Properties

Label 1296.2.i.d.433.1
Level $1296$
Weight $2$
Character 1296.433
Analytic conductor $10.349$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,2,Mod(433,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.i (of order \(3\), degree \(2\), not 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: \(10.3486121020\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 324)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 433.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1296.433
Dual form 1296.2.i.d.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 + 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{7} +O(q^{10})\) \(q+(-1.50000 + 2.59808i) q^{5} +(1.00000 + 1.73205i) q^{7} +(-3.00000 - 5.19615i) q^{11} +(-2.50000 + 4.33013i) q^{13} -3.00000 q^{17} -2.00000 q^{19} +(3.00000 - 5.19615i) q^{23} +(-2.00000 - 3.46410i) q^{25} +(-1.50000 - 2.59808i) q^{29} +(-2.00000 + 3.46410i) q^{31} -6.00000 q^{35} +5.00000 q^{37} +(3.00000 - 5.19615i) q^{41} +(-5.00000 - 8.66025i) q^{43} +(1.50000 - 2.59808i) q^{49} -6.00000 q^{53} +18.0000 q^{55} +(-6.00000 + 10.3923i) q^{59} +(-2.50000 - 4.33013i) q^{61} +(-7.50000 - 12.9904i) q^{65} +(1.00000 - 1.73205i) q^{67} -6.00000 q^{71} -1.00000 q^{73} +(6.00000 - 10.3923i) q^{77} +(-5.00000 - 8.66025i) q^{79} +(4.50000 - 7.79423i) q^{85} -3.00000 q^{89} -10.0000 q^{91} +(3.00000 - 5.19615i) q^{95} +(5.00000 + 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{5} + 2 q^{7} - 6 q^{11} - 5 q^{13} - 6 q^{17} - 4 q^{19} + 6 q^{23} - 4 q^{25} - 3 q^{29} - 4 q^{31} - 12 q^{35} + 10 q^{37} + 6 q^{41} - 10 q^{43} + 3 q^{49} - 12 q^{53} + 36 q^{55} - 12 q^{59} - 5 q^{61} - 15 q^{65} + 2 q^{67} - 12 q^{71} - 2 q^{73} + 12 q^{77} - 10 q^{79} + 9 q^{85} - 6 q^{89} - 20 q^{91} + 6 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −1.50000 + 2.59808i −0.670820 + 1.16190i 0.306851 + 0.951757i \(0.400725\pi\)
−0.977672 + 0.210138i \(0.932609\pi\)
\(6\) 0 0
\(7\) 1.00000 + 1.73205i 0.377964 + 0.654654i 0.990766 0.135583i \(-0.0432908\pi\)
−0.612801 + 0.790237i \(0.709957\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.00000 5.19615i −0.904534 1.56670i −0.821541 0.570149i \(-0.806886\pi\)
−0.0829925 0.996550i \(-0.526448\pi\)
\(12\) 0 0
\(13\) −2.50000 + 4.33013i −0.693375 + 1.20096i 0.277350 + 0.960769i \(0.410544\pi\)
−0.970725 + 0.240192i \(0.922790\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000 5.19615i 0.625543 1.08347i −0.362892 0.931831i \(-0.618211\pi\)
0.988436 0.151642i \(-0.0484560\pi\)
\(24\) 0 0
\(25\) −2.00000 3.46410i −0.400000 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.50000 2.59808i −0.278543 0.482451i 0.692480 0.721437i \(-0.256518\pi\)
−0.971023 + 0.238987i \(0.923185\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.00000 −1.01419
\(36\) 0 0
\(37\) 5.00000 0.821995 0.410997 0.911636i \(-0.365181\pi\)
0.410997 + 0.911636i \(0.365181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.00000 5.19615i 0.468521 0.811503i −0.530831 0.847477i \(-0.678120\pi\)
0.999353 + 0.0359748i \(0.0114536\pi\)
\(42\) 0 0
\(43\) −5.00000 8.66025i −0.762493 1.32068i −0.941562 0.336840i \(-0.890642\pi\)
0.179069 0.983836i \(-0.442691\pi\)
\(44\) 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
\(49\) 1.50000 2.59808i 0.214286 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 18.0000 2.42712
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 + 10.3923i −0.781133 + 1.35296i 0.150148 + 0.988663i \(0.452025\pi\)
−0.931282 + 0.364299i \(0.881308\pi\)
\(60\) 0 0
\(61\) −2.50000 4.33013i −0.320092 0.554416i 0.660415 0.750901i \(-0.270381\pi\)
−0.980507 + 0.196485i \(0.937047\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.50000 12.9904i −0.930261 1.61126i
\(66\) 0 0
\(67\) 1.00000 1.73205i 0.122169 0.211604i −0.798454 0.602056i \(-0.794348\pi\)
0.920623 + 0.390453i \(0.127682\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.00000 10.3923i 0.683763 1.18431i
\(78\) 0 0
\(79\) −5.00000 8.66025i −0.562544 0.974355i −0.997274 0.0737937i \(-0.976489\pi\)
