Properties

Label 336.2.q.e.289.1
Level $336$
Weight $2$
Character 336.289
Analytic conductor $2.683$
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] = [336,2,Mod(193,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 336.q (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: \(2.68297350792\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.289
Dual form 336.2.q.e.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-0.500000 + 2.59808i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(1.50000 + 2.59808i) q^{11} +4.00000 q^{13} +1.00000 q^{15} +(-2.00000 + 3.46410i) q^{19} +(-2.50000 + 0.866025i) q^{21} +(4.00000 - 6.92820i) q^{23} +(2.00000 + 3.46410i) q^{25} -1.00000 q^{27} -3.00000 q^{29} +(-2.50000 - 4.33013i) q^{31} +(-1.50000 + 2.59808i) q^{33} +(2.00000 + 1.73205i) q^{35} +(-4.00000 + 6.92820i) q^{37} +(2.00000 + 3.46410i) q^{39} +8.00000 q^{41} -6.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(5.00000 - 8.66025i) q^{47} +(-6.50000 - 2.59808i) q^{49} +(-4.50000 - 7.79423i) q^{53} +3.00000 q^{55} -4.00000 q^{57} +(-2.50000 - 4.33013i) q^{59} +(5.00000 - 8.66025i) q^{61} +(-2.00000 - 1.73205i) q^{63} +(2.00000 - 3.46410i) q^{65} +(3.00000 + 5.19615i) q^{67} +8.00000 q^{69} -10.0000 q^{71} +(-1.00000 - 1.73205i) q^{73} +(-2.00000 + 3.46410i) q^{75} +(-7.50000 + 2.59808i) q^{77} +(5.50000 - 9.52628i) q^{79} +(-0.500000 - 0.866025i) q^{81} -7.00000 q^{83} +(-1.50000 - 2.59808i) q^{87} +(9.00000 - 15.5885i) q^{89} +(-2.00000 + 10.3923i) q^{91} +(2.50000 - 4.33013i) q^{93} +(2.00000 + 3.46410i) q^{95} -17.0000 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + q^{5} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + q^{5} - q^{7} - q^{9} + 3 q^{11} + 8 q^{13} + 2 q^{15} - 4 q^{19} - 5 q^{21} + 8 q^{23} + 4 q^{25} - 2 q^{27} - 6 q^{29} - 5 q^{31} - 3 q^{33} + 4 q^{35} - 8 q^{37} + 4 q^{39} + 16 q^{41} - 12 q^{43} + q^{45} + 10 q^{47} - 13 q^{49} - 9 q^{53} + 6 q^{55} - 8 q^{57} - 5 q^{59} + 10 q^{61} - 4 q^{63} + 4 q^{65} + 6 q^{67} + 16 q^{69} - 20 q^{71} - 2 q^{73} - 4 q^{75} - 15 q^{77} + 11 q^{79} - q^{81} - 14 q^{83} - 3 q^{87} + 18 q^{89} - 4 q^{91} + 5 q^{93} + 4 q^{95} - 34 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) 0.500000 + 0.866025i 0.288675 + 0.500000i
\(4\) 0 0
\(5\) 0.500000 0.866025i 0.223607 0.387298i −0.732294 0.680989i \(-0.761550\pi\)
0.955901 + 0.293691i \(0.0948835\pi\)
\(6\) 0 0
\(7\) −0.500000 + 2.59808i −0.188982 + 0.981981i
\(8\) 0 0
\(9\) −0.500000 + 0.866025i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 1.50000 + 2.59808i 0.452267 + 0.783349i 0.998526 0.0542666i \(-0.0172821\pi\)
−0.546259 + 0.837616i \(0.683949\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 0 0
\(17\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) −2.00000 + 3.46410i −0.458831 + 0.794719i −0.998899 0.0469020i \(-0.985065\pi\)
0.540068 + 0.841621i \(0.318398\pi\)
\(20\) 0 0
\(21\) −2.50000 + 0.866025i −0.545545 + 0.188982i
\(22\) 0 0
\(23\) 4.00000 6.92820i 0.834058 1.44463i −0.0607377 0.998154i \(-0.519345\pi\)
0.894795 0.446476i \(-0.147321\pi\)
\(24\) 0 0
\(25\) 2.00000 + 3.46410i 0.400000 + 0.692820i
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −2.50000 4.33013i −0.449013 0.777714i 0.549309 0.835619i \(-0.314891\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) −1.50000 + 2.59808i −0.261116 + 0.452267i
\(34\) 0 0
\(35\) 2.00000 + 1.73205i 0.338062 + 0.292770i
\(36\) 0 0
\(37\) −4.00000 + 6.92820i −0.657596 + 1.13899i 0.323640 + 0.946180i \(0.395093\pi\)
−0.981236 + 0.192809i \(0.938240\pi\)
\(38\) 0 0
\(39\) 2.00000 + 3.46410i 0.320256 + 0.554700i
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) 0.500000 + 0.866025i 0.0745356 + 0.129099i
\(46\) 0 0
