Properties

Label 1296.2.i.g.433.1
Level $1296$
Weight $2$
Character 1296.433
Analytic conductor $10.349$
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] = [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: \(0\)
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 216)
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.g.865.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +O(q^{10})\) \(q+(-0.500000 + 0.866025i) q^{5} +(1.50000 + 2.59808i) q^{7} +(-2.50000 - 4.33013i) q^{11} +(-2.00000 + 3.46410i) q^{13} +8.00000 q^{17} -2.00000 q^{19} +(-1.00000 + 1.73205i) q^{23} +(2.00000 + 3.46410i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-3.50000 + 6.06218i) q^{31} -3.00000 q^{35} -6.00000 q^{37} +(-3.00000 + 5.19615i) q^{41} +(-1.00000 - 1.73205i) q^{43} +(-3.00000 - 5.19615i) q^{47} +(-1.00000 + 1.73205i) q^{49} -5.00000 q^{53} +5.00000 q^{55} +(2.00000 - 3.46410i) q^{59} +(4.00000 + 6.92820i) q^{61} +(-2.00000 - 3.46410i) q^{65} +(-5.00000 + 8.66025i) q^{67} -8.00000 q^{71} +1.00000 q^{73} +(7.50000 - 12.9904i) q^{77} +(8.00000 + 13.8564i) q^{79} +(5.50000 + 9.52628i) q^{83} +(-4.00000 + 6.92820i) q^{85} -6.00000 q^{89} -12.0000 q^{91} +(1.00000 - 1.73205i) q^{95} +(0.500000 + 0.866025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 3 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 3 q^{7} - 5 q^{11} - 4 q^{13} + 16 q^{17} - 4 q^{19} - 2 q^{23} + 4 q^{25} + 6 q^{29} - 7 q^{31} - 6 q^{35} - 12 q^{37} - 6 q^{41} - 2 q^{43} - 6 q^{47} - 2 q^{49} - 10 q^{53} + 10 q^{55} + 4 q^{59} + 8 q^{61} - 4 q^{65} - 10 q^{67} - 16 q^{71} + 2 q^{73} + 15 q^{77} + 16 q^{79} + 11 q^{83} - 8 q^{85} - 12 q^{89} - 24 q^{91} + 2 q^{95} + q^{97}+O(q^{100}) \) 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
\(5\) −0.500000 + 0.866025i −0.223607 + 0.387298i −0.955901 0.293691i \(-0.905116\pi\)
0.732294 + 0.680989i \(0.238450\pi\)
\(6\) 0 0
\(7\) 1.50000 + 2.59808i 0.566947 + 0.981981i 0.996866 + 0.0791130i \(0.0252088\pi\)
−0.429919 + 0.902867i \(0.641458\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.50000 4.33013i −0.753778 1.30558i −0.945979 0.324227i \(-0.894896\pi\)
0.192201 0.981356i \(-0.438437\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000 1.94029 0.970143 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 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
\(23\) −1.00000 + 1.73205i −0.208514 + 0.361158i −0.951247 0.308431i \(-0.900196\pi\)
0.742732 + 0.669588i \(0.233529\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −6.00000 −0.986394 −0.493197 0.869918i \(-0.664172\pi\)
−0.493197 + 0.869918i \(0.664172\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.00000 + 5.19615i −0.468521 + 0.811503i −0.999353 0.0359748i \(-0.988546\pi\)
0.530831 + 0.847477i \(0.321880\pi\)
\(42\) 0 0
\(43\) −1.00000 1.73205i −0.152499 0.264135i 0.779647 0.626219i \(-0.215399\pi\)
−0.932145 + 0.362084i \(0.882065\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.00000 5.19615i −0.437595 0.757937i 0.559908 0.828554i \(-0.310836\pi\)
−0.997503 + 0.0706177i \(0.977503\pi\)
\(48\) 0 0
\(49\) −1.00000 + 1.73205i −0.142857 + 0.247436i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.00000 −0.686803 −0.343401 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.00000 3.46410i 0.260378 0.450988i −0.705965 0.708247i \(-0.749486\pi\)
0.966342 + 0.257260i \(0.0828195\pi\)
\(60\) 0 0
\(61\) 4.00000 + 6.92820i 0.512148 + 0.887066i 0.999901 + 0.0140840i \(0.00448323\pi\)
−0.487753 + 0.872982i \(0.662183\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.00000 3.46410i −0.248069 0.429669i
\(66\) 0 0
\(67\) −5.00000 + 8.66025i −0.610847 + 1.05802i 0.380251 + 0.924883i \(0.375838\pi\)
−0.991098 + 0.133135i \(0.957496\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.50000 12.9904i 0.854704 1.48039i
\(78\) 0 0
\(79\) 8.00000 + 13.8564i 0.900070 + 1.55897i 0.827401 + 0.561611i \(0.189818\pi\)
0.0726692 + 0.997356i \(0.476848\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.50000 + 9.52628i 0.603703 + 1.04565i 0.992255 + 0.124218i \(0.0396422\pi\)
−0.388552 + 0.921427i \(0.627024\pi\)
\(84\) 0 0
\(85\) −4.00000 + 6.92820i −0.433861 + 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00000 1.73205i 0.102598 0.177705i
\(96\) 0 0
