Properties

Label 1260.2.s.c.361.1
Level $1260$
Weight $2$
Character 1260.361
Analytic conductor $10.061$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1260,2,Mod(361,1260)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1260, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1260.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.s (of order \(3\), 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: \(10.0611506547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 140)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1260.361
Dual form 1260.2.s.c.541.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 - 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +O(q^{10})\) \(q+(-0.500000 - 0.866025i) q^{5} +(2.50000 - 0.866025i) q^{7} +(3.00000 - 5.19615i) q^{11} +2.00000 q^{13} +(-3.00000 + 5.19615i) q^{17} +(-4.00000 - 6.92820i) q^{19} +(1.50000 + 2.59808i) q^{23} +(-0.500000 + 0.866025i) q^{25} -3.00000 q^{29} +(-1.00000 + 1.73205i) q^{31} +(-2.00000 - 1.73205i) q^{35} +(-4.00000 - 6.92820i) q^{37} +3.00000 q^{41} +5.00000 q^{43} +(5.50000 - 4.33013i) q^{49} +(6.00000 - 10.3923i) q^{53} -6.00000 q^{55} +(0.500000 + 0.866025i) q^{61} +(-1.00000 - 1.73205i) q^{65} +(3.50000 - 6.06218i) q^{67} +(5.00000 - 8.66025i) q^{73} +(3.00000 - 15.5885i) q^{77} +(2.00000 + 3.46410i) q^{79} -3.00000 q^{83} +6.00000 q^{85} +(-1.50000 - 2.59808i) q^{89} +(5.00000 - 1.73205i) q^{91} +(-4.00000 + 6.92820i) q^{95} -10.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 5 q^{7} + 6 q^{11} + 4 q^{13} - 6 q^{17} - 8 q^{19} + 3 q^{23} - q^{25} - 6 q^{29} - 2 q^{31} - 4 q^{35} - 8 q^{37} + 6 q^{41} + 10 q^{43} + 11 q^{49} + 12 q^{53} - 12 q^{55} + q^{61} - 2 q^{65} + 7 q^{67} + 10 q^{73} + 6 q^{77} + 4 q^{79} - 6 q^{83} + 12 q^{85} - 3 q^{89} + 10 q^{91} - 8 q^{95} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1260\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(757\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.500000 0.866025i −0.223607 0.387298i
\(6\) 0 0
\(7\) 2.50000 0.866025i 0.944911 0.327327i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.00000 5.19615i 0.904534 1.56670i 0.0829925 0.996550i \(-0.473552\pi\)
0.821541 0.570149i \(-0.193114\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 + 5.19615i −0.727607 + 1.26025i 0.230285 + 0.973123i \(0.426034\pi\)
−0.957892 + 0.287129i \(0.907299\pi\)
\(18\) 0 0
\(19\) −4.00000 6.92820i −0.917663 1.58944i −0.802955 0.596040i \(-0.796740\pi\)
−0.114708 0.993399i \(-0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.50000 + 2.59808i 0.312772 + 0.541736i 0.978961 0.204046i \(-0.0654092\pi\)
−0.666190 + 0.745782i \(0.732076\pi\)
\(24\) 0 0
\(25\) −0.500000 + 0.866025i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −1.00000 + 1.73205i −0.179605 + 0.311086i −0.941745 0.336327i \(-0.890815\pi\)
0.762140 + 0.647412i \(0.224149\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 1.73205i −0.338062 0.292770i
\(36\) 0 0
\(37\) −4.00000 6.92820i −0.657596 1.13899i −0.981236 0.192809i \(-0.938240\pi\)
0.323640 0.946180i \(-0.395093\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 0.468521 0.234261 0.972174i \(-0.424733\pi\)
0.234261 + 0.972174i \(0.424733\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) 5.50000 4.33013i 0.785714 0.618590i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 10.3923i 0.824163 1.42749i −0.0783936 0.996922i \(-0.524979\pi\)
0.902557 0.430570i \(-0.141688\pi\)
\(54\) 0 0
\(55\) −6.00000 −0.809040
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0.500000 + 0.866025i 0.0640184 + 0.110883i 0.896258 0.443533i \(-0.146275\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.00000 1.73205i −0.124035 0.214834i
\(66\) 0 0
\(67\) 3.50000 6.06218i 0.427593 0.740613i −0.569066 0.822292i \(-0.692695\pi\)
0.996659 + 0.0816792i \(0.0260283\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 5.00000 8.66025i 0.585206 1.01361i −0.409644 0.912245i \(-0.634347\pi\)
0.994850 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.00000 15.5885i 0.341882 1.77647i
\(78\) 0 0
\(79\) 2.00000 + 3.46410i 0.225018 + 0.389742i 0.956325 0.292306i \(-0.0944227\pi\)
