Properties

Label 576.2.i.k.193.1
Level $576$
Weight $2$
Character 576.193
Analytic conductor $4.599$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,2,Mod(193,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, 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("576.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 576.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: \(4.59938315643\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 193.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 576.193
Dual form 576.2.i.k.385.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.73205 q^{3} +(-0.500000 + 0.866025i) q^{5} +(-0.866025 - 1.50000i) q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +(-0.500000 + 0.866025i) q^{5} +(-0.866025 - 1.50000i) q^{7} +3.00000 q^{9} +(-0.866025 - 1.50000i) q^{11} +(-1.50000 + 2.59808i) q^{13} +(0.866025 - 1.50000i) q^{15} +4.00000 q^{17} +6.92820 q^{19} +(1.50000 + 2.59808i) q^{21} +(-4.33013 + 7.50000i) q^{23} +(2.00000 + 3.46410i) q^{25} -5.19615 q^{27} +(0.500000 + 0.866025i) q^{29} +(2.59808 - 4.50000i) q^{31} +(1.50000 + 2.59808i) q^{33} +1.73205 q^{35} +8.00000 q^{37} +(2.59808 - 4.50000i) q^{39} +(-2.50000 + 4.33013i) q^{41} +(4.33013 + 7.50000i) q^{43} +(-1.50000 + 2.59808i) q^{45} +(6.06218 + 10.5000i) q^{47} +(2.00000 - 3.46410i) q^{49} -6.92820 q^{51} +8.00000 q^{53} +1.73205 q^{55} -12.0000 q^{57} +(-0.866025 + 1.50000i) q^{59} +(-3.50000 - 6.06218i) q^{61} +(-2.59808 - 4.50000i) q^{63} +(-1.50000 - 2.59808i) q^{65} +(4.33013 - 7.50000i) q^{67} +(7.50000 - 12.9904i) q^{69} -3.46410 q^{71} -12.0000 q^{73} +(-3.46410 - 6.00000i) q^{75} +(-1.50000 + 2.59808i) q^{77} +(-2.59808 - 4.50000i) q^{79} +9.00000 q^{81} +(4.33013 + 7.50000i) q^{83} +(-2.00000 + 3.46410i) q^{85} +(-0.866025 - 1.50000i) q^{87} -4.00000 q^{89} +5.19615 q^{91} +(-4.50000 + 7.79423i) q^{93} +(-3.46410 + 6.00000i) q^{95} +(1.50000 + 2.59808i) q^{97} +(-2.59808 - 4.50000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{5} + 12 q^{9} - 6 q^{13} + 16 q^{17} + 6 q^{21} + 8 q^{25} + 2 q^{29} + 6 q^{33} + 32 q^{37} - 10 q^{41} - 6 q^{45} + 8 q^{49} + 32 q^{53} - 48 q^{57} - 14 q^{61} - 6 q^{65} + 30 q^{69} - 48 q^{73} - 6 q^{77} + 36 q^{81} - 8 q^{85} - 16 q^{89} - 18 q^{93} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.73205 −1.00000
\(4\) 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
\(7\) −0.866025 1.50000i −0.327327 0.566947i 0.654654 0.755929i \(-0.272814\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) −0.866025 1.50000i −0.261116 0.452267i 0.705422 0.708787i \(-0.250757\pi\)
−0.966539 + 0.256520i \(0.917424\pi\)
\(12\) 0 0
\(13\) −1.50000 + 2.59808i −0.416025 + 0.720577i −0.995535 0.0943882i \(-0.969911\pi\)
0.579510 + 0.814965i \(0.303244\pi\)
\(14\) 0 0
\(15\) 0.866025 1.50000i 0.223607 0.387298i
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 1.50000 + 2.59808i 0.327327 + 0.566947i
\(22\) 0 0
\(23\) −4.33013 + 7.50000i −0.902894 + 1.56386i −0.0791743 + 0.996861i \(0.525228\pi\)
−0.823720 + 0.566997i \(0.808105\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) 0.500000 + 0.866025i 0.0928477 + 0.160817i 0.908708 0.417432i \(-0.137070\pi\)
−0.815861 + 0.578249i \(0.803736\pi\)
\(30\) 0 0
\(31\) 2.59808 4.50000i 0.466628 0.808224i −0.532645 0.846339i \(-0.678802\pi\)
0.999273 + 0.0381148i \(0.0121353\pi\)
\(32\) 0 0
\(33\) 1.50000 + 2.59808i 0.261116 + 0.452267i
\(34\) 0 0
\(35\) 1.73205 0.292770
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 2.59808 4.50000i 0.416025 0.720577i
\(40\) 0 0
\(41\) −2.50000 + 4.33013i −0.390434 + 0.676252i −0.992507 0.122189i \(-0.961009\pi\)
0.602072 + 0.798441i \(0.294342\pi\)
\(42\) 0 0
\(43\) 4.33013 + 7.50000i 0.660338 + 1.14374i 0.980527 + 0.196385i \(0.0629204\pi\)
−0.320189 + 0.947354i \(0.603746\pi\)
\(44\) 0 0
\(45\) −1.50000 + 2.59808i −0.223607 + 0.387298i
\(46\) 0 0
\(47\) 6.06218 + 10.5000i 0.884260 + 1.53158i 0.846560 + 0.532293i \(0.178670\pi\)
0.0376995 + 0.999289i \(0.487997\pi\)
\(48\) 0 0
\(49\) 2.00000 3.46410i 0.285714 0.494872i
\(50\) 0 0
\(51\) −6.92820 −0.970143
\(52\) 0 0
\(53\) 8.00000 1.09888 0.549442 0.835532i \(-0.314840\pi\)
0.549442 + 0.835532i \(0.314840\pi\)
\(54\) 0 0
\(55\) 1.73205 0.233550
\(56\) 0 0
\(57\) −12.0000 −1.58944
\(58\) 0 0
\(59\) −0.866025 + 1.50000i −0.112747 + 0.195283i −0.916877 0.399170i \(-0.869298\pi\)
0.804130 + 0.594454i \(0.202632\pi\)
\(60\) 0 0
\(61\) −3.50000 6.06218i −0.448129 0.776182i 0.550135 0.835076i \(-0.314576\pi\)
−0.998264 + 0.0588933i \(0.981243\pi\)
\(62\) 0 0
\(63\) −2.59808 4.50000i −0.327327 0.566947i
\(64\) 0 0
\(65\) −1.50000 2.59808i −0.186052 0.322252i
\(66\) 0 0
\(67\) 4.33013 7.50000i 0.529009 0.916271i −0.470418 0.882443i \(-0.655897\pi\)
0.999428 0.0338274i \(-0.0107696\pi\)
\(68\) 0 0
\(69\) 7.50000 12.9904i 0.902894 1.56386i
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) −3.46410 6.00000i −0.400000 0.692820i
\(76\) 0 0
\(77\) −1.50000 + 2.59808i −0.170941 + 0.296078i
\(78\) 0 0
\(79\) −2.59808 4.50000i −0.292306 0.506290i 0.682048 0.731307i \(-0.261089\pi\)