\(97\) 0.500000 + 0.866025i 0.0507673 + 0.0879316i 0.890292 0.455389i \(-0.150500\pi\)
−0.839525 + 0.543321i \(0.817167\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.50000 + 7.79423i 0.447767 + 0.775555i 0.998240 0.0592978i \(-0.0188862\pi\)
−0.550474 + 0.834853i \(0.685553\pi\)
\(102\) 0 0
\(103\) 2.00000 3.46410i 0.197066 0.341328i −0.750510 0.660859i \(-0.770192\pi\)
0.947576 + 0.319531i \(0.103525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 9.00000 0.870063 0.435031 0.900415i \(-0.356737\pi\)
0.435031 + 0.900415i \(0.356737\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −3.00000 + 5.19615i −0.282216 + 0.488813i −0.971930 0.235269i \(-0.924403\pi\)
0.689714 + 0.724082i \(0.257736\pi\)
\(114\) 0 0
\(115\) −1.00000 1.73205i −0.0932505 0.161515i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0000 + 20.7846i 1.10004 + 1.90532i
\(120\) 0 0
\(121\) −7.00000 + 12.1244i −0.636364 + 1.10221i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 11.0000 0.976092 0.488046 0.872818i \(-0.337710\pi\)
0.488046 + 0.872818i \(0.337710\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.500000 + 0.866025i −0.0436852 + 0.0756650i −0.887041 0.461690i \(-0.847243\pi\)
0.843356 + 0.537355i \(0.180577\pi\)
\(132\) 0 0
\(133\) −3.00000 5.19615i −0.260133 0.450564i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −9.00000 15.5885i −0.768922 1.33181i −0.938148 0.346235i \(-0.887460\pi\)
0.169226 0.985577i \(-0.445873\pi\)
\(138\) 0 0
\(139\) 6.00000 10.3923i 0.508913 0.881464i −0.491033 0.871141i \(-0.663381\pi\)
0.999947 0.0103230i \(-0.00328598\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 20.0000 1.67248
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.50000 6.06218i 0.286731 0.496633i −0.686296 0.727322i \(-0.740765\pi\)
0.973028 + 0.230689i \(0.0740980\pi\)
\(150\) 0 0
\(151\) 2.50000 + 4.33013i 0.203447 + 0.352381i 0.949637 0.313353i \(-0.101452\pi\)
−0.746190 + 0.665733i \(0.768119\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.50000 6.06218i −0.281127 0.486926i
\(156\) 0 0
\(157\) 10.0000 17.3205i 0.798087 1.38233i −0.122774 0.992435i \(-0.539179\pi\)
0.920860 0.389892i \(-0.127488\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.00000 −0.472866
\(162\) 0 0
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 5.19615i 0.232147 0.402090i −0.726293 0.687386i \(-0.758758\pi\)
0.958440 + 0.285295i \(0.0920916\pi\)
\(168\) 0 0
\(169\) −1.50000 2.59808i −0.115385 0.199852i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.5000 18.1865i −0.798300 1.38270i −0.920722 0.390218i \(-0.872399\pi\)
0.122422 0.992478i \(-0.460934\pi\)
\(174\) 0 0
\(175\) −6.00000 + 10.3923i −0.453557 + 0.785584i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −9.00000 −0.672692 −0.336346 0.941739i \(-0.609191\pi\)
−0.336346 + 0.941739i \(0.609191\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.00000 5.19615i 0.220564 0.382029i
\(186\) 0 0
\(187\) −20.0000 34.6410i −1.46254 2.53320i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.00000 10.3923i −0.434145 0.751961i 0.563081 0.826402i \(-0.309616\pi\)
−0.997225 + 0.0744412i \(0.976283\pi\)
\(192\) 0 0
\(193\) 9.50000 16.4545i 0.683825 1.18442i −0.289980 0.957033i \(-0.593649\pi\)
0.973805 0.227387i \(-0.0730182\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.00000 0.213741 0.106871 0.994273i \(-0.465917\pi\)
0.106871 + 0.994273i \(0.465917\pi\)
\(198\) 0 0
\(199\) 11.0000 0.779769 0.389885 0.920864i \(-0.372515\pi\)
0.389885 + 0.920864i \(0.372515\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.00000 + 15.5885i −0.631676 + 1.09410i
\(204\) 0 0
\(205\) −3.00000 5.19615i −0.209529 0.362915i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.00000 + 8.66025i 0.345857 + 0.599042i
\(210\) 0 0
\(211\) 5.00000 8.66025i 0.344214 0.596196i −0.640996 0.767544i \(-0.721479\pi\)
0.985211 + 0.171347i \(0.0548120\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −21.0000 −1.42557
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −16.0000 + 27.7128i −1.07628 + 1.86417i
\(222\) 0 0
\(223\) 8.00000 + 13.8564i 0.535720 + 0.927894i 0.999128 + 0.0417488i \(0.0132929\pi\)
−0.463409 + 0.886145i \(0.653374\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.00000 + 10.3923i 0.398234 + 0.689761i 0.993508 0.113761i \(-0.0362899\pi\)
