\(858\) 0 0
\(859\) 41.5692 1.41832 0.709162 0.705046i \(-0.249074\pi\)
0.709162 + 0.705046i \(0.249074\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.3712 + 18.3712i 0.625362 + 0.625362i 0.946898 0.321536i \(-0.104199\pi\)
−0.321536 + 0.946898i \(0.604199\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −13.4350 13.4350i −0.456278 0.456278i
\(868\) 0 0
\(869\) 3.00000i 0.101768i
\(870\) 0 0
\(871\) 24.2487i 0.821636i
\(872\) 0 0
\(873\) −2.82843 + 2.82843i −0.0957278 + 0.0957278i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9.79796 + 9.79796i −0.330854 + 0.330854i −0.852911 0.522057i \(-0.825165\pi\)
0.522057 + 0.852911i \(0.325165\pi\)
\(878\) 0 0
\(879\) 18.0000i 0.607125i
\(880\) 0 0
\(881\) 20.7846i 0.700251i 0.936703 + 0.350126i \(0.113861\pi\)
−0.936703 + 0.350126i \(0.886139\pi\)
\(882\) 0 0
\(883\) 13.4722 + 13.4722i 0.453375 + 0.453375i 0.896473 0.443098i \(-0.146121\pi\)
−0.443098 + 0.896473i \(0.646121\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 25.4558 + 25.4558i 0.854724 + 0.854724i 0.990711 0.135987i \(-0.0434205\pi\)
−0.135987 + 0.990711i \(0.543421\pi\)
\(888\) 0 0
\(889\) 10.3923 9.00000i 0.348547 0.301850i
\(890\) 0 0
\(891\) 3.00000 0.100504
\(892\) 0 0
\(893\) −14.6969 14.6969i −0.491814 0.491814i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 7.34847 7.34847i 0.245358 0.245358i
\(898\) 0 0
\(899\) 62.3538 2.07962
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 31.9957 + 2.29719i 1.06475 + 0.0764456i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −7.34847 + 7.34847i −0.244002 + 0.244002i −0.818503 0.574502i \(-0.805196\pi\)
0.574502 + 0.818503i \(0.305196\pi\)
\(908\) 0 0
\(909\) 10.3923 0.344691
\(910\) 0 0
\(911\) −51.0000 −1.68971 −0.844853 0.534999i \(-0.820312\pi\)
−0.844853 + 0.534999i \(0.820312\pi\)
\(912\) 0 0
\(913\) −25.4558 + 25.4558i −0.842465 + 0.842465i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.96902 + 27.4249i −0.0650226 + 0.905649i
\(918\) 0 0
\(919\) 55.0000i 1.81428i −0.420826 0.907141i \(-0.638260\pi\)
0.420826 0.907141i \(-0.361740\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) 4.24264 4.24264i 0.139648 0.139648i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −9.89949 9.89949i −0.325142 0.325142i
\(928\) 0 0
\(929\) −31.1769 −1.02288 −0.511441 0.859319i \(-0.670888\pi\)
−0.511441 + 0.859319i \(0.670888\pi\)
\(930\) 0 0
\(931\) 24.0000 + 3.46410i 0.786568 + 0.113531i
\(932\) 0 0
\(933\) −22.0454 22.0454i −0.721734 0.721734i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.65685 + 5.65685i 0.184801 + 0.184801i 0.793444 0.608643i \(-0.208286\pi\)
−0.608643 + 0.793444i \(0.708286\pi\)
\(938\) 0 0
\(939\) 14.0000i 0.456873i
\(940\) 0 0
\(941\) 20.7846i 0.677559i −0.940866 0.338779i \(-0.889986\pi\)
0.940866 0.338779i \(-0.110014\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.0454 + 22.0454i −0.716379 + 0.716379i −0.967862 0.251482i \(-0.919082\pi\)
0.251482 + 0.967862i \(0.419082\pi\)
\(948\) 0 0
\(949\) 4.00000i 0.129845i
\(950\) 0 0
\(951\) 5.19615i 0.168497i
\(952\) 0 0
\(953\) −40.4166 40.4166i −1.30922 1.30922i −0.921977 0.387244i \(-0.873427\pi\)
−0.387244 0.921977i \(-0.626573\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −19.0919 19.0919i −0.617153 0.617153i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −17.0000 −0.548387
\(962\) 0 0
\(963\) −7.34847 7.34847i −0.236801 0.236801i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 22.0454 22.0454i 0.708933 0.708933i −0.257378 0.966311i \(-0.582859\pi\)
0.966311 + 0.257378i \(0.0828586\pi\)
\(968\) 0 0
\(969\) 20.7846 0.667698
\(970\) 0 0
\(971\) 10.3923i 0.333505i 0.985999 + 0.166752i \(0.0533281\pi\)
−0.985999 + 0.166752i \(0.946672\pi\)
\(972\) 0 0
\(973\) 27.4249 + 1.96902i 0.879201 + 0.0631238i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.0227 11.0227i 0.352648 0.352648i −0.508446 0.861094i \(-0.669780\pi\)
0.861094 + 0.508446i \(0.169780\pi\)
\(978\) 0 0
\(979\) 31.1769 0.996419
\(980\) 0 0
\(981\) −17.0000 −0.542768
\(982\) 0 0
\(983\) −33.9411 + 33.9411i −1.08255 + 1.08255i −0.0862831 + 0.996271i \(0.527499\pi\)
−0.996271 + 0.0862831i \(0.972501\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 15.8338 + 1.13681i 0.503994 + 0.0361851i
\(988\) 0 0
\(989\) 63.0000i 2.00328i
\(990\) 0 0
\(991\) 55.0000 1.74713 0.873566 0.486705i \(-0.161801\pi\)
0.873566 + 0.486705i \(0.161801\pi\)
\(992\) 0 0
\(993\) −13.4350 + 13.4350i −0.426348 + 0.426348i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −7.07107 7.07107i −0.223943 0.223943i 0.586214 0.810157i \(-0.300618\pi\)
−0.810157 + 0.586214i \(0.800618\pi\)
\(998\) 0 0
\(999\) −5.19615 −0.164399
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.a.1693.2 yes 8
5.2 odd 4 inner 2100.2.x.a.1357.4 yes 8
5.3 odd 4 inner 2100.2.x.a.1357.1 8
5.4 even 2 inner 2100.2.x.a.1693.3 yes 8
7.6 odd 2 inner 2100.2.x.a.1693.4 yes 8
35.13 even 4 inner 2100.2.x.a.1357.3 yes 8
35.27 even 4 inner 2100.2.x.a.1357.2 yes 8
35.34 odd 2 inner 2100.2.x.a.1693.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2100.2.x.a.1357.1 8 5.3 odd 4 inner
2100.2.x.a.1357.2 yes 8 35.27 even 4 inner
2100.2.x.a.1357.3 yes 8 35.13 even 4 inner
2100.2.x.a.1357.4 yes 8 5.2 odd 4 inner
2100.2.x.a.1693.1 yes 8 35.34 odd 2 inner
2100.2.x.a.1693.2 yes 8 1.1 even 1 trivial
2100.2.x.a.1693.3 yes 8 5.4 even 2 inner
2100.2.x.a.1693.4 yes 8 7.6 odd 2 inner