−0.974355 + 0.225018i \(0.927756\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 4.33013 + 7.50000i 0.475293 + 0.823232i 0.999600 0.0282978i \(-0.00900866\pi\)
−0.524306 + 0.851530i \(0.675675\pi\)
\(84\) 0 0
\(85\) −2.00000 + 3.46410i −0.216930 + 0.375735i
\(86\) 0 0
\(87\) −0.866025 1.50000i −0.0928477 0.160817i
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 5.19615 0.544705
\(92\) 0 0
\(93\) −4.50000 + 7.79423i −0.466628 + 0.808224i
\(94\) 0 0
\(95\) −3.46410 + 6.00000i −0.355409 + 0.615587i
\(96\) 0 0
\(97\) 1.50000 + 2.59808i 0.152302 + 0.263795i 0.932073 0.362270i \(-0.117998\pi\)
−0.779771 + 0.626064i \(0.784665\pi\)
\(98\) 0 0
\(99\) −2.59808 4.50000i −0.261116 0.452267i
\(100\) 0 0
\(101\) −6.50000 11.2583i −0.646774 1.12025i −0.983889 0.178782i \(-0.942784\pi\)
0.337115 0.941464i \(-0.390549\pi\)
\(102\) 0 0
\(103\) 0.866025 1.50000i 0.0853320 0.147799i −0.820201 0.572076i \(-0.806138\pi\)
0.905533 + 0.424277i \(0.139472\pi\)
\(104\) 0 0
\(105\) −3.00000 −0.292770
\(106\) 0 0
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) −13.8564 −1.31519
\(112\) 0 0
\(113\) 0.500000 0.866025i 0.0470360 0.0814688i −0.841549 0.540181i \(-0.818356\pi\)
0.888585 + 0.458712i \(0.151689\pi\)
\(114\) 0 0
\(115\) −4.33013 7.50000i −0.403786 0.699379i
\(116\) 0 0
\(117\) −4.50000 + 7.79423i −0.416025 + 0.720577i
\(118\) 0 0
\(119\) −3.46410 6.00000i −0.317554 0.550019i
\(120\) 0 0
\(121\) 4.00000 6.92820i 0.363636 0.629837i
\(122\) 0 0
\(123\) 4.33013 7.50000i 0.390434 0.676252i
\(124\) 0 0
\(125\) −9.00000 −0.804984
\(126\) 0 0
\(127\) 13.8564 1.22956 0.614779 0.788700i \(-0.289245\pi\)
0.614779 + 0.788700i \(0.289245\pi\)
\(128\) 0 0
\(129\) −7.50000 12.9904i −0.660338 1.14374i
\(130\) 0 0
\(131\) −7.79423 + 13.5000i −0.680985 + 1.17950i 0.293696 + 0.955899i \(0.405115\pi\)
−0.974681 + 0.223602i \(0.928219\pi\)
\(132\) 0 0
\(133\) −6.00000 10.3923i −0.520266 0.901127i
\(134\) 0 0
\(135\) 2.59808 4.50000i 0.223607 0.387298i
\(136\) 0 0
\(137\) −6.50000 11.2583i −0.555332 0.961864i −0.997878 0.0651178i \(-0.979258\pi\)
0.442545 0.896746i \(-0.354076\pi\)
\(138\) 0 0
\(139\) −0.866025 + 1.50000i −0.0734553 + 0.127228i −0.900414 0.435035i \(-0.856736\pi\)
0.826958 + 0.562263i \(0.190069\pi\)
\(140\) 0 0
\(141\) −10.5000 18.1865i −0.884260 1.53158i
\(142\) 0 0
\(143\) 5.19615 0.434524
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) −3.46410 + 6.00000i −0.285714 + 0.494872i
\(148\) 0 0
\(149\) −5.50000 + 9.52628i −0.450578 + 0.780423i −0.998422 0.0561570i \(-0.982115\pi\)
0.547844 + 0.836580i \(0.315449\pi\)
\(150\) 0 0
\(151\) 11.2583 + 19.5000i 0.916190 + 1.58689i 0.805150 + 0.593072i \(0.202085\pi\)
0.111040 + 0.993816i \(0.464582\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) 2.59808 + 4.50000i 0.208683 + 0.361449i
\(156\) 0 0
\(157\) −8.50000 + 14.7224i −0.678374 + 1.17498i 0.297097 + 0.954847i \(0.403982\pi\)
−0.975470 + 0.220131i \(0.929352\pi\)
\(158\) 0 0
\(159\) −13.8564 −1.09888
\(160\) 0 0
\(161\) 15.0000 1.18217
\(162\) 0 0
\(163\) −6.92820 −0.542659 −0.271329 0.962487i \(-0.587463\pi\)
−0.271329 + 0.962487i \(0.587463\pi\)
\(164\) 0 0
\(165\) −3.00000 −0.233550
\(166\) 0 0
\(167\) 7.79423 13.5000i 0.603136 1.04466i −0.389208 0.921150i \(-0.627251\pi\)
0.992343 0.123511i \(-0.0394155\pi\)
\(168\) 0 0
\(169\) 2.00000 + 3.46410i 0.153846 + 0.266469i
\(170\) 0 0
\(171\) 20.7846 1.58944
\(172\) 0 0
\(173\) −6.50000 11.2583i −0.494186 0.855955i 0.505792 0.862656i \(-0.331200\pi\)
−0.999978 + 0.00670064i \(0.997867\pi\)
\(174\) 0 0
\(175\) 3.46410 6.00000i 0.261861 0.453557i
\(176\) 0 0
\(177\) 1.50000 2.59808i 0.112747 0.195283i
\(178\) 0 0
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 6.06218 + 10.5000i 0.448129 + 0.776182i
\(184\) 0 0
\(185\) −4.00000 + 6.92820i −0.294086 + 0.509372i
\(186\) 0 0
\(187\) −3.46410 6.00000i −0.253320 0.438763i
\(188\) 0 0
\(189\) 4.50000 + 7.79423i 0.327327 + 0.566947i
\(190\) 0 0
\(191\) −0.866025 1.50000i −0.0626634 0.108536i 0.832992 0.553285i \(-0.186626\pi\)
−0.895655 + 0.444749i \(0.853293\pi\)
\(192\) 0 0
\(193\) 5.50000 9.52628i 0.395899 0.685717i −0.597317 0.802005i \(-0.703766\pi\)
0.993215 + 0.116289i \(0.0370998\pi\)
\(194\) 0 0
\(195\) 2.59808 + 4.50000i 0.186052 + 0.322252i
\(196\) 0 0
\(197\) 22.0000 1.56744 0.783718 0.621117i \(-0.213321\pi\)
0.783718 + 0.621117i \(0.213321\pi\)
\(198\) 0 0
\(199\) 3.46410 0.245564 0.122782 0.992434i \(-0.460818\pi\)
0.122782 + 0.992434i \(0.460818\pi\)
\(200\) 0 0
\(201\) −7.50000 + 12.9904i −0.529009 + 0.916271i
\(202\) 0 0
\(203\) 0.866025 1.50000i 0.0607831 0.105279i
\(204\) 0 0
\(205\) −2.50000 4.33013i −0.174608 0.302429i
\(206\) 0 0
\(207\) −12.9904 + 22.5000i −0.902894 + 1.56386i
\(208\) 0 0
\(209\) −6.00000 10.3923i −0.415029 0.718851i
\(210\) 0 0
\(211\) 4.33013 7.50000i 0.298098 0.516321i −0.677603 0.735428i \(-0.736981\pi\)
0.975701 + 0.219107i \(0.0703144\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) −8.66025 −0.590624
\(216\) 0 0
\(217\) −9.00000 −0.610960
\(218\) 0 0