−0.595274 + 0.803523i \(0.702957\pi\)
\(228\) 0 0
\(229\) 1.00000 1.73205i 0.0660819 0.114457i −0.831092 0.556136i \(-0.812283\pi\)
0.897173 + 0.441679i \(0.145617\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 22.0000 1.44127 0.720634 0.693316i \(-0.243851\pi\)
0.720634 + 0.693316i \(0.243851\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −11.0000 + 19.0526i −0.711531 + 1.23241i 0.252752 + 0.967531i \(0.418664\pi\)
−0.964282 + 0.264876i \(0.914669\pi\)
\(240\) 0 0
\(241\) −3.00000 5.19615i −0.193247 0.334714i 0.753077 0.657932i \(-0.228569\pi\)
−0.946324 + 0.323218i \(0.895235\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.00000 1.73205i −0.0638877 0.110657i
\(246\) 0 0
\(247\) 4.00000 6.92820i 0.254514 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 16.0000 1.00991 0.504956 0.863145i \(-0.331509\pi\)
0.504956 + 0.863145i \(0.331509\pi\)
\(252\) 0 0
\(253\) 10.0000 0.628695
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 3.46410i 0.124757 0.216085i −0.796881 0.604136i \(-0.793518\pi\)
0.921638 + 0.388051i \(0.126852\pi\)
\(258\) 0 0
\(259\) −9.00000 15.5885i −0.559233 0.968620i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −13.0000 22.5167i −0.801614 1.38844i −0.918553 0.395298i \(-0.870641\pi\)
0.116939 0.993139i \(-0.462692\pi\)
\(264\) 0 0
\(265\) 2.50000 4.33013i 0.153574 0.265998i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.00000 0.121942 0.0609711 0.998140i \(-0.480580\pi\)
0.0609711 + 0.998140i \(0.480580\pi\)
\(270\) 0 0
\(271\) 13.0000 0.789694 0.394847 0.918747i \(-0.370798\pi\)
0.394847 + 0.918747i \(0.370798\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.0000 17.3205i 0.603023 1.04447i
\(276\) 0 0
\(277\) 4.00000 + 6.92820i 0.240337 + 0.416275i 0.960810 0.277207i \(-0.0894088\pi\)
−0.720473 + 0.693482i \(0.756075\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.00000 + 13.8564i 0.477240 + 0.826604i 0.999660 0.0260845i \(-0.00830391\pi\)
−0.522420 + 0.852688i \(0.674971\pi\)
\(282\) 0 0
\(283\) 7.00000 12.1244i 0.416107 0.720718i −0.579437 0.815017i \(-0.696728\pi\)
0.995544 + 0.0942988i \(0.0300609\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −18.0000 −1.06251
\(288\) 0 0
\(289\) 47.0000 2.76471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3.00000 + 5.19615i −0.175262 + 0.303562i −0.940252 0.340480i \(-0.889411\pi\)
0.764990 + 0.644042i \(0.222744\pi\)
\(294\) 0 0
\(295\) 2.00000 + 3.46410i 0.116445 + 0.201688i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.00000 6.92820i −0.231326 0.400668i
\(300\) 0 0
\(301\) 3.00000 5.19615i 0.172917 0.299501i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) −24.0000 −1.36975 −0.684876 0.728659i \(-0.740144\pi\)
−0.684876 + 0.728659i \(0.740144\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.0000 25.9808i 0.850572 1.47323i −0.0301210 0.999546i \(-0.509589\pi\)
0.880693 0.473688i \(-0.157077\pi\)
\(312\) 0 0
\(313\) −10.5000 18.1865i −0.593495 1.02796i −0.993757 0.111563i \(-0.964414\pi\)
0.400262 0.916401i \(-0.368919\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.50000 + 7.79423i 0.252745 + 0.437767i 0.964281 0.264883i \(-0.0853332\pi\)
−0.711535 + 0.702650i \(0.752000\pi\)
\(318\) 0 0
\(319\) 15.0000 25.9808i 0.839839 1.45464i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 9.00000 15.5885i 0.496186 0.859419i
\(330\) 0 0
\(331\) 7.00000 + 12.1244i 0.384755 + 0.666415i 0.991735 0.128302i \(-0.0409527\pi\)
−0.606980 + 0.794717i \(0.707619\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −5.00000 8.66025i −0.273179 0.473160i
\(336\) 0 0
\(337\) 3.00000 5.19615i 0.163420 0.283052i −0.772673 0.634804i \(-0.781081\pi\)
0.936093 + 0.351752i \(0.114414\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 35.0000 1.89536
\(342\) 0 0
\(343\) 15.0000 0.809924
\(344\) 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
\(349\) −15.0000 25.9808i −0.802932 1.39072i −0.917679 0.397324i \(-0.869939\pi\)
0.114747 0.993395i \(-0.463394\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.0000 + 19.0526i 0.585471 + 1.01407i 0.994817 + 0.101686i \(0.0324237\pi\)
−0.409346 + 0.912379i \(0.634243\pi\)
\(354\) 0 0
\(355\) 4.00000 6.92820i 0.212298 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −30.0000 −1.58334 −0.791670 0.610949i \(-0.790788\pi\)