\(47\) 5.00000 8.66025i 0.729325 1.26323i −0.227844 0.973698i \(-0.573168\pi\)
0.957169 0.289530i \(-0.0934991\pi\)
\(48\) 0 0
\(49\) −6.50000 2.59808i −0.928571 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.50000 7.79423i −0.618123 1.07062i −0.989828 0.142269i \(-0.954560\pi\)
0.371706 0.928351i \(-0.378773\pi\)
\(54\) 0 0
\(55\) 3.00000 0.404520
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −2.50000 4.33013i −0.325472 0.563735i 0.656136 0.754643i \(-0.272190\pi\)
−0.981608 + 0.190909i \(0.938857\pi\)
\(60\) 0 0
\(61\) 5.00000 8.66025i 0.640184 1.10883i −0.345207 0.938527i \(-0.612191\pi\)
0.985391 0.170305i \(-0.0544754\pi\)
\(62\) 0 0
\(63\) −2.00000 1.73205i −0.251976 0.218218i
\(64\) 0 0
\(65\) 2.00000 3.46410i 0.248069 0.429669i
\(66\) 0 0
\(67\) 3.00000 + 5.19615i 0.366508 + 0.634811i 0.989017 0.147802i \(-0.0472198\pi\)
−0.622509 + 0.782613i \(0.713886\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) 0 0
\(73\) −1.00000 1.73205i −0.117041 0.202721i 0.801553 0.597924i \(-0.204008\pi\)
−0.918594 + 0.395203i \(0.870674\pi\)
\(74\) 0 0
\(75\) −2.00000 + 3.46410i −0.230940 + 0.400000i
\(76\) 0 0
\(77\) −7.50000 + 2.59808i −0.854704 + 0.296078i
\(78\) 0 0
\(79\) 5.50000 9.52628i 0.618798 1.07179i −0.370907 0.928670i \(-0.620953\pi\)
0.989705 0.143120i \(-0.0457135\pi\)
\(80\) 0 0
\(81\) −0.500000 0.866025i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −7.00000 −0.768350 −0.384175 0.923260i \(-0.625514\pi\)
−0.384175 + 0.923260i \(0.625514\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.50000 2.59808i −0.160817 0.278543i
\(88\) 0 0
\(89\) 9.00000 15.5885i 0.953998 1.65237i 0.217354 0.976093i \(-0.430258\pi\)
0.736644 0.676280i \(-0.236409\pi\)
\(90\) 0 0
\(91\) −2.00000 + 10.3923i −0.209657 + 1.08941i
\(92\) 0 0
\(93\) 2.50000 4.33013i 0.259238 0.449013i
\(94\) 0 0
\(95\) 2.00000 + 3.46410i 0.205196 + 0.355409i
\(96\) 0 0
\(97\) −17.0000 −1.72609 −0.863044 0.505128i \(-0.831445\pi\)
−0.863044 + 0.505128i \(0.831445\pi\)
\(98\) 0 0
\(99\) −3.00000 −0.301511
\(100\) 0 0
\(101\) 1.00000 + 1.73205i 0.0995037 + 0.172345i 0.911479 0.411346i \(-0.134941\pi\)
−0.811976 + 0.583691i \(0.801608\pi\)
\(102\) 0 0
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 0 0
\(105\) −0.500000 + 2.59808i −0.0487950 + 0.253546i
\(106\) 0 0
\(107\) −5.50000 + 9.52628i −0.531705 + 0.920940i 0.467610 + 0.883935i \(0.345115\pi\)
−0.999315 + 0.0370053i \(0.988218\pi\)
\(108\) 0 0
\(109\) 5.00000 + 8.66025i 0.478913 + 0.829502i 0.999708 0.0241802i \(-0.00769755\pi\)
−0.520794 + 0.853682i \(0.674364\pi\)
\(110\) 0 0
\(111\) −8.00000 −0.759326
\(112\) 0 0
\(113\) −8.00000 −0.752577 −0.376288 0.926503i \(-0.622800\pi\)
−0.376288 + 0.926503i \(0.622800\pi\)
\(114\) 0 0
\(115\) −4.00000 6.92820i −0.373002 0.646058i
\(116\) 0 0
\(117\) −2.00000 + 3.46410i −0.184900 + 0.320256i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 1.73205i 0.0909091 0.157459i
\(122\) 0 0
\(123\) 4.00000 + 6.92820i 0.360668 + 0.624695i
\(124\) 0 0
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 0 0
\(129\) −3.00000 5.19615i −0.264135 0.457496i
\(130\) 0 0
\(131\) 7.50000 12.9904i 0.655278 1.13497i −0.326546 0.945181i \(-0.605885\pi\)
0.981824 0.189794i \(-0.0607819\pi\)
\(132\) 0 0
\(133\) −8.00000 6.92820i −0.693688 0.600751i
\(134\) 0 0
\(135\) −0.500000 + 0.866025i −0.0430331 + 0.0745356i
\(136\) 0 0
\(137\) 7.00000 + 12.1244i 0.598050 + 1.03585i 0.993109 + 0.117198i \(0.0373911\pi\)
−0.395058 + 0.918656i \(0.629276\pi\)
\(138\) 0 0
\(139\) 22.0000 1.86602 0.933008 0.359856i \(-0.117174\pi\)
0.933008 + 0.359856i \(0.117174\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 0 0
\(143\) 6.00000 + 10.3923i 0.501745 + 0.869048i
\(144\) 0 0
\(145\) −1.50000 + 2.59808i −0.124568 + 0.215758i
\(146\) 0 0