\(219\) 20.7846 1.40449
\(220\) 0 0
\(221\) −6.00000 + 10.3923i −0.403604 + 0.699062i
\(222\) 0 0
\(223\) −0.866025 1.50000i −0.0579934 0.100447i 0.835571 0.549382i \(-0.185137\pi\)
−0.893565 + 0.448935i \(0.851804\pi\)
\(224\) 0 0
\(225\) 6.00000 + 10.3923i 0.400000 + 0.692820i
\(226\) 0 0
\(227\) 6.06218 + 10.5000i 0.402361 + 0.696909i 0.994010 0.109286i \(-0.0348564\pi\)
−0.591649 + 0.806195i \(0.701523\pi\)
\(228\) 0 0
\(229\) 7.50000 12.9904i 0.495614 0.858429i −0.504373 0.863486i \(-0.668276\pi\)
0.999987 + 0.00505719i \(0.00160976\pi\)
\(230\) 0 0
\(231\) 2.59808 4.50000i 0.170941 0.296078i
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) −12.1244 −0.790906
\(236\) 0 0
\(237\) 4.50000 + 7.79423i 0.292306 + 0.506290i
\(238\) 0 0
\(239\) 9.52628 16.5000i 0.616204 1.06730i −0.373968 0.927442i \(-0.622003\pi\)
0.990172 0.139855i \(-0.0446636\pi\)
\(240\) 0 0
\(241\) 4.50000 + 7.79423i 0.289870 + 0.502070i 0.973779 0.227498i \(-0.0730544\pi\)
−0.683908 + 0.729568i \(0.739721\pi\)
\(242\) 0 0
\(243\) −15.5885 −1.00000
\(244\) 0 0
\(245\) 2.00000 + 3.46410i 0.127775 + 0.221313i
\(246\) 0 0
\(247\) −10.3923 + 18.0000i −0.661247 + 1.14531i
\(248\) 0 0
\(249\) −7.50000 12.9904i −0.475293 0.823232i
\(250\) 0 0
\(251\) −20.7846 −1.31191 −0.655956 0.754799i \(-0.727735\pi\)
−0.655956 + 0.754799i \(0.727735\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 0 0
\(255\) 3.46410 6.00000i 0.216930 0.375735i
\(256\) 0 0
\(257\) −15.5000 + 26.8468i −0.966863 + 1.67466i −0.262341 + 0.964975i \(0.584494\pi\)
−0.704523 + 0.709681i \(0.748839\pi\)
\(258\) 0 0
\(259\) −6.92820 12.0000i −0.430498 0.745644i
\(260\) 0 0
\(261\) 1.50000 + 2.59808i 0.0928477 + 0.160817i
\(262\) 0 0
\(263\) −9.52628 16.5000i −0.587416 1.01743i −0.994570 0.104074i \(-0.966812\pi\)
0.407154 0.913360i \(-0.366521\pi\)
\(264\) 0 0
\(265\) −4.00000 + 6.92820i −0.245718 + 0.425596i
\(266\) 0 0
\(267\) 6.92820 0.423999
\(268\) 0 0
\(269\) 16.0000 0.975537 0.487769 0.872973i \(-0.337811\pi\)
0.487769 + 0.872973i \(0.337811\pi\)
\(270\) 0 0
\(271\) −13.8564 −0.841717 −0.420858 0.907126i \(-0.638271\pi\)
−0.420858 + 0.907126i \(0.638271\pi\)
\(272\) 0 0
\(273\) −9.00000 −0.544705
\(274\) 0 0
\(275\) 3.46410 6.00000i 0.208893 0.361814i
\(276\) 0 0
\(277\) 13.5000 + 23.3827i 0.811136 + 1.40493i 0.912069 + 0.410036i \(0.134484\pi\)
−0.100933 + 0.994893i \(0.532183\pi\)
\(278\) 0 0
\(279\) 7.79423 13.5000i 0.466628 0.808224i
\(280\) 0 0
\(281\) 9.50000 + 16.4545i 0.566722 + 0.981592i 0.996887 + 0.0788417i \(0.0251222\pi\)
−0.430165 + 0.902750i \(0.641545\pi\)
\(282\) 0 0
\(283\) −14.7224 + 25.5000i −0.875158 + 1.51582i −0.0185631 + 0.999828i \(0.505909\pi\)
−0.856595 + 0.515990i \(0.827424\pi\)
\(284\) 0 0
\(285\) 6.00000 10.3923i 0.355409 0.615587i
\(286\) 0 0
\(287\) 8.66025 0.511199
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −2.59808 4.50000i −0.152302 0.263795i
\(292\) 0 0
\(293\) 3.50000 6.06218i 0.204472 0.354156i −0.745492 0.666514i \(-0.767786\pi\)
0.949964 + 0.312358i \(0.101119\pi\)
\(294\) 0 0
\(295\) −0.866025 1.50000i −0.0504219 0.0873334i
\(296\) 0 0
\(297\) 4.50000 + 7.79423i 0.261116 + 0.452267i
\(298\) 0 0
\(299\) −12.9904 22.5000i −0.751253 1.30121i
\(300\) 0 0
\(301\) 7.50000 12.9904i 0.432293 0.748753i
\(302\) 0 0
\(303\) 11.2583 + 19.5000i 0.646774 + 1.12025i
\(304\) 0 0
\(305\) 7.00000 0.400819
\(306\) 0 0
\(307\) −6.92820 −0.395413 −0.197707 0.980261i \(-0.563349\pi\)
−0.197707 + 0.980261i \(0.563349\pi\)
\(308\) 0 0
\(309\) −1.50000 + 2.59808i −0.0853320 + 0.147799i
\(310\) 0 0
\(311\) 2.59808 4.50000i 0.147323 0.255172i −0.782914 0.622130i \(-0.786268\pi\)
0.930237 + 0.366958i \(0.119601\pi\)
\(312\) 0 0
\(313\) 0.500000 + 0.866025i 0.0282617 + 0.0489506i 0.879810 0.475325i \(-0.157669\pi\)
−0.851549 + 0.524276i \(0.824336\pi\)
\(314\) 0 0
\(315\) 5.19615 0.292770
\(316\) 0 0
\(317\) 8.50000 + 14.7224i 0.477408 + 0.826894i 0.999665 0.0258939i \(-0.00824321\pi\)
−0.522257 + 0.852788i \(0.674910\pi\)
\(318\) 0 0
\(319\) 0.866025 1.50000i 0.0484881 0.0839839i
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 27.7128 1.54198
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.5000 18.1865i 0.578884 1.00266i
\(330\) 0 0
\(331\) 6.06218 + 10.5000i 0.333207 + 0.577132i 0.983139 0.182861i \(-0.0585357\pi\)
−0.649931 + 0.759993i \(0.725202\pi\)
\(332\) 0 0
\(333\) 24.0000 1.31519
\(334\) 0 0
\(335\) 4.33013 + 7.50000i 0.236580 + 0.409769i
\(336\) 0 0
\(337\) −7.50000 + 12.9904i −0.408551 + 0.707631i −0.994728 0.102552i \(-0.967299\pi\)
0.586177 + 0.810183i \(0.300632\pi\)
\(338\) 0 0
\(339\) −0.866025 + 1.50000i −0.0470360 + 0.0814688i
\(340\) 0 0
\(341\) −9.00000 −0.487377
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) 7.50000 + 12.9904i 0.403786 + 0.699379i
\(346\) 0 0
\(347\) 11.2583 19.5000i 0.604379 1.04681i −0.387770 0.921756i \(-0.626755\pi\)
0.992149 0.125059i \(-0.0399120\pi\)
\(348\) 0 0
\(349\) 5.50000 + 9.52628i 0.294408 + 0.509930i 0.974847 0.222875i \(-0.0715441\pi\)
−0.680439 + 0.732805i \(0.738211\pi\)
\(350\) 0 0