−0.791670 + 0.610949i \(0.790788\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −0.500000 + 0.866025i −0.0261712 + 0.0453298i
\(366\) 0 0
\(367\) −5.50000 9.52628i −0.287098 0.497268i 0.686018 0.727585i \(-0.259357\pi\)
−0.973116 + 0.230317i \(0.926024\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −7.50000 12.9904i −0.389381 0.674427i
\(372\) 0 0
\(373\) −4.00000 + 6.92820i −0.207112 + 0.358729i −0.950804 0.309794i \(-0.899740\pi\)
0.743691 + 0.668523i \(0.233073\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −24.0000 −1.23606
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(384\) 0 0
\(385\) 7.50000 + 12.9904i 0.382235 + 0.662051i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.500000 0.866025i −0.0253510 0.0439092i 0.853072 0.521794i \(-0.174737\pi\)
−0.878423 + 0.477885i \(0.841404\pi\)
\(390\) 0 0
\(391\) −8.00000 + 13.8564i −0.404577 + 0.700749i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −16.0000 −0.805047
\(396\) 0 0
\(397\) −4.00000 −0.200754 −0.100377 0.994949i \(-0.532005\pi\)
−0.100377 + 0.994949i \(0.532005\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 10.3923i 0.299626 0.518967i −0.676425 0.736512i \(-0.736472\pi\)
0.976050 + 0.217545i \(0.0698049\pi\)
\(402\) 0 0
\(403\) −14.0000 24.2487i −0.697390 1.20791i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.0000 + 25.9808i 0.743522 + 1.28782i
\(408\) 0 0
\(409\) 4.50000 7.79423i 0.222511 0.385400i −0.733059 0.680165i \(-0.761908\pi\)
0.955570 + 0.294765i \(0.0952414\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 12.0000 0.590481
\(414\) 0 0
\(415\) −11.0000 −0.539969
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.0000 31.1769i 0.879358 1.52309i 0.0273103 0.999627i \(-0.491306\pi\)
0.852047 0.523465i \(-0.175361\pi\)
\(420\) 0 0
\(421\) −4.00000 6.92820i −0.194948 0.337660i 0.751935 0.659237i \(-0.229121\pi\)
−0.946883 + 0.321577i \(0.895787\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.0000 + 27.7128i 0.776114 + 1.34427i
\(426\) 0 0
\(427\) −12.0000 + 20.7846i −0.580721 + 1.00584i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 34.0000 1.63772 0.818861 0.573992i \(-0.194606\pi\)
0.818861 + 0.573992i \(0.194606\pi\)
\(432\) 0 0
\(433\) 13.0000 0.624740 0.312370 0.949960i \(-0.398877\pi\)
0.312370 + 0.949960i \(0.398877\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.00000 3.46410i 0.0956730 0.165710i
\(438\) 0 0
\(439\) 4.50000 + 7.79423i 0.214773 + 0.371998i 0.953202 0.302333i \(-0.0977654\pi\)
−0.738429 + 0.674331i \(0.764432\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.00000 3.46410i −0.0950229 0.164584i 0.814595 0.580030i \(-0.196959\pi\)
−0.909618 + 0.415445i \(0.863626\pi\)
\(444\) 0 0
\(445\) 3.00000 5.19615i 0.142214 0.246321i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −14.0000 −0.660701 −0.330350 0.943858i \(-0.607167\pi\)
−0.330350 + 0.943858i \(0.607167\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.00000 10.3923i 0.281284 0.487199i
\(456\) 0 0
\(457\) 0.500000 + 0.866025i 0.0233890 + 0.0405110i 0.877483 0.479608i \(-0.159221\pi\)
−0.854094 + 0.520119i \(0.825888\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.500000 0.866025i −0.0232873 0.0403348i 0.854147 0.520032i \(-0.174080\pi\)
−0.877434 + 0.479697i \(0.840747\pi\)
\(462\) 0 0
\(463\) −0.500000 + 0.866025i −0.0232370 + 0.0402476i −0.877410 0.479741i \(-0.840731\pi\)
0.854173 + 0.519989i \(0.174064\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.00000 −0.138823 −0.0694117 0.997588i \(-0.522112\pi\)
−0.0694117 + 0.997588i \(0.522112\pi\)
\(468\) 0 0
\(469\) −30.0000 −1.38527
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −5.00000 + 8.66025i −0.229900 + 0.398199i
\(474\) 0 0
\(475\) −4.00000 6.92820i −0.183533 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 13.0000 + 22.5167i 0.593985 + 1.02881i 0.993689 + 0.112168i \(0.0357796\pi\)
−0.399704 + 0.916644i \(0.630887\pi\)
\(480\) 0 0
\(481\) 12.0000 20.7846i 0.547153 0.947697i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.00000 −0.0454077
\(486\) 0 0
\(487\) 8.00000 0.362515 0.181257 0.983436i \(-0.441983\pi\)
0.181257 + 0.983436i \(0.441983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.50000 7.79423i 0.203082 0.351749i −0.746438 0.665455i \(-0.768237\pi\)