\(147\) −1.00000 6.92820i −0.0824786 0.571429i
\(148\) 0 0
\(149\) −9.00000 + 15.5885i −0.737309 + 1.27706i 0.216394 + 0.976306i \(0.430570\pi\)
−0.953703 + 0.300750i \(0.902763\pi\)
\(150\) 0 0
\(151\) −2.50000 4.33013i −0.203447 0.352381i 0.746190 0.665733i \(-0.231881\pi\)
−0.949637 + 0.313353i \(0.898548\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.00000 −0.401610
\(156\) 0 0
\(157\) 2.00000 + 3.46410i 0.159617 + 0.276465i 0.934731 0.355357i \(-0.115641\pi\)
−0.775113 + 0.631822i \(0.782307\pi\)
\(158\) 0 0
\(159\) 4.50000 7.79423i 0.356873 0.618123i
\(160\) 0 0
\(161\) 16.0000 + 13.8564i 1.26098 + 1.09204i
\(162\) 0 0
\(163\) 4.00000 6.92820i 0.313304 0.542659i −0.665771 0.746156i \(-0.731897\pi\)
0.979076 + 0.203497i \(0.0652307\pi\)
\(164\) 0 0
\(165\) 1.50000 + 2.59808i 0.116775 + 0.202260i
\(166\) 0 0
\(167\) 2.00000 0.154765 0.0773823 0.997001i \(-0.475344\pi\)
0.0773823 + 0.997001i \(0.475344\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −2.00000 3.46410i −0.152944 0.264906i
\(172\) 0 0
\(173\) −1.00000 + 1.73205i −0.0760286 + 0.131685i −0.901533 0.432710i \(-0.857557\pi\)
0.825505 + 0.564396i \(0.190891\pi\)
\(174\) 0 0
\(175\) −10.0000 + 3.46410i −0.755929 + 0.261861i
\(176\) 0 0
\(177\) 2.50000 4.33013i 0.187912 0.325472i
\(178\) 0 0
\(179\) 6.00000 + 10.3923i 0.448461 + 0.776757i 0.998286 0.0585225i \(-0.0186389\pi\)
−0.549825 + 0.835280i \(0.685306\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 4.00000 + 6.92820i 0.294086 + 0.509372i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.500000 2.59808i 0.0363696 0.188982i
\(190\) 0 0
\(191\) −8.00000 + 13.8564i −0.578860 + 1.00261i 0.416751 + 0.909021i \(0.363169\pi\)
−0.995610 + 0.0935936i \(0.970165\pi\)
\(192\) 0 0
\(193\) 13.5000 + 23.3827i 0.971751 + 1.68312i 0.690264 + 0.723558i \(0.257494\pi\)
0.281487 + 0.959565i \(0.409172\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −26.0000 −1.85242 −0.926212 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) 6.00000 + 10.3923i 0.425329 + 0.736691i 0.996451 0.0841740i \(-0.0268252\pi\)
−0.571122 + 0.820865i \(0.693492\pi\)
\(200\) 0 0
\(201\) −3.00000 + 5.19615i −0.211604 + 0.366508i
\(202\) 0 0
\(203\) 1.50000 7.79423i 0.105279 0.547048i
\(204\) 0 0
\(205\) 4.00000 6.92820i 0.279372 0.483887i
\(206\) 0 0
\(207\) 4.00000 + 6.92820i 0.278019 + 0.481543i
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) −6.00000 −0.413057 −0.206529 0.978441i \(-0.566217\pi\)
−0.206529 + 0.978441i \(0.566217\pi\)
\(212\) 0 0
\(213\) −5.00000 8.66025i −0.342594 0.593391i
\(214\) 0 0
\(215\) −3.00000 + 5.19615i −0.204598 + 0.354375i
\(216\) 0 0
\(217\) 12.5000 4.33013i 0.848555 0.293948i
\(218\) 0 0
\(219\) 1.00000 1.73205i 0.0675737 0.117041i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 0 0
\(227\) 10.5000 + 18.1865i 0.696909 + 1.20708i 0.969533 + 0.244962i \(0.0787754\pi\)
−0.272623 + 0.962121i \(0.587891\pi\)
\(228\) 0 0
\(229\) 6.00000 10.3923i 0.396491 0.686743i −0.596799 0.802391i \(-0.703561\pi\)
0.993290 + 0.115648i \(0.0368944\pi\)
\(230\) 0 0
\(231\) −6.00000 5.19615i −0.394771 0.341882i
\(232\) 0 0
\(233\) 12.0000 20.7846i 0.786146 1.36165i −0.142166 0.989843i \(-0.545407\pi\)
0.928312 0.371802i \(-0.121260\pi\)
\(234\) 0 0
\(235\) −5.00000 8.66025i −0.326164 0.564933i
\(236\) 0 0
\(237\) 11.0000 0.714527
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −11.5000 19.9186i −0.740780 1.28307i −0.952141 0.305661i \(-0.901123\pi\)
0.211360 0.977408i \(-0.432211\pi\)
\(242\) 0 0
\(243\) 0.500000 0.866025i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) −5.50000 + 4.33013i −0.351382 + 0.276642i
\(246\) 0 0
\(247\) −8.00000 + 13.8564i −0.509028 + 0.881662i
\(248\) 0 0
\(249\) −3.50000 6.06218i −0.221803 0.384175i