\(351\) 7.79423 13.5000i 0.416025 0.720577i
\(352\) 0 0
\(353\) −14.5000 25.1147i −0.771757 1.33672i −0.936599 0.350403i \(-0.886045\pi\)
0.164842 0.986320i \(-0.447289\pi\)
\(354\) 0 0
\(355\) 1.73205 3.00000i 0.0919277 0.159223i
\(356\) 0 0
\(357\) 6.00000 + 10.3923i 0.317554 + 0.550019i
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) −6.92820 + 12.0000i −0.363636 + 0.629837i
\(364\) 0 0
\(365\) 6.00000 10.3923i 0.314054 0.543958i
\(366\) 0 0
\(367\) −16.4545 28.5000i −0.858917 1.48769i −0.872963 0.487787i \(-0.837804\pi\)
0.0140459 0.999901i \(-0.495529\pi\)
\(368\) 0 0
\(369\) −7.50000 + 12.9904i −0.390434 + 0.676252i
\(370\) 0 0
\(371\) −6.92820 12.0000i −0.359694 0.623009i
\(372\) 0 0
\(373\) −9.50000 + 16.4545i −0.491891 + 0.851981i −0.999956 0.00933789i \(-0.997028\pi\)
0.508065 + 0.861319i \(0.330361\pi\)
\(374\) 0 0
\(375\) 15.5885 0.804984
\(376\) 0 0
\(377\) −3.00000 −0.154508
\(378\) 0 0
\(379\) −10.3923 −0.533817 −0.266908 0.963722i \(-0.586002\pi\)
−0.266908 + 0.963722i \(0.586002\pi\)
\(380\) 0 0
\(381\) −24.0000 −1.22956
\(382\) 0 0
\(383\) −12.9904 + 22.5000i −0.663777 + 1.14970i 0.315838 + 0.948813i \(0.397714\pi\)
−0.979615 + 0.200883i \(0.935619\pi\)
\(384\) 0 0
\(385\) −1.50000 2.59808i −0.0764471 0.132410i
\(386\) 0 0
\(387\) 12.9904 + 22.5000i 0.660338 + 1.14374i
\(388\) 0 0
\(389\) −14.5000 25.1147i −0.735179 1.27337i −0.954645 0.297747i \(-0.903765\pi\)
0.219465 0.975620i \(-0.429569\pi\)
\(390\) 0 0
\(391\) −17.3205 + 30.0000i −0.875936 + 1.51717i
\(392\) 0 0
\(393\) 13.5000 23.3827i 0.680985 1.17950i
\(394\) 0 0
\(395\) 5.19615 0.261447
\(396\) 0 0
\(397\) −8.00000 −0.401508 −0.200754 0.979642i \(-0.564339\pi\)
−0.200754 + 0.979642i \(0.564339\pi\)
\(398\) 0 0
\(399\) 10.3923 + 18.0000i 0.520266 + 0.901127i
\(400\) 0 0
\(401\) −6.50000 + 11.2583i −0.324595 + 0.562214i −0.981430 0.191820i \(-0.938561\pi\)
0.656836 + 0.754034i \(0.271895\pi\)
\(402\) 0 0
\(403\) 7.79423 + 13.5000i 0.388258 + 0.672483i
\(404\) 0 0
\(405\) −4.50000 + 7.79423i −0.223607 + 0.387298i
\(406\) 0 0
\(407\) −6.92820 12.0000i −0.343418 0.594818i
\(408\) 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
\(411\) 11.2583 + 19.5000i 0.555332 + 0.961864i
\(412\) 0 0
\(413\) 3.00000 0.147620
\(414\) 0 0
\(415\) −8.66025 −0.425115
\(416\) 0 0
\(417\) 1.50000 2.59808i 0.0734553 0.127228i
\(418\) 0 0
\(419\) 11.2583 19.5000i 0.550005 0.952637i −0.448268 0.893899i \(-0.647959\pi\)
0.998273 0.0587381i \(-0.0187077\pi\)
\(420\) 0 0
\(421\) −7.50000 12.9904i −0.365528 0.633112i 0.623333 0.781956i \(-0.285778\pi\)
−0.988861 + 0.148844i \(0.952445\pi\)
\(422\) 0 0
\(423\) 18.1865 + 31.5000i 0.884260 + 1.53158i
\(424\) 0 0
\(425\) 8.00000 + 13.8564i 0.388057 + 0.672134i
\(426\) 0 0
\(427\) −6.06218 + 10.5000i −0.293369 + 0.508131i
\(428\) 0 0
\(429\) −9.00000 −0.434524
\(430\) 0 0
\(431\) 27.7128 1.33488 0.667440 0.744664i \(-0.267390\pi\)
0.667440 + 0.744664i \(0.267390\pi\)
\(432\) 0 0
\(433\) 4.00000 0.192228 0.0961139 0.995370i \(-0.469359\pi\)
0.0961139 + 0.995370i \(0.469359\pi\)
\(434\) 0 0
\(435\) 1.73205 0.0830455
\(436\) 0 0
\(437\) −30.0000 + 51.9615i −1.43509 + 2.48566i
\(438\) 0 0
\(439\) −2.59808 4.50000i −0.123999 0.214773i 0.797342 0.603528i \(-0.206239\pi\)
−0.921341 + 0.388755i \(0.872905\pi\)
\(440\) 0 0
\(441\) 6.00000 10.3923i 0.285714 0.494872i
\(442\) 0 0
\(443\) −16.4545 28.5000i −0.781776 1.35408i −0.930906 0.365258i \(-0.880981\pi\)
0.149130 0.988818i \(-0.452353\pi\)
\(444\) 0 0
\(445\) 2.00000 3.46410i 0.0948091 0.164214i
\(446\) 0 0
\(447\) 9.52628 16.5000i 0.450578 0.780423i
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 8.66025 0.407795
\(452\) 0 0
\(453\) −19.5000 33.7750i −0.916190 1.58689i
\(454\) 0 0
\(455\) −2.59808 + 4.50000i −0.121800 + 0.210963i
\(456\) 0 0
\(457\) −10.5000 18.1865i −0.491169 0.850730i 0.508779 0.860897i \(-0.330097\pi\)
−0.999948 + 0.0101670i \(0.996764\pi\)
\(458\) 0 0
\(459\) −20.7846 −0.970143
\(460\) 0 0
\(461\) 0.500000 + 0.866025i 0.0232873 + 0.0403348i 0.877434 0.479697i \(-0.159253\pi\)
−0.854147 + 0.520032i \(0.825920\pi\)
\(462\) 0 0
\(463\) 16.4545 28.5000i 0.764705 1.32451i −0.175698 0.984444i \(-0.556218\pi\)
0.940403 0.340063i \(-0.110448\pi\)
\(464\) 0 0
\(465\) −4.50000 7.79423i −0.208683 0.361449i
\(466\) 0 0
\(467\) 6.92820 0.320599 0.160300 0.987068i \(-0.448754\pi\)
0.160300 + 0.987068i \(0.448754\pi\)
\(468\) 0 0
\(469\) −15.0000 −0.692636
\(470\) 0 0
\(471\) 14.7224 25.5000i 0.678374 1.17498i
\(472\) 0 0
\(473\) 7.50000 12.9904i 0.344850 0.597298i
\(474\) 0 0
\(475\) 13.8564 + 24.0000i 0.635776 + 1.10120i
\(476\) 0 0
\(477\) 24.0000 1.09888
\(478\) 0 0
\(479\) 4.33013 + 7.50000i 0.197849 + 0.342684i 0.947831 0.318774i \(-0.103271\pi\)
−0.749982 + 0.661458i \(0.769938\pi\)
\(480\) 0 0
\(481\) −12.0000 + 20.7846i −0.547153 + 0.947697i
\(482\) 0 0
\(483\) −25.9808 −1.18217
\(484\) 0 0
\(485\) −3.00000 −0.136223
\(486\) 0 0
\(487\) −27.7128 −1.25579 −0.627894 0.778299i \(-0.716083\pi\)