0.949520 + 0.313707i \(0.101571\pi\)
\(492\) 0 0
\(493\) 24.0000 + 41.5692i 1.08091 + 1.87218i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.0000 20.7846i −0.538274 0.932317i
\(498\) 0 0
\(499\) 3.00000 5.19615i 0.134298 0.232612i −0.791031 0.611776i \(-0.790455\pi\)
0.925329 + 0.379165i \(0.123789\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 0 0
\(505\) −9.00000 −0.400495
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 10.5000 18.1865i 0.465404 0.806104i −0.533815 0.845601i \(-0.679242\pi\)
0.999220 + 0.0394971i \(0.0125756\pi\)
\(510\) 0 0
\(511\) 1.50000 + 2.59808i 0.0663561 + 0.114932i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.00000 + 3.46410i 0.0881305 + 0.152647i
\(516\) 0 0
\(517\) −15.0000 + 25.9808i −0.659699 + 1.14263i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −28.0000 −1.22670 −0.613351 0.789810i \(-0.710179\pi\)
−0.613351 + 0.789810i \(0.710179\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −28.0000 + 48.4974i −1.21970 + 2.11258i
\(528\) 0 0
\(529\) 9.50000 + 16.4545i 0.413043 + 0.715412i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −12.0000 20.7846i −0.519778 0.900281i
\(534\) 0 0
\(535\) −4.50000 + 7.79423i −0.194552 + 0.336974i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0000 0.430730
\(540\) 0 0
\(541\) 36.0000 1.54776 0.773880 0.633332i \(-0.218313\pi\)
0.773880 + 0.633332i \(0.218313\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.00000 + 8.66025i −0.214176 + 0.370965i
\(546\) 0 0
\(547\) 16.0000 + 27.7128i 0.684111 + 1.18491i 0.973715 + 0.227768i \(0.0731428\pi\)
−0.289605 + 0.957146i \(0.593524\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 10.3923i −0.255609 0.442727i
\(552\) 0 0
\(553\) −24.0000 + 41.5692i −1.02058 + 1.76770i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.00000 0.0423714 0.0211857 0.999776i \(-0.493256\pi\)
0.0211857 + 0.999776i \(0.493256\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −5.50000 + 9.52628i −0.231797 + 0.401485i −0.958337 0.285640i \(-0.907794\pi\)
0.726540 + 0.687124i \(0.241127\pi\)
\(564\) 0 0
\(565\) −3.00000 5.19615i −0.126211 0.218604i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −14.0000 24.2487i −0.586911 1.01656i −0.994634 0.103454i \(-0.967011\pi\)
0.407724 0.913105i \(-0.366323\pi\)
\(570\) 0 0
\(571\) −6.00000 + 10.3923i −0.251092 + 0.434904i −0.963827 0.266529i \(-0.914123\pi\)
0.712735 + 0.701434i \(0.247456\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −10.0000 −0.416305 −0.208153 0.978096i \(-0.566745\pi\)
−0.208153 + 0.978096i \(0.566745\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.5000 + 28.5788i −0.684535 + 1.18565i
\(582\) 0 0
\(583\) 12.5000 + 21.6506i 0.517697 + 0.896678i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.50000 + 16.4545i 0.392107 + 0.679149i 0.992727 0.120385i \(-0.0384130\pi\)
−0.600620 + 0.799534i \(0.705080\pi\)
\(588\) 0 0
\(589\) 7.00000 12.1244i 0.288430 0.499575i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 22.0000 0.903432 0.451716 0.892162i \(-0.350812\pi\)
0.451716 + 0.892162i \(0.350812\pi\)
\(594\) 0 0
\(595\) −24.0000 −0.983904
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.00000 + 5.19615i −0.122577 + 0.212309i −0.920783 0.390075i \(-0.872449\pi\)
0.798206 + 0.602384i \(0.205782\pi\)
\(600\) 0 0
\(601\) 2.50000 + 4.33013i 0.101977 + 0.176630i 0.912499 0.409079i \(-0.134150\pi\)
−0.810522 + 0.585708i \(0.800816\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.00000 12.1244i −0.284590 0.492925i
\(606\) 0 0
\(607\) −20.0000 + 34.6410i −0.811775 + 1.40604i 0.0998457 + 0.995003i \(0.468165\pi\)
−0.911621 + 0.411033i \(0.865168\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.00000 5.19615i 0.120775 0.209189i −0.799298 0.600935i \(-0.794795\pi\)
0.920074 + 0.391745i \(0.128129\pi\)
\(618\) 0 0
\(619\) 14.0000 + 24.2487i 0.562708 + 0.974638i 0.997259 + 0.0739910i \(0.0235736\pi\)
−0.434551 + 0.900647i \(0.643093\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.00000 15.5885i −0.360577 0.624538i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −48.0000 −1.91389
\(630\) 0 0
\(631\) −43.0000 −1.71180 −0.855901 0.517139i \(-0.826997\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.50000 + 9.52628i −0.218261 + 0.378039i