−0.731307 + 0.682048i \(0.761089\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.50000 2.59808i −0.159000 0.275396i 0.775509 0.631337i \(-0.217494\pi\)
−0.934508 + 0.355942i \(0.884160\pi\)
\(90\) 0 0
\(91\) 5.00000 1.73205i 0.524142 0.181568i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 + 6.92820i −0.410391 + 0.710819i
\(96\) 0 0
\(97\) −10.0000 −1.01535 −0.507673 0.861550i \(-0.669494\pi\)
−0.507673 + 0.861550i \(0.669494\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.50000 + 2.59808i −0.149256 + 0.258518i −0.930953 0.365140i \(-0.881021\pi\)
0.781697 + 0.623658i \(0.214354\pi\)
\(102\) 0 0
\(103\) 3.50000 + 6.06218i 0.344865 + 0.597324i 0.985329 0.170664i \(-0.0545913\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.50000 2.59808i −0.145010 0.251166i 0.784366 0.620298i \(-0.212988\pi\)
−0.929377 + 0.369132i \(0.879655\pi\)
\(108\) 0 0
\(109\) −8.50000 + 14.7224i −0.814152 + 1.41015i 0.0957826 + 0.995402i \(0.469465\pi\)
−0.909935 + 0.414751i \(0.863869\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.0000 1.12887 0.564433 0.825479i \(-0.309095\pi\)
0.564433 + 0.825479i \(0.309095\pi\)
\(114\) 0 0
\(115\) 1.50000 2.59808i 0.139876 0.242272i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 + 15.5885i −0.275010 + 1.42899i
\(120\) 0 0
\(121\) −12.5000 21.6506i −1.13636 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 6.00000 + 10.3923i 0.524222 + 0.907980i 0.999602 + 0.0281993i \(0.00897729\pi\)
−0.475380 + 0.879781i \(0.657689\pi\)
\(132\) 0 0
\(133\) −16.0000 13.8564i −1.38738 1.20150i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) 2.00000 0.169638 0.0848189 0.996396i \(-0.472969\pi\)
0.0848189 + 0.996396i \(0.472969\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000 10.3923i 0.501745 0.869048i
\(144\) 0 0
\(145\) 1.50000 + 2.59808i 0.124568 + 0.215758i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.50000 12.9904i −0.614424 1.06421i −0.990485 0.137619i \(-0.956055\pi\)
0.376061 0.926595i \(-0.377278\pi\)
\(150\) 0 0
\(151\) 5.00000 8.66025i 0.406894 0.704761i −0.587646 0.809118i \(-0.699945\pi\)
0.994540 + 0.104357i \(0.0332784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.00000 0.160644
\(156\) 0 0
\(157\) −7.00000 + 12.1244i −0.558661 + 0.967629i 0.438948 + 0.898513i \(0.355351\pi\)
−0.997609 + 0.0691164i \(0.977982\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.00000 + 5.19615i 0.472866 + 0.409514i
\(162\) 0 0
\(163\) 8.00000 + 13.8564i 0.626608 + 1.08532i 0.988227 + 0.152992i \(0.0488907\pi\)
−0.361619 + 0.932326i \(0.617776\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.0000 1.62503 0.812514 0.582941i \(-0.198098\pi\)
0.812514 + 0.582941i \(0.198098\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(174\) 0 0
\(175\) −0.500000 + 2.59808i −0.0377964 + 0.196396i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.00000 + 5.19615i −0.224231 + 0.388379i −0.956088 0.293079i \(-0.905320\pi\)
0.731858 + 0.681457i \(0.238654\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) 18.0000 + 31.1769i 1.31629 + 2.27988i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −9.00000 15.5885i −0.651217 1.12794i −0.982828 0.184525i \(-0.940925\pi\)
0.331611 0.943416i \(-0.392408\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.73205i −0.0719816 + 0.124676i −0.899770 0.436365i \(-0.856266\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.0000 1.28245 0.641223 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 0 0
\(199\) −10.0000 + 17.3205i −0.708881 + 1.22782i 0.256391 + 0.966573i \(0.417466\pi\)
−0.965272 + 0.261245i \(0.915867\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.50000 + 2.59808i −0.526397 + 0.182349i
\(204\) 0 0
\(205\) −1.50000 2.59808i −0.104765 0.181458i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −48.0000 −3.32023
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2.50000 4.33013i −0.170499 0.295312i
\(216\) 0 0
\(217\) −1.00000 + 5.19615i −0.0678844 + 0.352738i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 + 10.3923i −0.403604 + 0.699062i