\(250\) 0 0
\(251\) −11.0000 −0.694314 −0.347157 0.937807i \(-0.612853\pi\)
−0.347157 + 0.937807i \(0.612853\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −7.00000 + 12.1244i −0.436648 + 0.756297i −0.997429 0.0716680i \(-0.977168\pi\)
0.560781 + 0.827964i \(0.310501\pi\)
\(258\) 0 0
\(259\) −16.0000 13.8564i −0.994192 0.860995i
\(260\) 0 0
\(261\) 1.50000 2.59808i 0.0928477 0.160817i
\(262\) 0 0
\(263\) −7.00000 12.1244i −0.431638 0.747620i 0.565376 0.824833i \(-0.308731\pi\)
−0.997015 + 0.0772134i \(0.975398\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −0.500000 0.866025i −0.0304855 0.0528025i 0.850380 0.526169i \(-0.176372\pi\)
−0.880866 + 0.473366i \(0.843039\pi\)
\(270\) 0 0
\(271\) −8.50000 + 14.7224i −0.516338 + 0.894324i 0.483482 + 0.875354i \(0.339372\pi\)
−0.999820 + 0.0189696i \(0.993961\pi\)
\(272\) 0 0
\(273\) −10.0000 + 3.46410i −0.605228 + 0.209657i
\(274\) 0 0
\(275\) −6.00000 + 10.3923i −0.361814 + 0.626680i
\(276\) 0 0
\(277\) −4.00000 6.92820i −0.240337 0.416275i 0.720473 0.693482i \(-0.243925\pi\)
−0.960810 + 0.277207i \(0.910591\pi\)
\(278\) 0 0
\(279\) 5.00000 0.299342
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) −9.00000 15.5885i −0.534994 0.926638i −0.999164 0.0408910i \(-0.986980\pi\)
0.464169 0.885747i \(-0.346353\pi\)
\(284\) 0 0
\(285\) −2.00000 + 3.46410i −0.118470 + 0.205196i
\(286\) 0 0
\(287\) −4.00000 + 20.7846i −0.236113 + 1.22688i
\(288\) 0 0
\(289\) 8.50000 14.7224i 0.500000 0.866025i
\(290\) 0 0
\(291\) −8.50000 14.7224i −0.498279 0.863044i
\(292\) 0 0
\(293\) −19.0000 −1.10999 −0.554996 0.831853i \(-0.687280\pi\)
−0.554996 + 0.831853i \(0.687280\pi\)
\(294\) 0 0
\(295\) −5.00000 −0.291111
\(296\) 0 0
\(297\) −1.50000 2.59808i −0.0870388 0.150756i
\(298\) 0 0
\(299\) 16.0000 27.7128i 0.925304 1.60267i
\(300\) 0 0
\(301\) 3.00000 15.5885i 0.172917 0.898504i
\(302\) 0 0
\(303\) −1.00000 + 1.73205i −0.0574485 + 0.0995037i
\(304\) 0 0
\(305\) −5.00000 8.66025i −0.286299 0.495885i
\(306\) 0 0
\(307\) 16.0000 0.913168 0.456584 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(312\) 0 0
\(313\) −4.50000 + 7.79423i −0.254355 + 0.440556i −0.964720 0.263278i \(-0.915197\pi\)
0.710365 + 0.703833i \(0.248530\pi\)
\(314\) 0 0
\(315\) −2.50000 + 0.866025i −0.140859 + 0.0487950i
\(316\) 0 0
\(317\) −6.50000 + 11.2583i −0.365076 + 0.632331i −0.988788 0.149323i \(-0.952290\pi\)
0.623712 + 0.781654i \(0.285624\pi\)
\(318\) 0 0
\(319\) −4.50000 7.79423i −0.251952 0.436393i
\(320\) 0 0
\(321\) −11.0000 −0.613960
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 8.00000 + 13.8564i 0.443760 + 0.768615i
\(326\) 0 0
\(327\) −5.00000 + 8.66025i −0.276501 + 0.478913i
\(328\) 0 0
\(329\) 20.0000 + 17.3205i 1.10264 + 0.954911i
\(330\) 0 0
\(331\) 2.00000 3.46410i 0.109930 0.190404i −0.805812 0.592172i \(-0.798271\pi\)
0.915742 + 0.401768i \(0.131604\pi\)
\(332\) 0 0
\(333\) −4.00000 6.92820i −0.219199 0.379663i
\(334\) 0 0
\(335\) 6.00000 0.327815
\(336\) 0 0
\(337\) −15.0000 −0.817102 −0.408551 0.912735i \(-0.633966\pi\)
−0.408551 + 0.912735i \(0.633966\pi\)
\(338\) 0 0
\(339\) −4.00000 6.92820i −0.217250 0.376288i
\(340\) 0 0
\(341\) 7.50000 12.9904i 0.406148 0.703469i
\(342\) 0 0
\(343\) 10.0000 15.5885i 0.539949 0.841698i
\(344\) 0 0
\(345\) 4.00000 6.92820i 0.215353 0.373002i
\(346\) 0 0
\(347\) −10.0000 17.3205i −0.536828 0.929814i −0.999072 0.0430610i \(-0.986289\pi\)
0.462244 0.886753i \(-0.347044\pi\)
\(348\) 0 0
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −4.00000 6.92820i −0.212899 0.368751i 0.739722 0.672913i \(-0.234957\pi\)
−0.952620 + 0.304162i \(0.901624\pi\)
\(354\) 0 0
\(355\) −5.00000 + 8.66025i −0.265372 + 0.459639i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.00000 5.19615i 0.158334 0.274242i −0.775934 0.630814i \(-0.782721\pi\)