\(636\) 0 0
\(637\) −4.00000 6.92820i −0.158486 0.274505i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.0000 19.0526i −0.434474 0.752531i 0.562779 0.826608i \(-0.309732\pi\)
−0.997253 + 0.0740768i \(0.976399\pi\)
\(642\) 0 0
\(643\) 6.00000 10.3923i 0.236617 0.409832i −0.723124 0.690718i \(-0.757295\pi\)
0.959741 + 0.280885i \(0.0906280\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.0000 0.629025 0.314512 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) −20.0000 −0.785069
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.50000 2.59808i 0.0586995 0.101671i −0.835182 0.549973i \(-0.814638\pi\)
0.893882 + 0.448303i \(0.147971\pi\)
\(654\) 0 0
\(655\) −0.500000 0.866025i −0.0195366 0.0338384i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.5000 + 18.1865i 0.409022 + 0.708447i 0.994780 0.102039i \(-0.0325366\pi\)
−0.585758 + 0.810486i \(0.699203\pi\)
\(660\) 0 0
\(661\) −19.0000 + 32.9090i −0.739014 + 1.28001i 0.213925 + 0.976850i \(0.431375\pi\)
−0.952940 + 0.303160i \(0.901958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.00000 0.232670
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 20.0000 34.6410i 0.772091 1.33730i
\(672\) 0 0
\(673\) −6.50000 11.2583i −0.250557 0.433977i 0.713123 0.701039i \(-0.247280\pi\)
−0.963679 + 0.267063i \(0.913947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −11.0000 19.0526i −0.422764 0.732249i 0.573444 0.819244i \(-0.305607\pi\)
−0.996209 + 0.0869952i \(0.972274\pi\)
\(678\) 0 0
\(679\) −1.50000 + 2.59808i −0.0575647 + 0.0997050i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0000 1.68361 0.841807 0.539779i \(-0.181492\pi\)
0.841807 + 0.539779i \(0.181492\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.0000 17.3205i 0.380970 0.659859i
\(690\) 0 0
\(691\) −2.00000 3.46410i −0.0760836 0.131781i 0.825473 0.564441i \(-0.190908\pi\)
−0.901557 + 0.432660i \(0.857575\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.00000 + 10.3923i 0.227593 + 0.394203i
\(696\) 0 0
\(697\) −24.0000 + 41.5692i −0.909065 + 1.57455i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.00000 0.113308 0.0566542 0.998394i \(-0.481957\pi\)
0.0566542 + 0.998394i \(0.481957\pi\)
\(702\) 0 0
\(703\) 12.0000 0.452589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.5000 + 23.3827i −0.507720 + 0.879396i
\(708\) 0 0
\(709\) 2.00000 + 3.46410i 0.0751116 + 0.130097i 0.901135 0.433539i \(-0.142735\pi\)
−0.826023 + 0.563636i \(0.809402\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.00000 12.1244i −0.262152 0.454061i
\(714\) 0 0
\(715\) −10.0000 + 17.3205i −0.373979 + 0.647750i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.0000 0.745874 0.372937 0.927857i \(-0.378351\pi\)
0.372937 + 0.927857i \(0.378351\pi\)
\(720\) 0 0
\(721\) 12.0000 0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −12.0000 + 20.7846i −0.445669 + 0.771921i
\(726\) 0 0
\(727\) 1.50000 + 2.59808i 0.0556319 + 0.0963573i 0.892500 0.451047i \(-0.148949\pi\)
−0.836868 + 0.547404i \(0.815616\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −8.00000 13.8564i −0.295891 0.512498i
\(732\) 0 0
\(733\) 23.0000 39.8372i 0.849524 1.47142i −0.0321090 0.999484i \(-0.510222\pi\)
0.881633 0.471935i \(-0.156444\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 50.0000 1.84177
\(738\) 0 0
\(739\) 40.0000 1.47142 0.735712 0.677295i \(-0.236848\pi\)
0.735712 + 0.677295i \(0.236848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10.0000 + 17.3205i −0.366864 + 0.635428i −0.989073 0.147423i \(-0.952902\pi\)
0.622209 + 0.782851i \(0.286235\pi\)
\(744\) 0 0
\(745\) 3.50000 + 6.06218i 0.128230 + 0.222101i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 13.5000 + 23.3827i 0.493279 + 0.854385i
\(750\) 0 0
\(751\) 22.5000 38.9711i 0.821037 1.42208i −0.0838743 0.996476i \(-0.526729\pi\)
0.904911 0.425601i \(-0.139937\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) −46.0000 −1.67190 −0.835949 0.548807i \(-0.815082\pi\)
−0.835949 + 0.548807i \(0.815082\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −16.0000 + 27.7128i −0.580000 + 1.00459i 0.415479 + 0.909603i \(0.363614\pi\)
−0.995479 + 0.0949859i \(0.969719\pi\)
\(762\) 0 0
\(763\) 15.0000 + 25.9808i 0.543036 + 0.940567i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.00000 + 13.8564i 0.288863 + 0.500326i