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.00000 + 10.3923i −0.398234 + 0.689761i −0.993508 0.113761i \(-0.963710\pi\)
0.595274 + 0.803523i \(0.297043\pi\)
\(228\) 0 0
\(229\) −1.00000 1.73205i −0.0660819 0.114457i 0.831092 0.556136i \(-0.187717\pi\)
−0.897173 + 0.441679i \(0.854383\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.00000 + 5.19615i 0.196537 + 0.340411i 0.947403 0.320043i \(-0.103697\pi\)
−0.750867 + 0.660454i \(0.770364\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) −13.0000 + 22.5167i −0.837404 + 1.45043i 0.0546547 + 0.998505i \(0.482594\pi\)
−0.892058 + 0.451920i \(0.850739\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.50000 2.59808i −0.415270 0.165985i
\(246\) 0 0
\(247\) −8.00000 13.8564i −0.509028 0.881662i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.0000 1.13615 0.568075 0.822977i \(-0.307688\pi\)
0.568075 + 0.822977i \(0.307688\pi\)
\(252\) 0 0
\(253\) 18.0000 1.13165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −12.0000 20.7846i −0.748539 1.29651i −0.948523 0.316709i \(-0.897422\pi\)
0.199983 0.979799i \(-0.435911\pi\)
\(258\) 0 0
\(259\) −16.0000 13.8564i −0.994192 0.860995i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.50000 + 2.59808i −0.0924940 + 0.160204i −0.908560 0.417755i \(-0.862817\pi\)
0.816066 + 0.577959i \(0.196151\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.50000 + 12.9904i −0.457283 + 0.792038i −0.998816 0.0486418i \(-0.984511\pi\)
0.541533 + 0.840679i \(0.317844\pi\)
\(270\) 0 0
\(271\) 8.00000 + 13.8564i 0.485965 + 0.841717i 0.999870 0.0161307i \(-0.00513477\pi\)
−0.513905 + 0.857847i \(0.671801\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.00000 + 5.19615i 0.180907 + 0.313340i
\(276\) 0 0
\(277\) 11.0000 19.0526i 0.660926 1.14476i −0.319447 0.947604i \(-0.603497\pi\)
0.980373 0.197153i \(-0.0631696\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −10.0000 + 17.3205i −0.594438 + 1.02960i 0.399188 + 0.916869i \(0.369292\pi\)
−0.993626 + 0.112728i \(0.964041\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.50000 2.59808i 0.442711 0.153360i
\(288\) 0 0
\(289\) −9.50000 16.4545i −0.558824 0.967911i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.00000 + 5.19615i 0.173494 + 0.300501i
\(300\) 0 0
\(301\) 12.5000 4.33013i 0.720488 0.249584i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.500000 0.866025i 0.0286299 0.0495885i
\(306\) 0 0
\(307\) −19.0000 −1.08439 −0.542194 0.840254i \(-0.682406\pi\)
−0.542194 + 0.840254i \(0.682406\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 2.00000 + 3.46410i 0.113047 + 0.195803i 0.916997 0.398894i \(-0.130606\pi\)
−0.803951 + 0.594696i \(0.797272\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.00000 + 15.5885i 0.505490 + 0.875535i 0.999980 + 0.00635137i \(0.00202172\pi\)
−0.494489 + 0.869184i \(0.664645\pi\)
\(318\) 0 0
\(319\) −9.00000 + 15.5885i −0.503903 + 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000 2.67079
\(324\) 0 0
\(325\) −1.00000 + 1.73205i −0.0554700 + 0.0960769i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 11.0000 + 19.0526i 0.604615 + 1.04722i 0.992112 + 0.125353i \(0.0400062\pi\)
−0.387498 + 0.921871i \(0.626660\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −7.00000 −0.382451
\(336\) 0 0
\(337\) 20.0000 1.08947 0.544735 0.838608i \(-0.316630\pi\)
0.544735 + 0.838608i \(0.316630\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.00000 + 10.3923i 0.324918 + 0.562775i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −13.5000 + 23.3827i −0.724718 + 1.25525i 0.234372 + 0.972147i \(0.424697\pi\)
−0.959090 + 0.283101i \(0.908637\pi\)
\(348\) 0 0
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6.00000 + 10.3923i −0.319348 + 0.553127i −0.980352 0.197256i \(-0.936797\pi\)
0.661004 + 0.750382i \(0.270130\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 + 5.19615i 0.158334 + 0.274242i 0.934268 0.356572i \(-0.116054\pi\)
−0.775934 + 0.630814i \(0.782721\pi\)
\(360\) 0 0
\(361\) −22.5000 + 38.9711i −1.18421 + 2.05111i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −10.0000 −0.523424
\(366\) 0 0
\(367\) −2.50000 + 4.33013i −0.130499 + 0.226031i −0.923869 0.382709i \(-0.874991\pi\)