0.434730 0.900561i \(-0.356844\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 4.50000 7.79423i 0.488094 0.845403i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.00000 −0.317999 −0.159000 0.987279i \(-0.550827\pi\)
−0.159000 + 0.987279i \(0.550827\pi\)
\(90\) 0 0
\(91\) −10.0000 −1.04828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 5.19615i 0.307794 0.533114i
\(96\) 0 0
\(97\) 5.00000 + 8.66025i 0.507673 + 0.879316i 0.999961 + 0.00888289i \(0.00282755\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 5.19615i −0.298511 0.517036i 0.677284 0.735721i \(-0.263157\pi\)
−0.975796 + 0.218685i \(0.929823\pi\)
\(102\) 0 0
\(103\) −8.00000 + 13.8564i −0.788263 + 1.36531i 0.138767 + 0.990325i \(0.455686\pi\)
−0.927030 + 0.374987i \(0.877647\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −7.00000 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −4.50000 + 7.79423i −0.423324 + 0.733219i −0.996262 0.0863794i \(-0.972470\pi\)
0.572938 + 0.819599i \(0.305804\pi\)
\(114\) 0 0
\(115\) 9.00000 + 15.5885i 0.839254 + 1.45363i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.00000 5.19615i −0.275010 0.476331i
\(120\) 0 0
\(121\) −12.5000 + 21.6506i −1.13636 + 1.96824i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.00000 −0.268328
\(126\) 0 0
\(127\) −2.00000 −0.177471 −0.0887357 0.996055i \(-0.528283\pi\)
−0.0887357 + 0.996055i \(0.528283\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.00000 + 5.19615i −0.262111 + 0.453990i −0.966803 0.255524i \(-0.917752\pi\)
0.704692 + 0.709514i \(0.251085\pi\)
\(132\) 0 0
\(133\) −2.00000 3.46410i −0.173422 0.300376i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.50000 + 12.9904i 0.640768 + 1.10984i 0.985262 + 0.171054i \(0.0547174\pi\)
−0.344493 + 0.938789i \(0.611949\pi\)
\(138\) 0 0
\(139\) 4.00000 6.92820i 0.339276 0.587643i −0.645021 0.764165i \(-0.723151\pi\)
0.984297 + 0.176522i \(0.0564848\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 30.0000 2.50873
\(144\) 0 0
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.50000 + 12.9904i −0.614424 + 1.06421i 0.376061 + 0.926595i \(0.377278\pi\)
−0.990485 + 0.137619i \(0.956055\pi\)
\(150\) 0 0
\(151\) −2.00000 3.46410i −0.162758 0.281905i 0.773099 0.634285i \(-0.218706\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −6.00000 10.3923i −0.481932 0.834730i
\(156\) 0 0
\(157\) −2.50000 + 4.33013i −0.199522 + 0.345582i −0.948373 0.317156i \(-0.897272\pi\)
0.748852 + 0.662738i \(0.230606\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.0000 0.945732
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.00000 15.5885i 0.696441 1.20627i −0.273252 0.961943i \(-0.588099\pi\)
0.969693 0.244328i \(-0.0785675\pi\)
\(168\) 0 0
\(169\) −6.00000 10.3923i −0.461538 0.799408i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4.50000 + 7.79423i 0.342129 + 0.592584i 0.984828 0.173534i \(-0.0555188\pi\)
−0.642699 + 0.766119i \(0.722185\pi\)
\(174\) 0 0
\(175\) 4.00000 6.92820i 0.302372 0.523723i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.50000 + 12.9904i −0.551411 + 0.955072i
\(186\) 0 0
\(187\) 9.00000 + 15.5885i 0.658145 + 1.13994i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.00000 + 5.19615i 0.217072 + 0.375980i 0.953912 0.300088i \(-0.0970159\pi\)
−0.736839 + 0.676068i \(0.763683\pi\)
\(192\) 0 0
\(193\) 12.5000 21.6506i 0.899770 1.55845i 0.0719816 0.997406i \(-0.477068\pi\)
0.827788 0.561041i \(-0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.0000 1.06871 0.534353 0.845262i \(-0.320555\pi\)
0.534353 + 0.845262i \(0.320555\pi\)
\(198\) 0 0
\(199\) −20.0000 −1.41776 −0.708881 0.705328i \(-0.750800\pi\)
−0.708881 + 0.705328i \(0.750800\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.00000 5.19615i 0.210559 0.364698i
\(204\) 0 0
\(205\) 9.00000 + 15.5885i 0.628587 + 1.08875i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 6.00000 + 10.3923i 0.415029 + 0.718851i
\(210\) 0 0
\(211\) 7.00000 12.1244i 0.481900 0.834675i −0.517884 0.855451i \(-0.673280\pi\)