0.934268 + 0.356572i \(0.116054\pi\)
\(360\) 0 0
\(361\) 1.50000 + 2.59808i 0.0789474 + 0.136741i
\(362\) 0 0
\(363\) 2.00000 0.104973
\(364\) 0 0
\(365\) −2.00000 −0.104685
\(366\) 0 0
\(367\) −3.50000 6.06218i −0.182699 0.316443i 0.760100 0.649806i \(-0.225150\pi\)
−0.942799 + 0.333363i \(0.891817\pi\)
\(368\) 0 0
\(369\) −4.00000 + 6.92820i −0.208232 + 0.360668i
\(370\) 0 0
\(371\) 22.5000 7.79423i 1.16814 0.404656i
\(372\) 0 0
\(373\) −8.00000 + 13.8564i −0.414224 + 0.717458i −0.995347 0.0963587i \(-0.969280\pi\)
0.581122 + 0.813816i \(0.302614\pi\)
\(374\) 0 0
\(375\) 4.50000 + 7.79423i 0.232379 + 0.402492i
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 3.50000 + 6.06218i 0.179310 + 0.310575i
\(382\) 0 0
\(383\) 1.00000 1.73205i 0.0510976 0.0885037i −0.839345 0.543599i \(-0.817061\pi\)
0.890443 + 0.455095i \(0.150395\pi\)
\(384\) 0 0
\(385\) −1.50000 + 7.79423i −0.0764471 + 0.397231i
\(386\) 0 0
\(387\) 3.00000 5.19615i 0.152499 0.264135i
\(388\) 0 0
\(389\) −13.0000 22.5167i −0.659126 1.14164i −0.980842 0.194804i \(-0.937593\pi\)
0.321716 0.946836i \(-0.395740\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 15.0000 0.756650
\(394\) 0 0
\(395\) −5.50000 9.52628i −0.276735 0.479319i
\(396\) 0 0
\(397\) −14.0000 + 24.2487i −0.702640 + 1.21701i 0.264897 + 0.964277i \(0.414662\pi\)
−0.967537 + 0.252731i \(0.918671\pi\)
\(398\) 0 0
\(399\) 2.00000 10.3923i 0.100125 0.520266i
\(400\) 0 0
\(401\) 4.00000 6.92820i 0.199750 0.345978i −0.748697 0.662912i \(-0.769320\pi\)
0.948447 + 0.316934i \(0.102654\pi\)
\(402\) 0 0
\(403\) −10.0000 17.3205i −0.498135 0.862796i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −24.0000 −1.18964
\(408\) 0 0
\(409\) −3.50000 6.06218i −0.173064 0.299755i 0.766426 0.642333i \(-0.222033\pi\)
−0.939490 + 0.342578i \(0.888700\pi\)
\(410\) 0 0
\(411\) −7.00000 + 12.1244i −0.345285 + 0.598050i
\(412\) 0 0
\(413\) 12.5000 4.33013i 0.615085 0.213072i
\(414\) 0 0
\(415\) −3.50000 + 6.06218i −0.171808 + 0.297581i
\(416\) 0 0
\(417\) 11.0000 + 19.0526i 0.538672 + 0.933008i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 5.00000 + 8.66025i 0.243108 + 0.421076i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 20.0000 + 17.3205i 0.967868 + 0.838198i
\(428\) 0 0
\(429\) −6.00000 + 10.3923i −0.289683 + 0.501745i
\(430\) 0 0
\(431\) −6.00000 10.3923i −0.289010 0.500580i 0.684564 0.728953i \(-0.259993\pi\)
−0.973574 + 0.228373i \(0.926659\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) 0 0
\(435\) −3.00000 −0.143839
\(436\) 0 0
\(437\) 16.0000 + 27.7128i 0.765384 + 1.32568i
\(438\) 0 0
\(439\) −4.50000 + 7.79423i −0.214773 + 0.371998i −0.953202 0.302333i \(-0.902235\pi\)
0.738429 + 0.674331i \(0.235568\pi\)
\(440\) 0 0
\(441\) 5.50000 4.33013i 0.261905 0.206197i
\(442\) 0 0
\(443\) −0.500000 + 0.866025i −0.0237557 + 0.0411461i −0.877659 0.479286i \(-0.840896\pi\)
0.853903 + 0.520432i \(0.174229\pi\)
\(444\) 0 0
\(445\) −9.00000 15.5885i −0.426641 0.738964i
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) 12.0000 + 20.7846i 0.565058 + 0.978709i
\(452\) 0 0
\(453\) 2.50000 4.33013i 0.117460 0.203447i
\(454\) 0 0
\(455\) 8.00000 + 6.92820i 0.375046 + 0.324799i
\(456\) 0 0
\(457\) 8.50000 14.7224i 0.397613 0.688686i −0.595818 0.803120i \(-0.703172\pi\)
0.993431 + 0.114433i \(0.0365053\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 30.0000 1.39724 0.698620 0.715493i \(-0.253798\pi\)
0.698620 + 0.715493i \(0.253798\pi\)
\(462\) 0 0
\(463\) 16.0000 0.743583 0.371792 0.928316i \(-0.378744\pi\)
0.371792 + 0.928316i \(0.378744\pi\)
\(464\) 0 0
\(465\) −2.50000 4.33013i −0.115935 0.200805i
\(466\) 0 0
\(467\) −14.0000 + 24.2487i −0.647843 + 1.12210i 0.335794 + 0.941935i \(0.390995\pi\)