0.793370 + 0.608740i \(0.208325\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.00000 31.1769i 0.311504 1.61862i
\(372\) 0 0
\(373\) 8.00000 + 13.8564i 0.414224 + 0.717458i 0.995347 0.0963587i \(-0.0307196\pi\)
−0.581122 + 0.813816i \(0.697386\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.50000 + 2.59808i 0.0766464 + 0.132755i 0.901801 0.432151i \(-0.142245\pi\)
−0.825155 + 0.564907i \(0.808912\pi\)
\(384\) 0 0
\(385\) −15.0000 + 5.19615i −0.764471 + 0.264820i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 3.00000 5.19615i 0.152106 0.263455i −0.779895 0.625910i \(-0.784728\pi\)
0.932002 + 0.362454i \(0.118061\pi\)
\(390\) 0 0
\(391\) −18.0000 −0.910299
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.00000 3.46410i 0.100631 0.174298i
\(396\) 0 0
\(397\) −13.0000 22.5167i −0.652451 1.13008i −0.982526 0.186124i \(-0.940407\pi\)
0.330075 0.943955i \(-0.392926\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.50000 + 2.59808i 0.0749064 + 0.129742i 0.901046 0.433724i \(-0.142801\pi\)
−0.826139 + 0.563466i \(0.809468\pi\)
\(402\) 0 0
\(403\) −2.00000 + 3.46410i −0.0996271 + 0.172559i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −48.0000 −2.37927
\(408\) 0 0
\(409\) −5.50000 + 9.52628i −0.271957 + 0.471044i −0.969363 0.245633i \(-0.921004\pi\)
0.697406 + 0.716677i \(0.254338\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.50000 + 2.59808i 0.0736321 + 0.127535i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −24.0000 −1.17248 −0.586238 0.810139i \(-0.699392\pi\)
−0.586238 + 0.810139i \(0.699392\pi\)
\(420\) 0 0
\(421\) 23.0000 1.12095 0.560476 0.828171i \(-0.310618\pi\)
0.560476 + 0.828171i \(0.310618\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 5.19615i −0.145521 0.252050i
\(426\) 0 0
\(427\) 2.00000 + 1.73205i 0.0967868 + 0.0838198i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.0000 25.9808i 0.722525 1.25145i −0.237460 0.971397i \(-0.576315\pi\)
0.959985 0.280052i \(-0.0903517\pi\)
\(432\) 0 0
\(433\) −28.0000 −1.34559 −0.672797 0.739827i \(-0.734907\pi\)
−0.672797 + 0.739827i \(0.734907\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 12.0000 20.7846i 0.574038 0.994263i
\(438\) 0 0
\(439\) 14.0000 + 24.2487i 0.668184 + 1.15733i 0.978412 + 0.206666i \(0.0662612\pi\)
−0.310228 + 0.950662i \(0.600405\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.50000 + 7.79423i 0.213801 + 0.370315i 0.952901 0.303281i \(-0.0980821\pi\)
−0.739100 + 0.673596i \(0.764749\pi\)
\(444\) 0 0
\(445\) −1.50000 + 2.59808i −0.0711068 + 0.123161i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.00000 0.424736 0.212368 0.977190i \(-0.431882\pi\)
0.212368 + 0.977190i \(0.431882\pi\)
\(450\) 0 0
\(451\) 9.00000 15.5885i 0.423793 0.734032i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −4.00000 3.46410i −0.187523 0.162400i
\(456\) 0 0
\(457\) 2.00000 + 3.46410i 0.0935561 + 0.162044i 0.909005 0.416785i \(-0.136843\pi\)
−0.815449 + 0.578829i \(0.803510\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 29.0000 1.34774 0.673872 0.738848i \(-0.264630\pi\)
0.673872 + 0.738848i \(0.264630\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 16.5000 + 28.5788i 0.763529 + 1.32247i 0.941021 + 0.338349i \(0.109868\pi\)
−0.177492 + 0.984122i \(0.556798\pi\)
\(468\) 0 0
\(469\) 3.50000 18.1865i 0.161615 0.839776i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 15.0000 25.9808i 0.689701 1.19460i
\(474\) 0 0
\(475\) 8.00000 0.367065
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −15.0000 + 25.9808i −0.685367 + 1.18709i 0.287954 + 0.957644i \(0.407025\pi\)
−0.973321 + 0.229447i \(0.926308\pi\)
\(480\) 0 0
\(481\) −8.00000 13.8564i −0.364769 0.631798i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5.00000 + 8.66025i 0.227038 + 0.393242i
\(486\) 0 0
\(487\) −4.00000 + 6.92820i −0.181257 + 0.313947i −0.942309 0.334744i \(-0.891350\pi\)
0.761052 + 0.648691i \(0.224683\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.00000 −0.270776 −0.135388 0.990793i \(-0.543228\pi\)
−0.135388 + 0.990793i \(0.543228\pi\)
\(492\) 0 0