0.999784 + 0.0207756i \(0.00661356\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 30.0000 2.04598
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.50000 12.9904i 0.504505 0.873828i
\(222\) 0 0
\(223\) −5.00000 8.66025i −0.334825 0.579934i 0.648626 0.761107i \(-0.275344\pi\)
−0.983451 + 0.181173i \(0.942010\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.00000 + 15.5885i 0.597351 + 1.03464i 0.993210 + 0.116331i \(0.0371134\pi\)
−0.395860 + 0.918311i \(0.629553\pi\)
\(228\) 0 0
\(229\) −2.50000 + 4.33013i −0.165205 + 0.286143i −0.936728 0.350058i \(-0.886162\pi\)
0.771523 + 0.636201i \(0.219495\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −15.0000 −0.982683 −0.491341 0.870967i \(-0.663493\pi\)
−0.491341 + 0.870967i \(0.663493\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(240\) 0 0
\(241\) −5.50000 9.52628i −0.354286 0.613642i 0.632709 0.774389i \(-0.281943\pi\)
−0.986996 + 0.160748i \(0.948609\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 4.50000 + 7.79423i 0.287494 + 0.497955i
\(246\) 0 0
\(247\) 5.00000 8.66025i 0.318142 0.551039i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −30.0000 −1.89358 −0.946792 0.321847i \(-0.895696\pi\)
−0.946792 + 0.321847i \(0.895696\pi\)
\(252\) 0 0
\(253\) −36.0000 −2.26330
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.5000 23.3827i 0.842107 1.45857i −0.0460033 0.998941i \(-0.514648\pi\)
0.888110 0.459631i \(-0.152018\pi\)
\(258\) 0 0
\(259\) 5.00000 + 8.66025i 0.310685 + 0.538122i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.0000 + 20.7846i 0.739952 + 1.28163i 0.952517 + 0.304487i \(0.0984850\pi\)
−0.212565 + 0.977147i \(0.568182\pi\)
\(264\) 0 0
\(265\) 9.00000 15.5885i 0.552866 0.957591i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) 10.0000 0.607457 0.303728 0.952759i \(-0.401768\pi\)
0.303728 + 0.952759i \(0.401768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 + 20.7846i −0.723627 + 1.25336i
\(276\) 0 0
\(277\) 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i \(-0.0695395\pi\)
−0.675810 + 0.737075i \(0.736206\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.50000 + 12.9904i 0.447412 + 0.774941i 0.998217 0.0596933i \(-0.0190123\pi\)
−0.550804 + 0.834634i \(0.685679\pi\)
\(282\) 0 0
\(283\) −2.00000 + 3.46410i −0.118888 + 0.205919i −0.919327 0.393494i \(-0.871266\pi\)
0.800439 + 0.599414i \(0.204600\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −7.50000 + 12.9904i −0.438155 + 0.758906i −0.997547 0.0699967i \(-0.977701\pi\)
0.559393 + 0.828903i \(0.311034\pi\)
\(294\) 0 0
\(295\) −18.0000 31.1769i −1.04800 1.81519i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.0000 + 25.9808i 0.867472 + 1.50251i
\(300\) 0 0
\(301\) 10.0000 17.3205i 0.576390 0.998337i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 15.0000 0.858898
\(306\) 0 0
\(307\) −20.0000 −1.14146 −0.570730 0.821138i \(-0.693340\pi\)
−0.570730 + 0.821138i \(0.693340\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 + 20.7846i −0.680458 + 1.17859i 0.294384 + 0.955687i \(0.404886\pi\)
−0.974841 + 0.222900i \(0.928448\pi\)
\(312\) 0 0
\(313\) 12.5000 + 21.6506i 0.706542 + 1.22377i 0.966132 + 0.258047i \(0.0830791\pi\)
−0.259590 + 0.965719i \(0.583588\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.50000 12.9904i −0.421242 0.729612i 0.574819 0.818280i \(-0.305072\pi\)
−0.996061 + 0.0886679i \(0.971739\pi\)
\(318\) 0 0
\(319\) −9.00000 + 15.5885i −0.503903 + 0.872786i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.00000 0.333849
\(324\) 0 0
\(325\) 20.0000 1.10940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.0000 + 22.5167i 0.714545 + 1.23763i 0.963135 + 0.269019i \(0.0866994\pi\)
−0.248590 + 0.968609i \(0.579967\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.00000 + 5.19615i 0.163908 + 0.283896i
\(336\) 0 0
\(337\) 5.00000 8.66025i 0.272367 0.471754i −0.697100 0.716974i \(-0.745527\pi\)
0.969468 + 0.245220i \(0.0788601\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −9.00000 + 15.5885i −0.483145 + 0.836832i −0.999813 0.0193540i \(-0.993839\pi\)