−0.983637 + 0.180161i \(0.942338\pi\)
\(468\) 0 0
\(469\) −15.0000 + 5.19615i −0.692636 + 0.239936i
\(470\) 0 0
\(471\) −2.00000 + 3.46410i −0.0921551 + 0.159617i
\(472\) 0 0
\(473\) −9.00000 15.5885i −0.413820 0.716758i
\(474\) 0 0
\(475\) −16.0000 −0.734130
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) 3.00000 + 5.19615i 0.137073 + 0.237418i 0.926388 0.376571i \(-0.122897\pi\)
−0.789314 + 0.613990i \(0.789564\pi\)
\(480\) 0 0
\(481\) −16.0000 + 27.7128i −0.729537 + 1.26360i
\(482\) 0 0
\(483\) −4.00000 + 20.7846i −0.182006 + 0.945732i
\(484\) 0 0
\(485\) −8.50000 + 14.7224i −0.385965 + 0.668511i
\(486\) 0 0
\(487\) 18.5000 + 32.0429i 0.838315 + 1.45200i 0.891303 + 0.453409i \(0.149792\pi\)
−0.0529875 + 0.998595i \(0.516874\pi\)
\(488\) 0 0
\(489\) 8.00000 0.361773
\(490\) 0 0
\(491\) 9.00000 0.406164 0.203082 0.979162i \(-0.434904\pi\)
0.203082 + 0.979162i \(0.434904\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) −1.50000 + 2.59808i −0.0674200 + 0.116775i
\(496\) 0 0
\(497\) 5.00000 25.9808i 0.224281 1.16540i
\(498\) 0 0
\(499\) 5.00000 8.66025i 0.223831 0.387686i −0.732137 0.681157i \(-0.761477\pi\)
0.955968 + 0.293471i \(0.0948104\pi\)
\(500\) 0 0
\(501\) 1.00000 + 1.73205i 0.0446767 + 0.0773823i
\(502\) 0 0
\(503\) 12.0000 0.535054 0.267527 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 1.50000 + 2.59808i 0.0666173 + 0.115385i
\(508\) 0 0
\(509\) 19.5000 33.7750i 0.864322 1.49705i −0.00339621 0.999994i \(-0.501081\pi\)
0.867719 0.497056i \(-0.165586\pi\)
\(510\) 0 0
\(511\) 5.00000 1.73205i 0.221187 0.0766214i
\(512\) 0 0
\(513\) 2.00000 3.46410i 0.0883022 0.152944i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 30.0000 1.31940
\(518\) 0 0
\(519\) −2.00000 −0.0877903
\(520\) 0 0
\(521\) 9.00000 + 15.5885i 0.394297 + 0.682943i 0.993011 0.118020i \(-0.0376547\pi\)
−0.598714 + 0.800963i \(0.704321\pi\)
\(522\) 0 0
\(523\) −6.00000 + 10.3923i −0.262362 + 0.454424i −0.966869 0.255273i \(-0.917835\pi\)
0.704507 + 0.709697i \(0.251168\pi\)
\(524\) 0 0
\(525\) −8.00000 6.92820i −0.349149 0.302372i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −20.5000 35.5070i −0.891304 1.54378i
\(530\) 0 0
\(531\) 5.00000 0.216982
\(532\) 0 0
\(533\) 32.0000 1.38607
\(534\) 0 0
\(535\) 5.50000 + 9.52628i 0.237786 + 0.411857i
\(536\) 0 0
\(537\) −6.00000 + 10.3923i −0.258919 + 0.448461i
\(538\) 0 0
\(539\) −3.00000 20.7846i −0.129219 0.895257i
\(540\) 0 0
\(541\) −17.0000 + 29.4449i −0.730887 + 1.26593i 0.225617 + 0.974216i \(0.427560\pi\)
−0.956504 + 0.291718i \(0.905773\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 10.0000 0.428353
\(546\) 0 0
\(547\) 8.00000 0.342055 0.171028 0.985266i \(-0.445291\pi\)
0.171028 + 0.985266i \(0.445291\pi\)
\(548\) 0 0
\(549\) 5.00000 + 8.66025i 0.213395 + 0.369611i
\(550\) 0 0
\(551\) 6.00000 10.3923i 0.255609 0.442727i
\(552\) 0 0
\(553\) 22.0000 + 19.0526i 0.935535 + 0.810197i
\(554\) 0 0
\(555\) −4.00000 + 6.92820i −0.169791 + 0.294086i
\(556\) 0 0
\(557\) −7.50000 12.9904i −0.317785 0.550420i 0.662240 0.749291i \(-0.269606\pi\)
−0.980026 + 0.198871i \(0.936272\pi\)
\(558\) 0 0
\(559\) −24.0000 −1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.5000 + 33.7750i 0.821827 + 1.42345i 0.904320 + 0.426855i \(0.140378\pi\)
−0.0824933 + 0.996592i \(0.526288\pi\)
\(564\) 0 0
\(565\) −4.00000 + 6.92820i −0.168281 + 0.291472i
\(566\) 0 0
\(567\) 2.50000 0.866025i 0.104990 0.0363696i
\(568\) 0 0
\(569\) −6.00000 + 10.3923i −0.251533 + 0.435668i −0.963948 0.266090i \(-0.914268\pi\)
0.712415 + 0.701758i \(0.247601\pi\)
\(570\) 0 0
\(571\) 11.0000 + 19.0526i 0.460336 + 0.797325i 0.998978 0.0452101i \(-0.0143957\pi\)
−0.538642 + 0.842535i \(0.681062\pi\)
\(572\) 0 0
\(573\) −16.0000 −0.668410