\(493\) 9.00000 15.5885i 0.405340 0.702069i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.00000 1.73205i −0.0447661 0.0775372i 0.842774 0.538267i \(-0.180921\pi\)
−0.887540 + 0.460730i \(0.847588\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.00000 0.401290 0.200645 0.979664i \(-0.435696\pi\)
0.200645 + 0.979664i \(0.435696\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 19.5000 + 33.7750i 0.864322 + 1.49705i 0.867719 + 0.497056i \(0.165586\pi\)
−0.00339621 + 0.999994i \(0.501081\pi\)
\(510\) 0 0
\(511\) 5.00000 25.9808i 0.221187 1.14932i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.50000 6.06218i 0.154228 0.267131i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 15.0000 25.9808i 0.657162 1.13824i −0.324185 0.945994i \(-0.605090\pi\)
0.981347 0.192244i \(-0.0615766\pi\)
\(522\) 0 0
\(523\) −22.0000 38.1051i −0.961993 1.66622i −0.717486 0.696573i \(-0.754707\pi\)
−0.244507 0.969648i \(-0.578626\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −6.00000 10.3923i −0.261364 0.452696i
\(528\) 0 0
\(529\) 7.00000 12.1244i 0.304348 0.527146i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.00000 0.259889
\(534\) 0 0
\(535\) −1.50000 + 2.59808i −0.0648507 + 0.112325i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −6.00000 41.5692i −0.258438 1.79051i
\(540\) 0 0
\(541\) −8.50000 14.7224i −0.365444 0.632967i 0.623404 0.781900i \(-0.285749\pi\)
−0.988847 + 0.148933i \(0.952416\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 17.0000 0.728200
\(546\) 0 0
\(547\) −19.0000 −0.812381 −0.406191 0.913788i \(-0.633143\pi\)
−0.406191 + 0.913788i \(0.633143\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0000 + 20.7846i 0.511217 + 0.885454i
\(552\) 0 0
\(553\) 8.00000 + 6.92820i 0.340195 + 0.294617i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 25.9808i 0.635570 1.10084i −0.350824 0.936442i \(-0.614098\pi\)
0.986394 0.164399i \(-0.0525683\pi\)
\(558\) 0 0
\(559\) 10.0000 0.422955
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.50000 + 7.79423i −0.189652 + 0.328488i −0.945134 0.326682i \(-0.894069\pi\)
0.755482 + 0.655169i \(0.227403\pi\)
\(564\) 0 0
\(565\) −6.00000 10.3923i −0.252422 0.437208i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −9.00000 15.5885i −0.377300 0.653502i 0.613369 0.789797i \(-0.289814\pi\)
−0.990668 + 0.136295i \(0.956481\pi\)
\(570\) 0 0
\(571\) 8.00000 13.8564i 0.334790 0.579873i −0.648655 0.761083i \(-0.724668\pi\)
0.983444 + 0.181210i \(0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) 0 0
\(577\) 5.00000 8.66025i 0.208153 0.360531i −0.742980 0.669314i \(-0.766588\pi\)
0.951133 + 0.308783i \(0.0999216\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −7.50000 + 2.59808i −0.311152 + 0.107786i
\(582\) 0 0
\(583\) −36.0000 62.3538i −1.49097 2.58243i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −6.00000 10.3923i −0.246390 0.426761i 0.716131 0.697966i \(-0.245911\pi\)
−0.962522 + 0.271205i \(0.912578\pi\)
\(594\) 0 0
\(595\) 15.0000 5.19615i 0.614940 0.213021i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −12.5000 + 21.6506i −0.508197 + 0.880223i
\(606\) 0 0
\(607\) 21.5000 + 37.2391i 0.872658 + 1.51149i 0.859237 + 0.511578i \(0.170939\pi\)
0.0134214 + 0.999910i \(0.495728\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.00000 + 1.73205i −0.0403896 + 0.0699569i −0.885514 0.464614i \(-0.846193\pi\)
0.845124 + 0.534570i \(0.179527\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 12.0000 0.483102 0.241551 0.970388i \(-0.422344\pi\)
0.241551 + 0.970388i \(0.422344\pi\)
\(618\) 0 0
\(619\) −7.00000 + 12.1244i −0.281354 + 0.487319i −0.971718 0.236143i \(-0.924117\pi\)
0.690365 + 0.723462i \(0.257450\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.00000 5.19615i −0.240385 0.208179i
\(624\) 0 0
\(625\) −0.500000 0.866025i −0.0200000 0.0346410i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 48.0000 1.91389
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.00000 6.92820i −0.158735 0.274937i
\(636\) 0 0
\(637\) 11.0000 8.66025i 0.435836 0.343132i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −19.5000 + 33.7750i −0.770204 + 1.33403i 0.167247 + 0.985915i \(0.446512\pi\)