0.516667 + 0.856186i \(0.327172\pi\)
\(348\) 0 0
\(349\) −1.00000 1.73205i −0.0535288 0.0927146i 0.838019 0.545640i \(-0.183714\pi\)
−0.891548 + 0.452926i \(0.850380\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.0000 25.9808i −0.798369 1.38282i −0.920677 0.390324i \(-0.872363\pi\)
0.122308 0.992492i \(-0.460970\pi\)
\(354\) 0 0
\(355\) 9.00000 15.5885i 0.477670 0.827349i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.50000 2.59808i 0.0785136 0.135990i
\(366\) 0 0
\(367\) 16.0000 + 27.7128i 0.835193 + 1.44660i 0.893873 + 0.448320i \(0.147978\pi\)
−0.0586798 + 0.998277i \(0.518689\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.00000 10.3923i −0.311504 0.539542i
\(372\) 0 0
\(373\) −7.00000 + 12.1244i −0.362446 + 0.627775i −0.988363 0.152115i \(-0.951392\pi\)
0.625917 + 0.779890i \(0.284725\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.0000 0.772539
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.0000 25.9808i 0.766464 1.32755i −0.173005 0.984921i \(-0.555348\pi\)
0.939469 0.342634i \(-0.111319\pi\)
\(384\) 0 0
\(385\) 18.0000 + 31.1769i 0.917365 + 1.58892i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 15.0000 + 25.9808i 0.760530 + 1.31728i 0.942578 + 0.333987i \(0.108394\pi\)
−0.182047 + 0.983290i \(0.558272\pi\)
\(390\) 0 0
\(391\) −9.00000 + 15.5885i −0.455150 + 0.788342i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) −7.00000 −0.351320 −0.175660 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.50000 12.9904i 0.374532 0.648709i −0.615725 0.787961i \(-0.711137\pi\)
0.990257 + 0.139253i \(0.0444700\pi\)
\(402\) 0 0
\(403\) −10.0000 17.3205i −0.498135 0.862796i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.0000 25.9808i −0.743522 1.28782i
\(408\) 0 0
\(409\) −17.5000 + 30.3109i −0.865319 + 1.49878i 0.00141047 + 0.999999i \(0.499551\pi\)
−0.866730 + 0.498778i \(0.833782\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −24.0000 −1.18096
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(420\) 0 0
\(421\) −14.5000 25.1147i −0.706687 1.22402i −0.966079 0.258245i \(-0.916856\pi\)
0.259393 0.965772i \(-0.416478\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00000 + 10.3923i 0.291043 + 0.504101i
\(426\) 0 0
\(427\) 5.00000 8.66025i 0.241967 0.419099i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.00000 + 10.3923i −0.287019 + 0.497131i
\(438\) 0 0
\(439\) 4.00000 + 6.92820i 0.190910 + 0.330665i 0.945552 0.325471i \(-0.105523\pi\)
−0.754642 + 0.656136i \(0.772190\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(444\) 0 0
\(445\) 4.50000 7.79423i 0.213320 0.369482i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −36.0000 −1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.0000 25.9808i 0.703211 1.21800i
\(456\) 0 0
\(457\) −17.5000 30.3109i −0.818615 1.41788i −0.906702 0.421771i \(-0.861409\pi\)
0.0880870 0.996113i \(-0.471925\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.00000 + 5.19615i 0.139724 + 0.242009i 0.927392 0.374091i \(-0.122045\pi\)
−0.787668 + 0.616100i \(0.788712\pi\)
\(462\) 0 0
\(463\) −2.00000 + 3.46410i −0.0929479 + 0.160990i −0.908750 0.417340i \(-0.862962\pi\)
0.815802 + 0.578331i \(0.196296\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −30.0000 + 51.9615i −1.37940 + 2.38919i
\(474\) 0 0
\(475\) 4.00000 + 6.92820i 0.183533 + 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 9.00000 + 15.5885i 0.411220 + 0.712255i 0.995023 0.0996406i \(-0.0317693\pi\)
−0.583803 + 0.811895i \(0.698436\pi\)
\(480\) 0 0
\(481\) −12.5000 + 21.6506i −0.569951 + 0.987184i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −30.0000 −1.36223
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.00000 5.19615i 0.135388 0.234499i −0.790358 0.612646i \(-0.790105\pi\)
0.925746 + 0.378147i \(0.123439\pi\)
\(492\) 0 0
\(493\) 4.50000 + 7.79423i 0.202670 + 0.351034i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 10.3923i −0.269137 0.466159i
\(498\) 0 0
\(499\) −11.0000 + 19.0526i −0.492428 + 0.852910i −0.999962 0.00872186i \(-0.997224\pi\)