\(574\) 0 0
\(575\) 32.0000 1.33449
\(576\) 0 0
\(577\) 16.5000 + 28.5788i 0.686904 + 1.18975i 0.972834 + 0.231502i \(0.0743641\pi\)
−0.285930 + 0.958250i \(0.592303\pi\)
\(578\) 0 0
\(579\) −13.5000 + 23.3827i −0.561041 + 0.971751i
\(580\) 0 0
\(581\) 3.50000 18.1865i 0.145204 0.754505i
\(582\) 0 0
\(583\) 13.5000 23.3827i 0.559113 0.968412i
\(584\) 0 0
\(585\) 2.00000 + 3.46410i 0.0826898 + 0.143223i
\(586\) 0 0
\(587\) −45.0000 −1.85735 −0.928674 0.370896i \(-0.879051\pi\)
−0.928674 + 0.370896i \(0.879051\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) −13.0000 22.5167i −0.534749 0.926212i
\(592\) 0 0
\(593\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −6.00000 + 10.3923i −0.245564 + 0.425329i
\(598\) 0 0
\(599\) −21.0000 36.3731i −0.858037 1.48616i −0.873799 0.486287i \(-0.838351\pi\)
0.0157622 0.999876i \(-0.494983\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) −6.00000 −0.244339
\(604\) 0 0
\(605\) −1.00000 1.73205i −0.0406558 0.0704179i
\(606\) 0 0
\(607\) 6.50000 11.2583i 0.263827 0.456962i −0.703429 0.710766i \(-0.748349\pi\)
0.967256 + 0.253804i \(0.0816819\pi\)
\(608\) 0 0
\(609\) 7.50000 2.59808i 0.303915 0.105279i
\(610\) 0 0
\(611\) 20.0000 34.6410i 0.809113 1.40143i
\(612\) 0 0
\(613\) −4.00000 6.92820i −0.161558 0.279827i 0.773869 0.633345i \(-0.218319\pi\)
−0.935428 + 0.353518i \(0.884985\pi\)
\(614\) 0 0
\(615\) 8.00000 0.322591
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) 3.00000 + 5.19615i 0.120580 + 0.208851i 0.919997 0.391926i \(-0.128191\pi\)
−0.799416 + 0.600777i \(0.794858\pi\)
\(620\) 0 0
\(621\) −4.00000 + 6.92820i −0.160514 + 0.278019i
\(622\) 0 0
\(623\) 36.0000 + 31.1769i 1.44231 + 1.24908i
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) −6.00000 10.3923i −0.239617 0.415029i
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 11.0000 0.437903 0.218952 0.975736i \(-0.429736\pi\)
0.218952 + 0.975736i \(0.429736\pi\)
\(632\) 0 0
\(633\) −3.00000 5.19615i −0.119239 0.206529i
\(634\) 0 0
\(635\) 3.50000 6.06218i 0.138893 0.240570i
\(636\) 0 0
\(637\) −26.0000 10.3923i −1.03016 0.411758i
\(638\) 0 0
\(639\) 5.00000 8.66025i 0.197797 0.342594i
\(640\) 0 0
\(641\) 11.0000 + 19.0526i 0.434474 + 0.752531i 0.997253 0.0740768i \(-0.0236010\pi\)
−0.562779 + 0.826608i \(0.690268\pi\)
\(642\) 0 0
\(643\) −46.0000 −1.81406 −0.907031 0.421063i \(-0.861657\pi\)
−0.907031 + 0.421063i \(0.861657\pi\)
\(644\) 0 0
\(645\) −6.00000 −0.236250
\(646\) 0 0
\(647\) −25.0000 43.3013i −0.982851 1.70235i −0.651120 0.758975i \(-0.725701\pi\)
−0.331731 0.943374i \(-0.607633\pi\)
\(648\) 0 0
\(649\) 7.50000 12.9904i 0.294401 0.509917i
\(650\) 0 0
\(651\) 10.0000 + 8.66025i 0.391931 + 0.339422i
\(652\) 0 0
\(653\) −7.50000 + 12.9904i −0.293498 + 0.508353i −0.974634 0.223803i \(-0.928153\pi\)
0.681137 + 0.732156i \(0.261486\pi\)
\(654\) 0 0
\(655\) −7.50000 12.9904i −0.293049 0.507576i
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −19.0000 32.9090i −0.739014 1.28001i −0.952940 0.303160i \(-0.901958\pi\)
0.213925 0.976850i \(-0.431375\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −10.0000 + 3.46410i −0.387783 + 0.134332i
\(666\) 0 0
\(667\) −12.0000 + 20.7846i −0.464642 + 0.804783i
\(668\) 0 0
\(669\) −0.500000 0.866025i −0.0193311 0.0334825i
\(670\) 0 0
\(671\) 30.0000 1.15814
\(672\) 0 0
\(673\) 29.0000 1.11787 0.558934 0.829212i \(-0.311211\pi\)
0.558934 + 0.829212i \(0.311211\pi\)
\(674\) 0 0
\(675\) −2.00000 3.46410i −0.0769800 0.133333i
\(676\) 0 0
\(677\) −9.50000 + 16.4545i −0.365115 + 0.632397i −0.988795 0.149283i \(-0.952304\pi\)
0.623680 + 0.781680i \(0.285637\pi\)
\(678\) 0 0
\(679\) 8.50000 44.1673i 0.326200 1.69499i
\(680\) 0 0
\(681\) −10.5000 + 18.1865i −0.402361 + 0.696909i
\(682\) 0 0