−0.937451 + 0.348117i \(0.886821\pi\)
\(642\) 0 0
\(643\) 20.0000 0.788723 0.394362 0.918955i \(-0.370966\pi\)
0.394362 + 0.918955i \(0.370966\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 10.5000 18.1865i 0.412798 0.714986i −0.582397 0.812905i \(-0.697885\pi\)
0.995194 + 0.0979182i \(0.0312184\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −21.0000 36.3731i −0.821794 1.42339i −0.904345 0.426801i \(-0.859640\pi\)
0.0825519 0.996587i \(-0.473693\pi\)
\(654\) 0 0
\(655\) 6.00000 10.3923i 0.234439 0.406061i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 12.5000 21.6506i 0.486194 0.842112i −0.513680 0.857982i \(-0.671718\pi\)
0.999874 + 0.0158695i \(0.00505163\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −4.00000 + 20.7846i −0.155113 + 0.805993i
\(666\) 0 0
\(667\) −4.50000 7.79423i −0.174241 0.301794i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 6.00000 0.231627
\(672\) 0 0
\(673\) 32.0000 1.23351 0.616755 0.787155i \(-0.288447\pi\)
0.616755 + 0.787155i \(0.288447\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9.00000 15.5885i −0.345898 0.599113i 0.639618 0.768693i \(-0.279092\pi\)
−0.985517 + 0.169580i \(0.945759\pi\)
\(678\) 0 0
\(679\) −25.0000 + 8.66025i −0.959412 + 0.332350i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.50000 + 12.9904i −0.286980 + 0.497063i −0.973087 0.230437i \(-0.925985\pi\)
0.686108 + 0.727500i \(0.259318\pi\)
\(684\) 0 0
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.0000 20.7846i 0.457164 0.791831i
\(690\) 0 0
\(691\) −7.00000 12.1244i −0.266293 0.461232i 0.701609 0.712562i \(-0.252465\pi\)
−0.967901 + 0.251330i \(0.919132\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.00000 1.73205i −0.0379322 0.0657004i
\(696\) 0 0
\(697\) −9.00000 + 15.5885i −0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27.0000 −1.01978 −0.509888 0.860241i \(-0.670313\pi\)
−0.509888 + 0.860241i \(0.670313\pi\)
\(702\) 0 0
\(703\) −32.0000 + 55.4256i −1.20690 + 2.09042i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.50000 + 7.79423i −0.0564133 + 0.293132i
\(708\) 0 0
\(709\) 0.500000 + 0.866025i 0.0187779 + 0.0325243i 0.875262 0.483650i \(-0.160689\pi\)
−0.856484 + 0.516174i \(0.827356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −6.00000 −0.224702
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −15.0000 25.9808i −0.559406 0.968919i −0.997546 0.0700124i \(-0.977696\pi\)
0.438141 0.898906i \(-0.355637\pi\)
\(720\) 0 0
\(721\) 14.0000 + 12.1244i 0.521387 + 0.451535i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.50000 2.59808i 0.0557086 0.0964901i
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −15.0000 + 25.9808i −0.554795 + 0.960933i
\(732\) 0 0
\(733\) −1.00000 1.73205i −0.0369358 0.0639748i 0.846967 0.531646i \(-0.178426\pi\)
−0.883902 + 0.467671i \(0.845093\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.0000 36.3731i −0.773545 1.33982i
\(738\) 0 0
\(739\) 23.0000 39.8372i 0.846069 1.46543i −0.0386212 0.999254i \(-0.512297\pi\)
0.884690 0.466180i \(-0.154370\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 39.0000 1.43077 0.715386 0.698730i \(-0.246251\pi\)
0.715386 + 0.698730i \(0.246251\pi\)
\(744\) 0 0
\(745\) −7.50000 + 12.9904i −0.274779 + 0.475931i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −6.00000 5.19615i −0.219235 0.189863i
\(750\) 0 0
\(751\) −10.0000 17.3205i −0.364905 0.632034i 0.623856 0.781540i \(-0.285565\pi\)
−0.988761 + 0.149505i \(0.952232\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −10.0000 −0.363937
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 + 25.9808i 0.543750 + 0.941802i 0.998684 + 0.0512772i \(0.0163292\pi\)
−0.454935 + 0.890525i \(0.650337\pi\)
\(762\) 0 0
\(763\) −8.50000 + 44.1673i −0.307721 + 1.59896i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −21.0000 + 36.3731i −0.755318 + 1.30825i 0.189899 + 0.981804i \(0.439184\pi\)
−0.945216 + 0.326445i \(0.894149\pi\)
\(774\) 0 0
\(775\) −1.00000 1.73205i −0.0359211 0.0622171i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 20.7846i −0.429945 0.744686i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) 15.5000 26.8468i 0.552515 0.956985i −0.445577 0.895244i \(-0.647001\pi\)