0.507534 + 0.861632i \(0.330557\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.00000 5.19615i 0.132973 0.230315i −0.791849 0.610718i \(-0.790881\pi\)
0.924821 + 0.380402i \(0.124214\pi\)
\(510\) 0 0
\(511\) −1.00000 1.73205i −0.0442374 0.0766214i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −24.0000 41.5692i −1.05757 1.83176i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) 0 0
\(523\) 10.0000 0.437269 0.218635 0.975807i \(-0.429840\pi\)
0.218635 + 0.975807i \(0.429840\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.00000 10.3923i 0.261364 0.452696i
\(528\) 0 0
\(529\) −6.50000 11.2583i −0.282609 0.489493i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 15.0000 + 25.9808i 0.649722 + 1.12535i
\(534\) 0 0
\(535\) 18.0000 31.1769i 0.778208 1.34790i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −18.0000 −0.775315
\(540\) 0 0
\(541\) 41.0000 1.76273 0.881364 0.472438i \(-0.156626\pi\)
0.881364 + 0.472438i \(0.156626\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.5000 18.1865i 0.449771 0.779026i
\(546\) 0 0
\(547\) −20.0000 34.6410i −0.855138 1.48114i −0.876517 0.481371i \(-0.840139\pi\)
0.0213785 0.999771i \(-0.493195\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.00000 + 5.19615i 0.127804 + 0.221364i
\(552\) 0 0
\(553\) 10.0000 17.3205i 0.425243 0.736543i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.0000 0.635570 0.317785 0.948163i \(-0.397061\pi\)
0.317785 + 0.948163i \(0.397061\pi\)
\(558\) 0 0
\(559\) 50.0000 2.11477
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(564\) 0 0
\(565\) −13.5000 23.3827i −0.567949 0.983717i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.50000 + 2.59808i 0.0628833 + 0.108917i 0.895753 0.444552i \(-0.146637\pi\)
−0.832870 + 0.553469i \(0.813304\pi\)
\(570\) 0 0
\(571\) 22.0000 38.1051i 0.920671 1.59465i 0.122292 0.992494i \(-0.460975\pi\)
0.798379 0.602155i \(-0.205691\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −24.0000 −1.00087
\(576\) 0 0
\(577\) −37.0000 −1.54033 −0.770165 0.637845i \(-0.779826\pi\)
−0.770165 + 0.637845i \(0.779826\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 18.0000 + 31.1769i 0.745484 + 1.29122i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.0000 25.9808i −0.619116 1.07234i −0.989647 0.143521i \(-0.954158\pi\)
0.370531 0.928820i \(-0.379176\pi\)
\(588\) 0 0
\(589\) 4.00000 6.92820i 0.164817 0.285472i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −39.0000 −1.60154 −0.800769 0.598973i \(-0.795576\pi\)
−0.800769 + 0.598973i \(0.795576\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.00000 + 10.3923i −0.245153 + 0.424618i −0.962175 0.272433i \(-0.912172\pi\)
0.717021 + 0.697051i \(0.245505\pi\)
\(600\) 0 0
\(601\) 0.500000 + 0.866025i 0.0203954 + 0.0353259i 0.876043 0.482233i \(-0.160174\pi\)
−0.855648 + 0.517559i \(0.826841\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −37.5000 64.9519i −1.52459 2.64067i
\(606\) 0 0
\(607\) −5.00000 + 8.66025i −0.202944 + 0.351509i −0.949476 0.313841i \(-0.898384\pi\)
0.746532 + 0.665350i \(0.231718\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.50000 12.9904i 0.301939 0.522973i −0.674636 0.738150i \(-0.735700\pi\)
0.976575 + 0.215177i \(0.0690329\pi\)
\(618\) 0 0
\(619\) −20.0000 34.6410i −0.803868 1.39234i −0.917053 0.398766i \(-0.869439\pi\)
0.113185 0.993574i \(-0.463895\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.00000 5.19615i −0.120192 0.208179i
\(624\) 0 0
\(625\) 14.5000 25.1147i 0.580000 1.00459i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −15.0000 −0.598089
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.00000 5.19615i 0.119051 0.206203i
\(636\) 0 0
\(637\) 7.50000 + 12.9904i 0.297161 + 0.514698i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.5000 18.1865i −0.414725 0.718325i 0.580674 0.814136i \(-0.302789\pi\)
−0.995400 + 0.0958109i \(0.969456\pi\)
\(642\) 0 0
\(643\) −20.0000 + 34.6410i −0.788723 + 1.36611i 0.138027 + 0.990429i \(0.455924\pi\)