−0.627894 + 0.778299i \(0.716083\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −0.866025 + 1.50000i −0.0390832 + 0.0676941i −0.884905 0.465771i \(-0.845777\pi\)
0.845822 + 0.533465i \(0.179110\pi\)
\(492\) 0 0
\(493\) 2.00000 + 3.46410i 0.0900755 + 0.156015i
\(494\) 0 0
\(495\) 5.19615 0.233550
\(496\) 0 0
\(497\) 3.00000 + 5.19615i 0.134568 + 0.233079i
\(498\) 0 0
\(499\) −0.866025 + 1.50000i −0.0387686 + 0.0671492i −0.884759 0.466049i \(-0.845677\pi\)
0.845990 + 0.533199i \(0.179010\pi\)
\(500\) 0 0
\(501\) −13.5000 + 23.3827i −0.603136 + 1.04466i
\(502\) 0 0
\(503\) −24.2487 −1.08120 −0.540598 0.841281i \(-0.681802\pi\)
−0.540598 + 0.841281i \(0.681802\pi\)
\(504\) 0 0
\(505\) 13.0000 0.578492
\(506\) 0 0
\(507\) −3.46410 6.00000i −0.153846 0.266469i
\(508\) 0 0
\(509\) 18.5000 32.0429i 0.819998 1.42028i −0.0856847 0.996322i \(-0.527308\pi\)
0.905683 0.423956i \(-0.139359\pi\)
\(510\) 0 0
\(511\) 10.3923 + 18.0000i 0.459728 + 0.796273i
\(512\) 0 0
\(513\) −36.0000 −1.58944
\(514\) 0 0
\(515\) 0.866025 + 1.50000i 0.0381616 + 0.0660979i
\(516\) 0 0
\(517\) 10.5000 18.1865i 0.461789 0.799843i
\(518\) 0 0
\(519\) 11.2583 + 19.5000i 0.494186 + 0.855955i
\(520\) 0 0
\(521\) −26.0000 −1.13908 −0.569540 0.821963i \(-0.692879\pi\)
−0.569540 + 0.821963i \(0.692879\pi\)
\(522\) 0 0
\(523\) 10.3923 0.454424 0.227212 0.973845i \(-0.427039\pi\)
0.227212 + 0.973845i \(0.427039\pi\)
\(524\) 0 0
\(525\) −6.00000 + 10.3923i −0.261861 + 0.453557i
\(526\) 0 0
\(527\) 10.3923 18.0000i 0.452696 0.784092i
\(528\) 0 0
\(529\) −26.0000 45.0333i −1.13043 1.95797i
\(530\) 0 0
\(531\) −2.59808 + 4.50000i −0.112747 + 0.195283i
\(532\) 0 0
\(533\) −7.50000 12.9904i −0.324861 0.562676i
\(534\) 0 0
\(535\) 3.46410 6.00000i 0.149766 0.259403i
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −6.92820 −0.298419
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7.79423 13.5000i −0.333257 0.577218i 0.649891 0.760027i \(-0.274814\pi\)
−0.983148 + 0.182809i \(0.941481\pi\)
\(548\) 0 0
\(549\) −10.5000 18.1865i −0.448129 0.776182i
\(550\) 0 0
\(551\) 3.46410 + 6.00000i 0.147576 + 0.255609i
\(552\) 0 0
\(553\) −4.50000 + 7.79423i −0.191359 + 0.331444i
\(554\) 0 0
\(555\) 6.92820 12.0000i 0.294086 0.509372i
\(556\) 0 0
\(557\) −16.0000 −0.677942 −0.338971 0.940797i \(-0.610079\pi\)
−0.338971 + 0.940797i \(0.610079\pi\)
\(558\) 0 0
\(559\) −25.9808 −1.09887
\(560\) 0 0
\(561\) 6.00000 + 10.3923i 0.253320 + 0.438763i
\(562\) 0 0
\(563\) 11.2583 19.5000i 0.474482 0.821827i −0.525091 0.851046i \(-0.675969\pi\)
0.999573 + 0.0292191i \(0.00930205\pi\)
\(564\) 0 0
\(565\) 0.500000 + 0.866025i 0.0210352 + 0.0364340i
\(566\) 0 0
\(567\) −7.79423 13.5000i −0.327327 0.566947i
\(568\) 0 0
\(569\) −15.5000 26.8468i −0.649794 1.12548i −0.983172 0.182683i \(-0.941522\pi\)
0.333378 0.942793i \(-0.391811\pi\)
\(570\) 0 0
\(571\) 4.33013 7.50000i 0.181210 0.313865i −0.761083 0.648655i \(-0.775332\pi\)
0.942293 + 0.334790i \(0.108665\pi\)
\(572\) 0 0
\(573\) 1.50000 + 2.59808i 0.0626634 + 0.108536i
\(574\) 0 0
\(575\) −34.6410 −1.44463
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 0 0
\(579\) −9.52628 + 16.5000i −0.395899 + 0.685717i
\(580\) 0 0
\(581\) 7.50000 12.9904i 0.311152 0.538932i
\(582\) 0 0
\(583\) −6.92820 12.0000i −0.286937 0.496989i
\(584\) 0 0
\(585\) −4.50000 7.79423i −0.186052 0.322252i
\(586\) 0 0
\(587\) −2.59808 4.50000i −0.107234 0.185735i 0.807415 0.589984i \(-0.200866\pi\)
−0.914649 + 0.404249i \(0.867533\pi\)
\(588\) 0 0
\(589\) 18.0000 31.1769i 0.741677 1.28462i
\(590\) 0 0
\(591\) −38.1051 −1.56744
\(592\) 0 0
\(593\) −20.0000 −0.821302 −0.410651 0.911793i \(-0.634698\pi\)
−0.410651 + 0.911793i \(0.634698\pi\)
\(594\) 0 0
\(595\) 6.92820 0.284029
\(596\) 0 0
\(597\) −6.00000 −0.245564
\(598\) 0 0
\(599\) 0.866025 1.50000i 0.0353848 0.0612883i −0.847791 0.530331i \(-0.822068\pi\)
0.883175 + 0.469043i \(0.155401\pi\)
\(600\) 0 0
\(601\) 5.50000 + 9.52628i 0.224350 + 0.388585i 0.956124 0.292962i \(-0.0946409\pi\)
−0.731774 + 0.681547i \(0.761308\pi\)
\(602\) 0 0
\(603\) 12.9904 22.5000i 0.529009 0.916271i
\(604\) 0 0
\(605\) 4.00000 + 6.92820i 0.162623 + 0.281672i
\(606\) 0 0
\(607\) −6.06218 + 10.5000i −0.246056 + 0.426182i −0.962428 0.271537i \(-0.912468\pi\)
0.716372 + 0.697719i \(0.245801\pi\)
\(608\) 0 0
\(609\) −1.50000 + 2.59808i −0.0607831 + 0.105279i
\(610\) 0 0
\(611\) −36.3731 −1.47150
\(612\) 0 0
\(613\) −8.00000 −0.323117 −0.161558 0.986863i \(-0.551652\pi\)
−0.161558 + 0.986863i \(0.551652\pi\)
\(614\) 0 0
\(615\) 4.33013 + 7.50000i 0.174608 + 0.302429i
\(616\) 0 0
\(617\) −2.50000 + 4.33013i −0.100646 + 0.174324i −0.911951 0.410299i \(-0.865424\pi\)
0.811305 + 0.584623i \(0.198758\pi\)
\(618\) 0 0
\(619\) −7.79423 13.5000i −0.313276 0.542611i 0.665793 0.746136i \(-0.268093\pi\)
−0.979070 + 0.203526i \(0.934760\pi\)
\(620\) 0 0
\(621\) 22.5000 38.9711i 0.902894 1.56386i
\(622\) 0 0
\(623\) 3.46410 + 6.00000i 0.138786 + 0.240385i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 10.3923 + 18.0000i 0.415029 + 0.718851i
\(628\) 0 0
\(629\) 32.0000 1.27592
\(630\) 0 0
\(631\) 41.5692 1.65484 0.827422 0.561580i \(-0.189806\pi\)