\(768\) 0 0
\(769\) 3.50000 6.06218i 0.126213 0.218608i −0.795993 0.605305i \(-0.793051\pi\)
0.922207 + 0.386698i \(0.126384\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −30.0000 −1.07903 −0.539513 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) −28.0000 −1.00579
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.00000 10.3923i 0.214972 0.372343i
\(780\) 0 0
\(781\) 20.0000 + 34.6410i 0.715656 + 1.23955i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 + 17.3205i 0.356915 + 0.618195i
\(786\) 0 0
\(787\) −16.0000 + 27.7128i −0.570338 + 0.987855i 0.426193 + 0.904632i \(0.359855\pi\)
−0.996531 + 0.0832226i \(0.973479\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.5000 30.3109i 0.619882 1.07367i −0.369625 0.929181i \(-0.620514\pi\)
0.989507 0.144486i \(-0.0461528\pi\)
\(798\) 0 0
\(799\) −24.0000 41.5692i −0.849059 1.47061i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.50000 4.33013i −0.0882231 0.152807i
\(804\) 0 0
\(805\) 3.00000 5.19615i 0.105736 0.183140i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 38.0000 1.33601 0.668004 0.744157i \(-0.267149\pi\)
0.668004 + 0.744157i \(0.267149\pi\)
\(810\) 0 0
\(811\) 14.0000 0.491606 0.245803 0.969320i \(-0.420948\pi\)
0.245803 + 0.969320i \(0.420948\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 6.00000 10.3923i 0.210171 0.364027i
\(816\) 0 0
\(817\) 2.00000 + 3.46410i 0.0699711 + 0.121194i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 27.0000 + 46.7654i 0.942306 + 1.63212i 0.761056 + 0.648686i \(0.224681\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(822\) 0 0
\(823\) −21.5000 + 37.2391i −0.749443 + 1.29807i 0.198647 + 0.980071i \(0.436345\pi\)
−0.948090 + 0.318002i \(0.896988\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.0000 −1.11275 −0.556375 0.830932i \(-0.687808\pi\)
−0.556375 + 0.830932i \(0.687808\pi\)
\(828\) 0 0
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8.00000 + 13.8564i −0.277184 + 0.480096i
\(834\) 0 0
\(835\) 3.00000 + 5.19615i 0.103819 + 0.179820i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(840\) 0 0
\(841\) −3.50000 + 6.06218i −0.120690 + 0.209041i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −42.0000 −1.44314
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.00000 10.3923i 0.205677 0.356244i
\(852\) 0 0
\(853\) 5.00000 + 8.66025i 0.171197 + 0.296521i 0.938839 0.344358i \(-0.111903\pi\)
−0.767642 + 0.640879i \(0.778570\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −12.0000 20.7846i −0.409912 0.709989i 0.584967 0.811057i \(-0.301107\pi\)
−0.994880 + 0.101068i \(0.967774\pi\)
\(858\) 0 0
\(859\) −12.0000 + 20.7846i −0.409435 + 0.709162i −0.994826 0.101589i \(-0.967607\pi\)
0.585392 + 0.810751i \(0.300941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −52.0000 −1.77010 −0.885050 0.465495i \(-0.845876\pi\)
−0.885050 + 0.465495i \(0.845876\pi\)
\(864\) 0 0
\(865\) 21.0000 0.714021
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 40.0000 69.2820i 1.35691 2.35023i
\(870\) 0 0
\(871\) −20.0000 34.6410i −0.677674 1.17377i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −13.5000 23.3827i −0.456383 0.790479i
\(876\) 0 0
\(877\) −21.0000 + 36.3731i −0.709120 + 1.22823i 0.256064 + 0.966660i \(0.417574\pi\)
−0.965184 + 0.261571i \(0.915759\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −52.0000 −1.75192 −0.875962 0.482380i \(-0.839773\pi\)
−0.875962 + 0.482380i \(0.839773\pi\)
\(882\) 0 0
\(883\) 46.0000 1.54802 0.774012 0.633171i \(-0.218247\pi\)
0.774012 + 0.633171i \(0.218247\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.00000 + 3.46410i −0.0671534 + 0.116313i −0.897647 0.440715i \(-0.854725\pi\)
0.830494 + 0.557028i \(0.188058\pi\)
\(888\) 0 0
\(889\) 16.5000 + 28.5788i 0.553392 + 0.958503i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.00000 + 10.3923i 0.200782 + 0.347765i
\(894\) 0 0
\(895\) 4.50000 7.79423i 0.150418 0.260532i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −42.0000 −1.40078
\(900\) 0 0
\(901\) −40.0000 −1.33259
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 29.0000 + 50.2295i 0.962929 + 1.66784i 0.715079 + 0.699044i \(0.246391\pi\)
0.247851 + 0.968798i \(0.420276\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −19.0000 32.9090i −0.629498 1.09032i −0.987653 0.156660i \(-0.949927\pi\)