−0.926750 + 0.375680i \(0.877409\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.0000 −0.707653 −0.353827 0.935311i \(-0.615120\pi\)
−0.353827 + 0.935311i \(0.615120\pi\)
\(648\) 0 0
\(649\) 72.0000 2.82625
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 15.0000 25.9808i 0.586995 1.01671i −0.407628 0.913148i \(-0.633644\pi\)
0.994623 0.103558i \(-0.0330227\pi\)
\(654\) 0 0
\(655\) −9.00000 15.5885i −0.351659 0.609091i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.00000 + 5.19615i 0.116863 + 0.202413i 0.918523 0.395367i \(-0.129383\pi\)
−0.801660 + 0.597781i \(0.796049\pi\)
\(660\) 0 0
\(661\) −14.5000 + 25.1147i −0.563985 + 0.976850i 0.433159 + 0.901318i \(0.357399\pi\)
−0.997143 + 0.0755324i \(0.975934\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −18.0000 −0.696963
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −15.0000 + 25.9808i −0.579069 + 1.00298i
\(672\) 0 0
\(673\) 0.500000 + 0.866025i 0.0192736 + 0.0333828i 0.875501 0.483216i \(-0.160531\pi\)
−0.856228 + 0.516599i \(0.827198\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.0000 36.3731i −0.807096 1.39793i −0.914867 0.403755i \(-0.867705\pi\)
0.107772 0.994176i \(-0.465628\pi\)
\(678\) 0 0
\(679\) −10.0000 + 17.3205i −0.383765 + 0.664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −45.0000 −1.71936
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.0000 25.9808i 0.571454 0.989788i
\(690\) 0 0
\(691\) −5.00000 8.66025i −0.190209 0.329452i 0.755110 0.655598i \(-0.227583\pi\)
−0.945319 + 0.326146i \(0.894250\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 + 20.7846i 0.455186 + 0.788405i
\(696\) 0 0
\(697\) −9.00000 + 15.5885i −0.340899 + 0.590455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 39.0000 1.47301 0.736505 0.676432i \(-0.236475\pi\)
0.736505 + 0.676432i \(0.236475\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 6.00000 10.3923i 0.225653 0.390843i
\(708\) 0 0
\(709\) 9.50000 + 16.4545i 0.356780 + 0.617961i 0.987421 0.158114i \(-0.0505412\pi\)
−0.630641 + 0.776075i \(0.717208\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 12.0000 + 20.7846i 0.449404 + 0.778390i
\(714\) 0 0
\(715\) −45.0000 + 77.9423i −1.68290 + 2.91488i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −18.0000 −0.671287 −0.335643 0.941989i \(-0.608954\pi\)
−0.335643 + 0.941989i \(0.608954\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.00000 + 10.3923i −0.222834 + 0.385961i
\(726\) 0 0
\(727\) −5.00000 8.66025i −0.185440 0.321191i 0.758285 0.651923i \(-0.226038\pi\)
−0.943725 + 0.330732i \(0.892704\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 15.0000 + 25.9808i 0.554795 + 0.960933i
\(732\) 0 0
\(733\) −7.00000 + 12.1244i −0.258551 + 0.447823i −0.965854 0.259087i \(-0.916578\pi\)
0.707303 + 0.706910i \(0.249912\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −18.0000 + 31.1769i −0.660356 + 1.14377i 0.320166 + 0.947361i \(0.396261\pi\)
−0.980522 + 0.196409i \(0.937072\pi\)
\(744\) 0 0
\(745\) −22.5000 38.9711i −0.824336 1.42779i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −12.0000 20.7846i −0.438470 0.759453i
\(750\) 0 0
\(751\) −5.00000 + 8.66025i −0.182453 + 0.316017i −0.942715 0.333599i \(-0.891737\pi\)
0.760263 + 0.649616i \(0.225070\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −4.50000 + 7.79423i −0.163125 + 0.282541i −0.935988 0.352032i \(-0.885491\pi\)
0.772863 + 0.634573i \(0.218824\pi\)
\(762\) 0 0
\(763\) −7.00000 12.1244i −0.253417 0.438931i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −30.0000 51.9615i −1.08324 1.87622i
\(768\) 0 0
\(769\) −11.5000 + 19.9186i −0.414701 + 0.718283i −0.995397 0.0958377i \(-0.969447\pi\)
0.580696 + 0.814120i \(0.302780\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −45.0000 −1.61854 −0.809269 0.587439i \(-0.800136\pi\)
−0.809269 + 0.587439i \(0.800136\pi\)
\(774\) 0 0
\(775\) 16.0000 0.574737
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −6.00000 + 10.3923i −0.214972 + 0.372343i
\(780\) 0 0
\(781\) 18.0000 + 31.1769i 0.644091 + 1.11560i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −7.50000 12.9904i −0.267686 0.463647i