0.827422 + 0.561580i \(0.189806\pi\)
\(632\) 0 0
\(633\) −7.50000 + 12.9904i −0.298098 + 0.516321i
\(634\) 0 0
\(635\) −6.92820 + 12.0000i −0.274937 + 0.476205i
\(636\) 0 0
\(637\) 6.00000 + 10.3923i 0.237729 + 0.411758i
\(638\) 0 0
\(639\) −10.3923 −0.411113
\(640\) 0 0
\(641\) −3.50000 6.06218i −0.138242 0.239442i 0.788589 0.614920i \(-0.210812\pi\)
−0.926831 + 0.375478i \(0.877478\pi\)
\(642\) 0 0
\(643\) −2.59808 + 4.50000i −0.102458 + 0.177463i −0.912697 0.408637i \(-0.866004\pi\)
0.810239 + 0.586100i \(0.199337\pi\)
\(644\) 0 0
\(645\) 15.0000 0.590624
\(646\) 0 0
\(647\) 13.8564 0.544752 0.272376 0.962191i \(-0.412191\pi\)
0.272376 + 0.962191i \(0.412191\pi\)
\(648\) 0 0
\(649\) 3.00000 0.117760
\(650\) 0 0
\(651\) 15.5885 0.610960
\(652\) 0 0
\(653\) −0.500000 + 0.866025i −0.0195665 + 0.0338902i −0.875643 0.482959i \(-0.839562\pi\)
0.856076 + 0.516849i \(0.172895\pi\)
\(654\) 0 0
\(655\) −7.79423 13.5000i −0.304546 0.527489i
\(656\) 0 0
\(657\) −36.0000 −1.40449
\(658\) 0 0
\(659\) 19.9186 + 34.5000i 0.775918 + 1.34393i 0.934277 + 0.356548i \(0.116046\pi\)
−0.158359 + 0.987382i \(0.550620\pi\)
\(660\) 0 0
\(661\) −8.50000 + 14.7224i −0.330612 + 0.572636i −0.982632 0.185565i \(-0.940588\pi\)
0.652020 + 0.758202i \(0.273922\pi\)
\(662\) 0 0
\(663\) 10.3923 18.0000i 0.403604 0.699062i
\(664\) 0 0
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −8.66025 −0.335326
\(668\) 0 0
\(669\) 1.50000 + 2.59808i 0.0579934 + 0.100447i
\(670\) 0 0
\(671\) −6.06218 + 10.5000i −0.234028 + 0.405348i
\(672\) 0 0
\(673\) −2.50000 4.33013i −0.0963679 0.166914i 0.813811 0.581130i \(-0.197389\pi\)
−0.910179 + 0.414216i \(0.864056\pi\)
\(674\) 0 0
\(675\) −10.3923 18.0000i −0.400000 0.692820i
\(676\) 0 0
\(677\) 5.50000 + 9.52628i 0.211382 + 0.366125i 0.952147 0.305639i \(-0.0988702\pi\)
−0.740765 + 0.671764i \(0.765537\pi\)
\(678\) 0 0
\(679\) 2.59808 4.50000i 0.0997050 0.172694i
\(680\) 0 0
\(681\) −10.5000 18.1865i −0.402361 0.696909i
\(682\) 0 0
\(683\) 34.6410 1.32550 0.662751 0.748840i \(-0.269389\pi\)
0.662751 + 0.748840i \(0.269389\pi\)
\(684\) 0 0
\(685\) 13.0000 0.496704
\(686\) 0 0
\(687\) −12.9904 + 22.5000i −0.495614 + 0.858429i
\(688\) 0 0
\(689\) −12.0000 + 20.7846i −0.457164 + 0.791831i
\(690\) 0 0
\(691\) −9.52628 16.5000i −0.362397 0.627690i 0.625958 0.779857i \(-0.284708\pi\)
−0.988355 + 0.152167i \(0.951375\pi\)
\(692\) 0 0
\(693\) −4.50000 + 7.79423i −0.170941 + 0.296078i
\(694\) 0 0
\(695\) −0.866025 1.50000i −0.0328502 0.0568982i
\(696\) 0 0
\(697\) −10.0000 + 17.3205i −0.378777 + 0.656061i
\(698\) 0 0
\(699\) −17.3205 −0.655122
\(700\) 0 0
\(701\) −16.0000 −0.604312 −0.302156 0.953259i \(-0.597706\pi\)
−0.302156 + 0.953259i \(0.597706\pi\)
\(702\) 0 0
\(703\) 55.4256 2.09042
\(704\) 0 0
\(705\) 21.0000 0.790906
\(706\) 0 0
\(707\) −11.2583 + 19.5000i −0.423413 + 0.733373i
\(708\) 0 0
\(709\) 13.5000 + 23.3827i 0.507003 + 0.878155i 0.999967 + 0.00810550i \(0.00258009\pi\)
−0.492964 + 0.870050i \(0.664087\pi\)
\(710\) 0 0
\(711\) −7.79423 13.5000i −0.292306 0.506290i
\(712\) 0 0
\(713\) 22.5000 + 38.9711i 0.842632 + 1.45948i
\(714\) 0 0
\(715\) −2.59808 + 4.50000i −0.0971625 + 0.168290i
\(716\) 0 0
\(717\) −16.5000 + 28.5788i −0.616204 + 1.06730i
\(718\) 0 0
\(719\) 41.5692 1.55027 0.775135 0.631795i \(-0.217682\pi\)
0.775135 + 0.631795i \(0.217682\pi\)
\(720\) 0 0
\(721\) −3.00000 −0.111726
\(722\) 0 0
\(723\) −7.79423 13.5000i −0.289870 0.502070i
\(724\) 0 0
\(725\) −2.00000 + 3.46410i −0.0742781 + 0.128654i
\(726\) 0 0
\(727\) 11.2583 + 19.5000i 0.417548 + 0.723215i 0.995692 0.0927199i \(-0.0295561\pi\)
−0.578144 + 0.815935i \(0.696223\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 17.3205 + 30.0000i 0.640622 + 1.10959i
\(732\) 0 0
\(733\) 10.5000 18.1865i 0.387826 0.671735i −0.604331 0.796734i \(-0.706559\pi\)
0.992157 + 0.124999i \(0.0398927\pi\)
\(734\) 0 0
\(735\) −3.46410 6.00000i −0.127775 0.221313i
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) −3.46410 −0.127429 −0.0637145 0.997968i \(-0.520295\pi\)
−0.0637145 + 0.997968i \(0.520295\pi\)
\(740\) 0 0
\(741\) 18.0000 31.1769i 0.661247 1.14531i
\(742\) 0 0
\(743\) 0.866025 1.50000i 0.0317714 0.0550297i −0.849703 0.527262i \(-0.823218\pi\)
0.881474 + 0.472233i \(0.156552\pi\)
\(744\) 0 0
\(745\) −5.50000 9.52628i −0.201504 0.349016i
\(746\) 0 0
\(747\) 12.9904 + 22.5000i 0.475293 + 0.823232i
\(748\) 0 0
\(749\) 6.00000 + 10.3923i 0.219235 + 0.379727i
\(750\) 0 0
\(751\) −12.9904 + 22.5000i −0.474026 + 0.821037i −0.999558 0.0297372i \(-0.990533\pi\)
0.525532 + 0.850774i \(0.323866\pi\)
\(752\) 0 0
\(753\) 36.0000 1.31191
\(754\) 0 0
\(755\) −22.5167 −0.819465
\(756\) 0 0
\(757\) −6.00000 −0.218074 −0.109037 0.994038i \(-0.534777\pi\)
−0.109037 + 0.994038i \(0.534777\pi\)
\(758\) 0 0
\(759\) −25.9808 −0.943042
\(760\) 0 0
\(761\) 21.5000 37.2391i 0.779374 1.34992i −0.152928 0.988237i \(-0.548870\pi\)
0.932303 0.361679i \(-0.117796\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −6.00000 + 10.3923i −0.216930 + 0.375735i