0.998092 0.0617409i \(-0.0196653\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 30.0000 10.3923i 1.06668 0.369508i
\(792\) 0 0
\(793\) 1.00000 + 1.73205i 0.0355110 + 0.0615069i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 12.0000 0.425062 0.212531 0.977154i \(-0.431829\pi\)
0.212531 + 0.977154i \(0.431829\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −30.0000 51.9615i −1.05868 1.83368i
\(804\) 0 0
\(805\) 1.50000 7.79423i 0.0528681 0.274710i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −25.5000 + 44.1673i −0.896532 + 1.55284i −0.0646355 + 0.997909i \(0.520588\pi\)
−0.831897 + 0.554930i \(0.812745\pi\)
\(810\) 0 0
\(811\) 2.00000 0.0702295 0.0351147 0.999383i \(-0.488820\pi\)
0.0351147 + 0.999383i \(0.488820\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 8.00000 13.8564i 0.280228 0.485369i
\(816\) 0 0
\(817\) −20.0000 34.6410i −0.699711 1.21194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.00000 + 15.5885i 0.314102 + 0.544041i 0.979246 0.202674i \(-0.0649632\pi\)
−0.665144 + 0.746715i \(0.731630\pi\)
\(822\) 0 0
\(823\) 24.5000 42.4352i 0.854016 1.47920i −0.0235383 0.999723i \(-0.507493\pi\)
0.877555 0.479477i \(-0.159174\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 45.0000 1.56480 0.782402 0.622774i \(-0.213994\pi\)
0.782402 + 0.622774i \(0.213994\pi\)
\(828\) 0 0
\(829\) 5.00000 8.66025i 0.173657 0.300783i −0.766039 0.642795i \(-0.777775\pi\)
0.939696 + 0.342012i \(0.111108\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.00000 + 41.5692i 0.207888 + 1.44029i
\(834\) 0 0
\(835\) −10.5000 18.1865i −0.363367 0.629371i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 42.0000 1.45000 0.725001 0.688748i \(-0.241839\pi\)
0.725001 + 0.688748i \(0.241839\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 4.50000 + 7.79423i 0.154805 + 0.268130i
\(846\) 0 0
\(847\) −50.0000 43.3013i −1.71802 1.48785i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 20.7846i 0.411355 0.712487i
\(852\) 0 0
\(853\) 32.0000 1.09566 0.547830 0.836590i \(-0.315454\pi\)
0.547830 + 0.836590i \(0.315454\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.0000 36.3731i 0.717346 1.24248i −0.244701 0.969599i \(-0.578690\pi\)
0.962048 0.272882i \(-0.0879768\pi\)
\(858\) 0 0
\(859\) 2.00000 + 3.46410i 0.0682391 + 0.118194i 0.898126 0.439738i \(-0.144929\pi\)
−0.829887 + 0.557931i \(0.811595\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 16.5000 + 28.5788i 0.561667 + 0.972835i 0.997351 + 0.0727356i \(0.0231729\pi\)
−0.435685 + 0.900099i \(0.643494\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 24.0000 0.814144
\(870\) 0 0
\(871\) 7.00000 12.1244i 0.237186 0.410818i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.50000 0.866025i 0.0845154 0.0292770i
\(876\) 0 0
\(877\) 20.0000 + 34.6410i 0.675352 + 1.16974i 0.976366 + 0.216124i \(0.0693416\pi\)
−0.301014 + 0.953620i \(0.597325\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −9.00000 −0.303218 −0.151609 0.988441i \(-0.548445\pi\)
−0.151609 + 0.988441i \(0.548445\pi\)
\(882\) 0 0
\(883\) −28.0000 −0.942275 −0.471138 0.882060i \(-0.656156\pi\)
−0.471138 + 0.882060i \(0.656156\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 28.5000 + 49.3634i 0.956936 + 1.65746i 0.729873 + 0.683582i \(0.239579\pi\)
0.227063 + 0.973880i \(0.427088\pi\)
\(888\) 0 0
\(889\) 20.0000 6.92820i 0.670778 0.232364i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.00000 5.19615i 0.100056 0.173301i
\(900\) 0 0
\(901\) 36.0000 + 62.3538i 1.19933 + 2.07731i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −8.50000 14.7224i −0.282550 0.489390i
\(906\) 0 0
\(907\) −11.5000 + 19.9186i −0.381851 + 0.661386i −0.991327 0.131419i \(-0.958047\pi\)
0.609476 + 0.792805i \(0.291380\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −30.0000 −0.993944 −0.496972 0.867766i \(-0.665555\pi\)
−0.496972 + 0.867766i \(0.665555\pi\)
\(912\) 0 0
\(913\) −9.00000 + 15.5885i −0.297857 + 0.515903i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.0000 + 20.7846i 0.792550 + 0.686368i
\(918\) 0 0