\(786\) 0 0
\(787\) 19.0000 32.9090i 0.677277 1.17308i −0.298521 0.954403i \(-0.596493\pi\)
0.975798 0.218675i \(-0.0701734\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) 25.0000 0.887776
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.5000 38.9711i 0.796991 1.38043i −0.124576 0.992210i \(-0.539757\pi\)
0.921567 0.388219i \(-0.126909\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.00000 + 5.19615i 0.105868 + 0.183368i
\(804\) 0 0
\(805\) −18.0000 + 31.1769i −0.634417 + 1.09884i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −3.00000 −0.105474 −0.0527372 0.998608i \(-0.516795\pi\)
−0.0527372 + 0.998608i \(0.516795\pi\)
\(810\) 0 0
\(811\) 16.0000 0.561836 0.280918 0.959732i \(-0.409361\pi\)
0.280918 + 0.959732i \(0.409361\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −6.00000 + 10.3923i −0.210171 + 0.364027i
\(816\) 0 0
\(817\) 10.0000 + 17.3205i 0.349856 + 0.605968i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 10.5000 + 18.1865i 0.366453 + 0.634714i 0.989008 0.147861i \(-0.0472389\pi\)
−0.622556 + 0.782576i \(0.713906\pi\)
\(822\) 0 0
\(823\) −2.00000 + 3.46410i −0.0697156 + 0.120751i −0.898776 0.438408i \(-0.855543\pi\)
0.829060 + 0.559159i \(0.188876\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 30.0000 1.04320 0.521601 0.853189i \(-0.325335\pi\)
0.521601 + 0.853189i \(0.325335\pi\)
\(828\) 0 0
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.50000 + 7.79423i −0.155916 + 0.270054i
\(834\) 0 0
\(835\) 27.0000 + 46.7654i 0.934374 + 1.61838i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.00000 15.5885i −0.310715 0.538173i 0.667803 0.744338i \(-0.267235\pi\)
−0.978517 + 0.206165i \(0.933902\pi\)
\(840\) 0 0
\(841\) 10.0000 17.3205i 0.344828 0.597259i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 36.0000 1.23844
\(846\) 0 0
\(847\) −50.0000 −1.71802
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 15.0000 25.9808i 0.514193 0.890609i
\(852\) 0 0
\(853\) −7.00000 12.1244i −0.239675 0.415130i 0.720946 0.692992i \(-0.243708\pi\)
−0.960621 + 0.277862i \(0.910374\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −22.5000 38.9711i −0.768585 1.33123i −0.938330 0.345741i \(-0.887628\pi\)
0.169745 0.985488i \(-0.445706\pi\)
\(858\) 0 0
\(859\) 4.00000 6.92820i 0.136478 0.236387i −0.789683 0.613515i \(-0.789755\pi\)
0.926161 + 0.377128i \(0.123088\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) −27.0000 −0.918028
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −30.0000 + 51.9615i −1.01768 + 1.76267i
\(870\) 0 0
\(871\) 5.00000 + 8.66025i 0.169419 + 0.293442i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.00000 5.19615i −0.101419 0.175662i
\(876\) 0 0
\(877\) 27.5000 47.6314i 0.928609 1.60840i 0.142957 0.989729i \(-0.454339\pi\)
0.785652 0.618669i \(-0.212328\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −6.00000 −0.202145 −0.101073 0.994879i \(-0.532227\pi\)
−0.101073 + 0.994879i \(0.532227\pi\)
\(882\) 0 0
\(883\) 16.0000 0.538443 0.269221 0.963078i \(-0.413234\pi\)
0.269221 + 0.963078i \(0.413234\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 15.0000 25.9808i 0.503651 0.872349i −0.496340 0.868128i \(-0.665323\pi\)
0.999991 0.00422062i \(-0.00134347\pi\)
\(888\) 0 0
\(889\) −2.00000 3.46410i −0.0670778 0.116182i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 18.0000 31.1769i 0.601674 1.04213i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 18.0000 0.599667
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −21.0000 + 36.3731i −0.698064 + 1.20908i
\(906\) 0 0
\(907\) 4.00000 + 6.92820i 0.132818 + 0.230047i 0.924762 0.380547i \(-0.124264\pi\)
−0.791944 + 0.610594i \(0.790931\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −27.0000 46.7654i −0.894550 1.54941i −0.834361 0.551219i \(-0.814163\pi\)
−0.0601892 0.998187i \(-0.519170\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −38.0000 −1.25350 −0.626752 0.779219i \(-0.715616\pi\)