\(766\) 0 0
\(767\) −2.59808 4.50000i −0.0938111 0.162486i
\(768\) 0 0
\(769\) 0.500000 0.866025i 0.0180305 0.0312297i −0.856869 0.515534i \(-0.827594\pi\)
0.874900 + 0.484304i \(0.160927\pi\)
\(770\) 0 0
\(771\) 26.8468 46.5000i 0.966863 1.67466i
\(772\) 0 0
\(773\) −40.0000 −1.43870 −0.719350 0.694648i \(-0.755560\pi\)
−0.719350 + 0.694648i \(0.755560\pi\)
\(774\) 0 0
\(775\) 20.7846 0.746605
\(776\) 0 0
\(777\) 12.0000 + 20.7846i 0.430498 + 0.745644i
\(778\) 0 0
\(779\) −17.3205 + 30.0000i −0.620572 + 1.07486i
\(780\) 0 0
\(781\) 3.00000 + 5.19615i 0.107348 + 0.185933i
\(782\) 0 0
\(783\) −2.59808 4.50000i −0.0928477 0.160817i
\(784\) 0 0
\(785\) −8.50000 14.7224i −0.303378 0.525466i
\(786\) 0 0
\(787\) 12.9904 22.5000i 0.463057 0.802038i −0.536054 0.844183i \(-0.680086\pi\)
0.999112 + 0.0421450i \(0.0134192\pi\)
\(788\) 0 0
\(789\) 16.5000 + 28.5788i 0.587416 + 1.01743i
\(790\) 0 0
\(791\) −1.73205 −0.0615846
\(792\) 0 0
\(793\) 21.0000 0.745732
\(794\) 0 0
\(795\) 6.92820 12.0000i 0.245718 0.425596i
\(796\) 0 0
\(797\) 14.5000 25.1147i 0.513616 0.889610i −0.486259 0.873815i \(-0.661639\pi\)
0.999875 0.0157948i \(-0.00502786\pi\)
\(798\) 0 0
\(799\) 24.2487 + 42.0000i 0.857858 + 1.48585i
\(800\) 0 0
\(801\) −12.0000 −0.423999
\(802\) 0 0
\(803\) 10.3923 + 18.0000i 0.366736 + 0.635206i
\(804\) 0 0
\(805\) −7.50000 + 12.9904i −0.264340 + 0.457851i
\(806\) 0 0
\(807\) −27.7128 −0.975537
\(808\) 0 0
\(809\) −4.00000 −0.140633 −0.0703163 0.997525i \(-0.522401\pi\)
−0.0703163 + 0.997525i \(0.522401\pi\)
\(810\) 0 0
\(811\) 20.7846 0.729846 0.364923 0.931038i \(-0.381095\pi\)
0.364923 + 0.931038i \(0.381095\pi\)
\(812\) 0 0
\(813\) 24.0000 0.841717
\(814\) 0 0
\(815\) 3.46410 6.00000i 0.121342 0.210171i
\(816\) 0 0
\(817\) 30.0000 + 51.9615i 1.04957 + 1.81790i
\(818\) 0 0
\(819\) 15.5885 0.544705
\(820\) 0 0
\(821\) −3.50000 6.06218i −0.122151 0.211571i 0.798465 0.602042i \(-0.205646\pi\)
−0.920616 + 0.390470i \(0.872313\pi\)
\(822\) 0 0
\(823\) 23.3827 40.5000i 0.815069 1.41174i −0.0942090 0.995552i \(-0.530032\pi\)
0.909278 0.416189i \(-0.136634\pi\)
\(824\) 0 0
\(825\) −6.00000 + 10.3923i −0.208893 + 0.361814i
\(826\) 0 0
\(827\) 6.92820 0.240917 0.120459 0.992718i \(-0.461563\pi\)
0.120459 + 0.992718i \(0.461563\pi\)
\(828\) 0 0
\(829\) −24.0000 −0.833554 −0.416777 0.909009i \(-0.636840\pi\)
−0.416777 + 0.909009i \(0.636840\pi\)
\(830\) 0 0
\(831\) −23.3827 40.5000i −0.811136 1.40493i
\(832\) 0 0
\(833\) 8.00000 13.8564i 0.277184 0.480096i
\(834\) 0 0
\(835\) 7.79423 + 13.5000i 0.269730 + 0.467187i
\(836\) 0 0
\(837\) −13.5000 + 23.3827i −0.466628 + 0.808224i
\(838\) 0 0
\(839\) 12.9904 + 22.5000i 0.448478 + 0.776786i 0.998287 0.0585039i \(-0.0186330\pi\)
−0.549809 + 0.835290i \(0.685300\pi\)
\(840\) 0 0
\(841\) 14.0000 24.2487i 0.482759 0.836162i
\(842\) 0 0
\(843\) −16.4545 28.5000i −0.566722 0.981592i
\(844\) 0 0
\(845\) −4.00000 −0.137604
\(846\) 0 0
\(847\) −13.8564 −0.476112
\(848\) 0 0
\(849\) 25.5000 44.1673i 0.875158 1.51582i
\(850\) 0 0
\(851\) −34.6410 + 60.0000i −1.18748 + 2.05677i
\(852\) 0 0
\(853\) −11.5000 19.9186i −0.393753 0.681999i 0.599189 0.800608i \(-0.295490\pi\)
−0.992941 + 0.118609i \(0.962157\pi\)
\(854\) 0 0
\(855\) −10.3923 + 18.0000i −0.355409 + 0.615587i
\(856\) 0 0
\(857\) 21.5000 + 37.2391i 0.734426 + 1.27206i 0.954975 + 0.296687i \(0.0958819\pi\)
−0.220549 + 0.975376i \(0.570785\pi\)
\(858\) 0 0
\(859\) 18.1865 31.5000i 0.620517 1.07477i −0.368873 0.929480i \(-0.620256\pi\)
0.989390 0.145286i \(-0.0464103\pi\)
\(860\) 0 0
\(861\) −15.0000 −0.511199
\(862\) 0 0
\(863\) −55.4256 −1.88671 −0.943355 0.331785i \(-0.892349\pi\)
−0.943355 + 0.331785i \(0.892349\pi\)
\(864\) 0 0
\(865\) 13.0000 0.442013
\(866\) 0 0
\(867\) 1.73205 0.0588235
\(868\) 0 0
\(869\) −4.50000 + 7.79423i −0.152652 + 0.264401i
\(870\) 0 0
\(871\) 12.9904 + 22.5000i 0.440162 + 0.762383i
\(872\) 0 0
\(873\) 4.50000 + 7.79423i 0.152302 + 0.263795i
\(874\) 0 0
\(875\) 7.79423 + 13.5000i 0.263493 + 0.456383i
\(876\) 0 0
\(877\) −21.5000 + 37.2391i −0.726003 + 1.25747i 0.232556 + 0.972583i \(0.425291\pi\)
−0.958560 + 0.284892i \(0.908042\pi\)
\(878\) 0 0
\(879\) −6.06218 + 10.5000i −0.204472 + 0.354156i
\(880\) 0 0
\(881\) 52.0000 1.75192 0.875962 0.482380i \(-0.160227\pi\)
0.875962 + 0.482380i \(0.160227\pi\)
\(882\) 0 0
\(883\) −20.7846 −0.699458 −0.349729 0.936851i \(-0.613726\pi\)
−0.349729 + 0.936851i \(0.613726\pi\)
\(884\) 0 0
\(885\) 1.50000 + 2.59808i 0.0504219 + 0.0873334i
\(886\) 0 0
\(887\) 2.59808 4.50000i 0.0872349 0.151095i −0.819106 0.573641i \(-0.805530\pi\)
0.906341 + 0.422546i \(0.138864\pi\)
\(888\) 0 0
\(889\) −12.0000 20.7846i −0.402467 0.697093i
\(890\) 0 0
\(891\) −7.79423 13.5000i −0.261116 0.452267i
\(892\) 0 0
\(893\) 42.0000 + 72.7461i 1.40548 + 2.43436i
\(894\) 0 0
\(895\) −3.46410 + 6.00000i −0.115792 + 0.200558i
\(896\) 0 0
\(897\) 22.5000 + 38.9711i 0.751253 + 1.30121i
\(898\) 0 0
\(899\) 5.19615 0.173301
\(900\) 0 0
\(901\) 32.0000 1.06607
\(902\) 0 0