0.358154 0.933662i \(-0.383406\pi\)
\(912\) 0 0
\(913\) 27.5000 47.6314i 0.910117 1.57637i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.00000 −0.0990687
\(918\) 0 0
\(919\) 47.0000 1.55039 0.775193 0.631724i \(-0.217652\pi\)
0.775193 + 0.631724i \(0.217652\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.0000 27.7128i 0.526646 0.912178i
\(924\) 0 0
\(925\) −12.0000 20.7846i −0.394558 0.683394i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.00000 + 3.46410i 0.0656179 + 0.113653i 0.896968 0.442096i \(-0.145765\pi\)
−0.831350 + 0.555749i \(0.812431\pi\)
\(930\) 0 0
\(931\) 2.00000 3.46410i 0.0655474 0.113531i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 40.0000 1.30814
\(936\) 0 0
\(937\) −21.0000 −0.686040 −0.343020 0.939328i \(-0.611450\pi\)
−0.343020 + 0.939328i \(0.611450\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −22.5000 + 38.9711i −0.733479 + 1.27042i 0.221908 + 0.975068i \(0.428771\pi\)
−0.955387 + 0.295355i \(0.904562\pi\)
\(942\) 0 0
\(943\) −6.00000 10.3923i −0.195387 0.338420i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.50000 2.59808i −0.0487435 0.0844261i 0.840624 0.541619i \(-0.182188\pi\)
−0.889368 + 0.457193i \(0.848855\pi\)
\(948\) 0 0
\(949\) −2.00000 + 3.46410i −0.0649227 + 0.112449i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 36.0000 1.16615 0.583077 0.812417i \(-0.301849\pi\)
0.583077 + 0.812417i \(0.301849\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 27.0000 46.7654i 0.871875 1.51013i
\(960\) 0 0
\(961\) −9.00000 15.5885i −0.290323 0.502853i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 9.50000 + 16.4545i 0.305816 + 0.529689i
\(966\) 0 0
\(967\) 17.5000 30.3109i 0.562762 0.974732i −0.434492 0.900676i \(-0.643072\pi\)
0.997254 0.0740568i \(-0.0235946\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −51.0000 −1.63667 −0.818334 0.574743i \(-0.805102\pi\)
−0.818334 + 0.574743i \(0.805102\pi\)
\(972\) 0 0
\(973\) 36.0000 1.15411
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.00000 + 15.5885i −0.287936 + 0.498719i −0.973317 0.229465i \(-0.926302\pi\)
0.685381 + 0.728184i \(0.259636\pi\)
\(978\) 0 0
\(979\) 15.0000 + 25.9808i 0.479402 + 0.830349i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −3.00000 5.19615i −0.0956851 0.165732i 0.814209 0.580572i \(-0.197171\pi\)
−0.909894 + 0.414840i \(0.863838\pi\)
\(984\) 0 0
\(985\) −1.50000 + 2.59808i −0.0477940 + 0.0827816i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −19.0000 −0.603555 −0.301777 0.953378i \(-0.597580\pi\)
−0.301777 + 0.953378i \(0.597580\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −5.50000 + 9.52628i −0.174362 + 0.302003i
\(996\) 0 0
\(997\) −14.0000 24.2487i −0.443384 0.767964i 0.554554 0.832148i \(-0.312889\pi\)
−0.997938 + 0.0641836i \(0.979556\pi\)
\(998\) 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.g.433.1 2
3.2 odd 2 1296.2.i.l.433.1 2
4.3 odd 2 648.2.i.c.433.1 2
9.2 odd 6 1296.2.i.l.865.1 2
9.4 even 3 432.2.a.f.1.1 1
9.5 odd 6 432.2.a.c.1.1 1
9.7 even 3 inner 1296.2.i.g.865.1 2
12.11 even 2 648.2.i.e.433.1 2
36.7 odd 6 648.2.i.c.217.1 2
36.11 even 6 648.2.i.e.217.1 2
36.23 even 6 216.2.a.b.1.1 1
36.31 odd 6 216.2.a.c.1.1 yes 1
72.5 odd 6 1728.2.a.r.1.1 1
72.13 even 6 1728.2.a.i.1.1 1
72.59 even 6 1728.2.a.s.1.1 1
72.67 odd 6 1728.2.a.l.1.1 1
180.23 odd 12 5400.2.f.z.649.1 2
180.59 even 6 5400.2.a.h.1.1 1
180.67 even 12 5400.2.f.b.649.2 2
180.103 even 12 5400.2.f.b.649.1 2
180.139 odd 6 5400.2.a.e.1.1 1
180.167 odd 12 5400.2.f.z.649.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.a.b.1.1 1 36.23 even 6
216.2.a.c.1.1 yes 1 36.31 odd 6
432.2.a.c.1.1 1 9.5 odd 6
432.2.a.f.1.1 1 9.4 even 3
648.2.i.c.217.1 2 36.7 odd 6
648.2.i.c.433.1 2 4.3 odd 2
648.2.i.e.217.1 2 36.11 even 6
648.2.i.e.433.1 2 12.11 even 2
1296.2.i.g.433.1 2 1.1 even 1 trivial
1296.2.i.g.865.1 2 9.7 even 3 inner
1296.2.i.l.433.1 2 3.2 odd 2
1296.2.i.l.865.1 2 9.2 odd 6
1728.2.a.i.1.1 1 72.13 even 6
1728.2.a.l.1.1 1 72.67 odd 6
1728.2.a.r.1.1 1 72.5 odd 6
1728.2.a.s.1.1 1 72.59 even 6
5400.2.a.e.1.1 1 180.139 odd 6
5400.2.a.h.1.1 1 180.59 even 6
5400.2.f.b.649.1 2 180.103 even 12
5400.2.f.b.649.2 2 180.67 even 12
5400.2.f.z.649.1 2 180.23 odd 12
5400.2.f.z.649.2 2 180.167 odd 12