−0.626752 + 0.779219i \(0.715616\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 15.0000 25.9808i 0.493731 0.855167i
\(924\) 0 0
\(925\) −10.0000 17.3205i −0.328798 0.569495i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.5000 + 23.3827i 0.442921 + 0.767161i 0.997905 0.0646999i \(-0.0206090\pi\)
−0.554984 + 0.831861i \(0.687276\pi\)
\(930\) 0 0
\(931\) −3.00000 + 5.19615i −0.0983210 + 0.170297i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −54.0000 −1.76599
\(936\) 0 0
\(937\) 23.0000 0.751377 0.375689 0.926746i \(-0.377406\pi\)
0.375689 + 0.926746i \(0.377406\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −7.50000 + 12.9904i −0.244493 + 0.423474i −0.961989 0.273088i \(-0.911955\pi\)
0.717496 + 0.696563i \(0.245288\pi\)
\(942\) 0 0
\(943\) −18.0000 31.1769i −0.586161 1.01526i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9.00000 + 15.5885i 0.292461 + 0.506557i 0.974391 0.224860i \(-0.0721926\pi\)
−0.681930 + 0.731417i \(0.738859\pi\)
\(948\) 0 0
\(949\) 2.50000 4.33013i 0.0811534 0.140562i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 45.0000 1.45769 0.728846 0.684677i \(-0.240057\pi\)
0.728846 + 0.684677i \(0.240057\pi\)
\(954\) 0 0
\(955\) −18.0000 −0.582466
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15.0000 + 25.9808i −0.484375 + 0.838963i
\(960\) 0 0
\(961\) 7.50000 + 12.9904i 0.241935 + 0.419045i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37.5000 + 64.9519i 1.20717 + 2.09088i
\(966\) 0 0
\(967\) 19.0000 32.9090i 0.610999 1.05828i −0.380074 0.924956i \(-0.624101\pi\)
0.991072 0.133325i \(-0.0425653\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 54.0000 1.73294 0.866471 0.499227i \(-0.166383\pi\)
0.866471 + 0.499227i \(0.166383\pi\)
\(972\) 0 0
\(973\) 16.0000 0.512936
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.0000 25.9808i 0.479893 0.831198i −0.519841 0.854263i \(-0.674009\pi\)
0.999734 + 0.0230645i \(0.00734232\pi\)
\(978\) 0 0
\(979\) 9.00000 + 15.5885i 0.287641 + 0.498209i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 12.0000 + 20.7846i 0.382741 + 0.662926i 0.991453 0.130465i \(-0.0416470\pi\)
−0.608712 + 0.793391i \(0.708314\pi\)
\(984\) 0 0
\(985\) −22.5000 + 38.9711i −0.716910 + 1.24172i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −60.0000 −1.90789
\(990\) 0 0
\(991\) −26.0000 −0.825917 −0.412959 0.910750i \(-0.635505\pi\)
−0.412959 + 0.910750i \(0.635505\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 30.0000 51.9615i 0.951064 1.64729i
\(996\) 0 0
\(997\) −26.5000 45.8993i −0.839263 1.45365i −0.890511 0.454961i \(-0.849653\pi\)
0.0512480 0.998686i \(-0.483680\pi\)
\(998\) 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 1296.2.i.d.433.1 2
3.2 odd 2 1296.2.i.p.433.1 2
4.3 odd 2 324.2.e.a.109.1 2
9.2 odd 6 1296.2.i.p.865.1 2
9.4 even 3 1296.2.a.j.1.1 1
9.5 odd 6 1296.2.a.a.1.1 1
9.7 even 3 inner 1296.2.i.d.865.1 2
12.11 even 2 324.2.e.d.109.1 2
36.7 odd 6 324.2.e.a.217.1 2
36.11 even 6 324.2.e.d.217.1 2
36.23 even 6 324.2.a.b.1.1 1
36.31 odd 6 324.2.a.d.1.1 yes 1
72.5 odd 6 5184.2.a.z.1.1 1
72.13 even 6 5184.2.a.d.1.1 1
72.59 even 6 5184.2.a.bc.1.1 1
72.67 odd 6 5184.2.a.g.1.1 1
180.23 odd 12 8100.2.d.j.649.1 2
180.59 even 6 8100.2.a.f.1.1 1
180.67 even 12 8100.2.d.a.649.2 2
180.103 even 12 8100.2.d.a.649.1 2
180.139 odd 6 8100.2.a.a.1.1 1
180.167 odd 12 8100.2.d.j.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.a.b.1.1 1 36.23 even 6
324.2.a.d.1.1 yes 1 36.31 odd 6
324.2.e.a.109.1 2 4.3 odd 2
324.2.e.a.217.1 2 36.7 odd 6
324.2.e.d.109.1 2 12.11 even 2
324.2.e.d.217.1 2 36.11 even 6
1296.2.a.a.1.1 1 9.5 odd 6
1296.2.a.j.1.1 1 9.4 even 3
1296.2.i.d.433.1 2 1.1 even 1 trivial
1296.2.i.d.865.1 2 9.7 even 3 inner
1296.2.i.p.433.1 2 3.2 odd 2
1296.2.i.p.865.1 2 9.2 odd 6
5184.2.a.d.1.1 1 72.13 even 6
5184.2.a.g.1.1 1 72.67 odd 6
5184.2.a.z.1.1 1 72.5 odd 6
5184.2.a.bc.1.1 1 72.59 even 6
8100.2.a.a.1.1 1 180.139 odd 6
8100.2.a.f.1.1 1 180.59 even 6
8100.2.d.a.649.1 2 180.103 even 12
8100.2.d.a.649.2 2 180.67 even 12
8100.2.d.j.649.1 2 180.23 odd 12
8100.2.d.j.649.2 2 180.167 odd 12