\(903\) −12.9904 + 22.5000i −0.432293 + 0.748753i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −16.4545 28.5000i −0.546362 0.946327i −0.998520 0.0543890i \(-0.982679\pi\)
0.452158 0.891938i \(-0.350654\pi\)
\(908\) 0 0
\(909\) −19.5000 33.7750i −0.646774 1.12025i
\(910\) 0 0
\(911\) −16.4545 28.5000i −0.545161 0.944247i −0.998597 0.0529580i \(-0.983135\pi\)
0.453435 0.891289i \(-0.350198\pi\)
\(912\) 0 0
\(913\) 7.50000 12.9904i 0.248214 0.429919i
\(914\) 0 0
\(915\) −12.1244 −0.400819
\(916\) 0 0
\(917\) 27.0000 0.891619
\(918\) 0 0
\(919\) 13.8564 0.457081 0.228540 0.973534i \(-0.426605\pi\)
0.228540 + 0.973534i \(0.426605\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) 0 0
\(923\) 5.19615 9.00000i 0.171033 0.296239i
\(924\) 0 0
\(925\) 16.0000 + 27.7128i 0.526077 + 0.911192i
\(926\) 0 0
\(927\) 2.59808 4.50000i 0.0853320 0.147799i
\(928\) 0 0
\(929\) 17.5000 + 30.3109i 0.574156 + 0.994468i 0.996133 + 0.0878612i \(0.0280032\pi\)
−0.421976 + 0.906607i \(0.638663\pi\)
\(930\) 0 0
\(931\) 13.8564 24.0000i 0.454125 0.786568i
\(932\) 0 0
\(933\) −4.50000 + 7.79423i −0.147323 + 0.255172i
\(934\) 0 0
\(935\) 6.92820 0.226576
\(936\) 0 0
\(937\) 4.00000 0.130674 0.0653372 0.997863i \(-0.479188\pi\)
0.0653372 + 0.997863i \(0.479188\pi\)
\(938\) 0 0
\(939\) −0.866025 1.50000i −0.0282617 0.0489506i
\(940\) 0 0
\(941\) 3.50000 6.06218i 0.114097 0.197621i −0.803322 0.595545i \(-0.796936\pi\)
0.917418 + 0.397924i \(0.130269\pi\)
\(942\) 0 0
\(943\) −21.6506 37.5000i −0.705042 1.22117i
\(944\) 0 0
\(945\) −9.00000 −0.292770
\(946\) 0 0
\(947\) −7.79423 13.5000i −0.253278 0.438691i 0.711148 0.703042i \(-0.248176\pi\)
−0.964426 + 0.264351i \(0.914842\pi\)
\(948\) 0 0
\(949\) 18.0000 31.1769i 0.584305 1.01205i
\(950\) 0 0
\(951\) −14.7224 25.5000i −0.477408 0.826894i
\(952\) 0 0
\(953\) −44.0000 −1.42530 −0.712650 0.701520i \(-0.752505\pi\)
−0.712650 + 0.701520i \(0.752505\pi\)
\(954\) 0 0
\(955\) 1.73205 0.0560478
\(956\) 0 0
\(957\) −1.50000 + 2.59808i −0.0484881 + 0.0839839i
\(958\) 0 0
\(959\) −11.2583 + 19.5000i −0.363550 + 0.629688i
\(960\) 0 0
\(961\) 2.00000 + 3.46410i 0.0645161 + 0.111745i
\(962\) 0 0
\(963\) −20.7846 −0.669775
\(964\) 0 0
\(965\) 5.50000 + 9.52628i 0.177051 + 0.306662i
\(966\) 0 0
\(967\) 16.4545 28.5000i 0.529140 0.916498i −0.470282 0.882516i \(-0.655848\pi\)
0.999422 0.0339820i \(-0.0108189\pi\)
\(968\) 0 0
\(969\) −48.0000 −1.54198
\(970\) 0 0
\(971\) −38.1051 −1.22285 −0.611426 0.791302i \(-0.709404\pi\)
−0.611426 + 0.791302i \(0.709404\pi\)
\(972\) 0 0
\(973\) 3.00000 0.0961756
\(974\) 0 0
\(975\) 20.7846 0.665640
\(976\) 0 0
\(977\) 12.5000 21.6506i 0.399910 0.692665i −0.593804 0.804610i \(-0.702375\pi\)
0.993714 + 0.111945i \(0.0357079\pi\)
\(978\) 0 0
\(979\) 3.46410 + 6.00000i 0.110713 + 0.191761i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.33013 + 7.50000i 0.138110 + 0.239213i 0.926781 0.375602i \(-0.122564\pi\)
−0.788671 + 0.614815i \(0.789231\pi\)
\(984\) 0 0
\(985\) −11.0000 + 19.0526i −0.350489 + 0.607065i
\(986\) 0 0
\(987\) −18.1865 + 31.5000i −0.578884 + 1.00266i
\(988\) 0 0
\(989\) −75.0000 −2.38486
\(990\) 0 0
\(991\) 55.4256 1.76065 0.880327 0.474368i \(-0.157323\pi\)
0.880327 + 0.474368i \(0.157323\pi\)
\(992\) 0 0
\(993\) −10.5000 18.1865i −0.333207 0.577132i
\(994\) 0 0
\(995\) −1.73205 + 3.00000i −0.0549097 + 0.0951064i
\(996\) 0 0
\(997\) −6.50000 11.2583i −0.205857 0.356555i 0.744548 0.667568i \(-0.232665\pi\)
−0.950405 + 0.311014i \(0.899332\pi\)
\(998\) 0 0
\(999\) −41.5692 −1.31519
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 576.2.i.k.193.1 4
3.2 odd 2 1728.2.i.l.577.1 4
4.3 odd 2 inner 576.2.i.k.193.2 4
8.3 odd 2 288.2.i.d.193.1 yes 4
8.5 even 2 288.2.i.d.193.2 yes 4
9.2 odd 6 1728.2.i.l.1153.1 4
9.4 even 3 5184.2.a.bx.1.2 2
9.5 odd 6 5184.2.a.bl.1.2 2
9.7 even 3 inner 576.2.i.k.385.1 4
12.11 even 2 1728.2.i.l.577.2 4
24.5 odd 2 864.2.i.d.577.1 4
24.11 even 2 864.2.i.d.577.2 4
36.7 odd 6 inner 576.2.i.k.385.2 4
36.11 even 6 1728.2.i.l.1153.2 4
36.23 even 6 5184.2.a.bl.1.1 2
36.31 odd 6 5184.2.a.bx.1.1 2
72.5 odd 6 2592.2.a.p.1.2 2
72.11 even 6 864.2.i.d.289.2 4
72.13 even 6 2592.2.a.l.1.2 2
72.29 odd 6 864.2.i.d.289.1 4
72.43 odd 6 288.2.i.d.97.1 4
72.59 even 6 2592.2.a.p.1.1 2
72.61 even 6 288.2.i.d.97.2 yes 4
72.67 odd 6 2592.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.2.i.d.97.1 4 72.43 odd 6
288.2.i.d.97.2 yes 4 72.61 even 6
288.2.i.d.193.1 yes 4 8.3 odd 2
288.2.i.d.193.2 yes 4 8.5 even 2
576.2.i.k.193.1 4 1.1 even 1 trivial
576.2.i.k.193.2 4 4.3 odd 2 inner
576.2.i.k.385.1 4 9.7 even 3 inner
576.2.i.k.385.2 4 36.7 odd 6 inner
864.2.i.d.289.1 4 72.29 odd 6
864.2.i.d.289.2 4 72.11 even 6
864.2.i.d.577.1 4 24.5 odd 2
864.2.i.d.577.2 4 24.11 even 2
1728.2.i.l.577.1 4 3.2 odd 2
1728.2.i.l.577.2 4 12.11 even 2
1728.2.i.l.1153.1 4 9.2 odd 6
1728.2.i.l.1153.2 4 36.11 even 6
2592.2.a.l.1.1 2 72.67 odd 6
2592.2.a.l.1.2 2 72.13 even 6
2592.2.a.p.1.1 2 72.59 even 6
2592.2.a.p.1.2 2 72.5 odd 6
5184.2.a.bl.1.1 2 36.23 even 6
5184.2.a.bl.1.2 2 9.5 odd 6
5184.2.a.bx.1.1 2 36.31 odd 6
5184.2.a.bx.1.2 2 9.4 even 3