\(919\) −4.00000 6.92820i −0.131948 0.228540i 0.792480 0.609898i \(-0.208790\pi\)
−0.924427 + 0.381358i \(0.875456\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −22.5000 38.9711i −0.738201 1.27860i −0.953305 0.302010i \(-0.902342\pi\)
0.215104 0.976591i \(-0.430991\pi\)
\(930\) 0 0
\(931\) −52.0000 20.7846i −1.70423 0.681188i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 18.0000 31.1769i 0.588663 1.01959i
\(936\) 0 0
\(937\) 44.0000 1.43742 0.718709 0.695311i \(-0.244734\pi\)
0.718709 + 0.695311i \(0.244734\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.0000 46.7654i 0.880175 1.52451i 0.0290288 0.999579i \(-0.490759\pi\)
0.851146 0.524929i \(-0.175908\pi\)
\(942\) 0 0
\(943\) 4.50000 + 7.79423i 0.146540 + 0.253815i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.50000 + 7.79423i 0.146230 + 0.253278i 0.929831 0.367986i \(-0.119953\pi\)
−0.783601 + 0.621264i \(0.786619\pi\)
\(948\) 0 0
\(949\) 10.0000 17.3205i 0.324614 0.562247i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) −9.00000 + 15.5885i −0.291233 + 0.504431i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −6.00000 + 31.1769i −0.193750 + 1.00676i
\(960\) 0 0
\(961\) 13.5000 + 23.3827i 0.435484 + 0.754280i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −7.00000 −0.225105 −0.112552 0.993646i \(-0.535903\pi\)
−0.112552 + 0.993646i \(0.535903\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −6.00000 10.3923i −0.192549 0.333505i 0.753545 0.657396i \(-0.228342\pi\)
−0.946094 + 0.323891i \(0.895009\pi\)
\(972\) 0 0
\(973\) 5.00000 1.73205i 0.160293 0.0555270i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −27.0000 + 46.7654i −0.863807 + 1.49616i 0.00442082 + 0.999990i \(0.498593\pi\)
−0.868227 + 0.496167i \(0.834741\pi\)
\(978\) 0 0
\(979\) −18.0000 −0.575282
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −19.5000 + 33.7750i −0.621953 + 1.07725i 0.367168 + 0.930155i \(0.380327\pi\)
−0.989122 + 0.147100i \(0.953006\pi\)
\(984\) 0 0
\(985\) −9.00000 15.5885i −0.286764 0.496690i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 7.50000 + 12.9904i 0.238486 + 0.413070i
\(990\) 0 0
\(991\) −13.0000 + 22.5167i −0.412959 + 0.715265i −0.995212 0.0977423i \(-0.968838\pi\)
0.582253 + 0.813008i \(0.302171\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) −16.0000 + 27.7128i −0.506725 + 0.877674i 0.493245 + 0.869891i \(0.335811\pi\)
−0.999970 + 0.00778294i \(0.997523\pi\)
\(998\) 0 0
\(999\) 0 0
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 1260.2.s.c.361.1 2
3.2 odd 2 140.2.i.a.81.1 2
7.2 even 3 inner 1260.2.s.c.541.1 2
7.3 odd 6 8820.2.a.a.1.1 1
7.4 even 3 8820.2.a.p.1.1 1
12.11 even 2 560.2.q.f.81.1 2
15.2 even 4 700.2.r.a.249.2 4
15.8 even 4 700.2.r.a.249.1 4
15.14 odd 2 700.2.i.b.501.1 2
21.2 odd 6 140.2.i.a.121.1 yes 2
21.5 even 6 980.2.i.f.961.1 2
21.11 odd 6 980.2.a.g.1.1 1
21.17 even 6 980.2.a.e.1.1 1
21.20 even 2 980.2.i.f.361.1 2
84.11 even 6 3920.2.a.k.1.1 1
84.23 even 6 560.2.q.f.401.1 2
84.59 odd 6 3920.2.a.w.1.1 1
105.2 even 12 700.2.r.a.149.1 4
105.17 odd 12 4900.2.e.n.2549.2 2
105.23 even 12 700.2.r.a.149.2 4
105.32 even 12 4900.2.e.m.2549.1 2
105.38 odd 12 4900.2.e.n.2549.1 2
105.44 odd 6 700.2.i.b.401.1 2
105.53 even 12 4900.2.e.m.2549.2 2
105.59 even 6 4900.2.a.q.1.1 1
105.74 odd 6 4900.2.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
140.2.i.a.81.1 2 3.2 odd 2
140.2.i.a.121.1 yes 2 21.2 odd 6
560.2.q.f.81.1 2 12.11 even 2
560.2.q.f.401.1 2 84.23 even 6
700.2.i.b.401.1 2 105.44 odd 6
700.2.i.b.501.1 2 15.14 odd 2
700.2.r.a.149.1 4 105.2 even 12
700.2.r.a.149.2 4 105.23 even 12
700.2.r.a.249.1 4 15.8 even 4
700.2.r.a.249.2 4 15.2 even 4
980.2.a.e.1.1 1 21.17 even 6
980.2.a.g.1.1 1 21.11 odd 6
980.2.i.f.361.1 2 21.20 even 2
980.2.i.f.961.1 2 21.5 even 6
1260.2.s.c.361.1 2 1.1 even 1 trivial
1260.2.s.c.541.1 2 7.2 even 3 inner
3920.2.a.k.1.1 1 84.11 even 6
3920.2.a.w.1.1 1 84.59 odd 6
4900.2.a.i.1.1 1 105.74 odd 6
4900.2.a.q.1.1 1 105.59 even 6
4900.2.e.m.2549.1 2 105.32 even 12
4900.2.e.m.2549.2 2 105.53 even 12
4900.2.e.n.2549.1 2 105.38 odd 12
4900.2.e.n.2549.2 2 105.17 odd 12
8820.2.a.a.1.1 1 7.3 odd 6
8820.2.a.p.1.1 1 7.4 even 3