\(683\) 8.50000 + 14.7224i 0.325243 + 0.563338i 0.981562 0.191146i \(-0.0612204\pi\)
−0.656318 + 0.754484i \(0.727887\pi\)
\(684\) 0 0
\(685\) 14.0000 0.534913
\(686\) 0 0
\(687\) 12.0000 0.457829
\(688\) 0 0
\(689\) −18.0000 31.1769i −0.685745 1.18775i
\(690\) 0 0
\(691\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(692\) 0 0
\(693\) 1.50000 7.79423i 0.0569803 0.296078i
\(694\) 0 0
\(695\) 11.0000 19.0526i 0.417254 0.722705i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) 0 0
\(701\) 5.00000 0.188847 0.0944237 0.995532i \(-0.469899\pi\)
0.0944237 + 0.995532i \(0.469899\pi\)
\(702\) 0 0
\(703\) −16.0000 27.7128i −0.603451 1.04521i
\(704\) 0 0
\(705\) 5.00000 8.66025i 0.188311 0.326164i
\(706\) 0 0
\(707\) −5.00000 + 1.73205i −0.188044 + 0.0651405i
\(708\) 0 0
\(709\) −3.00000 + 5.19615i −0.112667 + 0.195146i −0.916845 0.399244i \(-0.869273\pi\)
0.804178 + 0.594389i \(0.202606\pi\)
\(710\) 0 0
\(711\) 5.50000 + 9.52628i 0.206266 + 0.357263i
\(712\) 0 0
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) 12.0000 0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −17.0000 + 29.4449i −0.633993 + 1.09811i 0.352735 + 0.935723i \(0.385252\pi\)
−0.986728 + 0.162385i \(0.948081\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.5000 19.9186i 0.427690 0.740780i
\(724\) 0 0
\(725\) −6.00000 10.3923i −0.222834 0.385961i
\(726\) 0 0
\(727\) −47.0000 −1.74313 −0.871567 0.490277i \(-0.836896\pi\)
−0.871567 + 0.490277i \(0.836896\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 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
\(735\) −6.50000 2.59808i −0.239756 0.0958315i
\(736\) 0 0
\(737\) −9.00000 + 15.5885i −0.331519 + 0.574208i
\(738\) 0 0
\(739\) 5.00000 + 8.66025i 0.183928 + 0.318573i 0.943215 0.332184i \(-0.107785\pi\)
−0.759287 + 0.650756i \(0.774452\pi\)
\(740\) 0 0
\(741\) −16.0000 −0.587775
\(742\) 0 0
\(743\) −34.0000 −1.24734 −0.623670 0.781688i \(-0.714359\pi\)
−0.623670 + 0.781688i \(0.714359\pi\)
\(744\) 0 0
\(745\) 9.00000 + 15.5885i 0.329734 + 0.571117i
\(746\) 0 0
\(747\) 3.50000 6.06218i 0.128058 0.221803i
\(748\) 0 0
\(749\) −22.0000 19.0526i −0.803863 0.696165i
\(750\) 0 0
\(751\) −13.5000 + 23.3827i −0.492622 + 0.853246i −0.999964 0.00849853i \(-0.997295\pi\)
0.507342 + 0.861745i \(0.330628\pi\)
\(752\) 0 0
\(753\) −5.50000 9.52628i −0.200431 0.347157i
\(754\) 0 0
\(755\) −5.00000 −0.181969
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 12.0000 + 20.7846i 0.435572 + 0.754434i
\(760\) 0 0
\(761\) 6.00000 10.3923i 0.217500 0.376721i −0.736543 0.676391i \(-0.763543\pi\)
0.954043 + 0.299670i \(0.0968765\pi\)
\(762\) 0 0
\(763\) −25.0000 + 8.66025i −0.905061 + 0.313522i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −10.0000 17.3205i −0.361079 0.625407i
\(768\) 0 0
\(769\) 37.0000 1.33425 0.667127 0.744944i \(-0.267524\pi\)
0.667127 + 0.744944i \(0.267524\pi\)
\(770\) 0 0
\(771\) −14.0000 −0.504198
\(772\) 0 0
\(773\) 13.0000 + 22.5167i 0.467578 + 0.809868i 0.999314 0.0370420i \(-0.0117935\pi\)
−0.531736 + 0.846910i \(0.678460\pi\)
\(774\) 0 0
\(775\) 10.0000 17.3205i 0.359211 0.622171i
\(776\) 0 0
\(777\) 4.00000 20.7846i 0.143499 0.745644i
\(778\) 0 0
\(779\) −16.0000 + 27.7128i −0.573259 + 0.992915i
\(780\) 0 0
\(781\) −15.0000 25.9808i −0.536742 0.929665i
\(782\) 0 0
\(783\) 3.00000 0.107211
\(784\) 0 0
\(785\) 4.00000 0.142766
\(786\) 0 0
\(787\) −11.0000 19.0526i −0.392108 0.679150i 0.600620 0.799535i \(-0.294921\pi\)
−0.992727 + 0.120384i \(0.961587\pi\)
\(788\) 0 0
\(789\) 7.00000 12.1244i 0.249207 0.431638i
\(790\) 0 0
\(791\) 4.00000 20.7846i 0.142224 0.739016i
\(792\) 0 0
\(793\) 20.0000 34.6410i 0.710221 1.23014i
\(794\) 0 0
\(795\) −4.50000 7.79423i −0.159599 0.276433i
\(796\) 0 0
\(797\) 3.00000 0.106265 0.0531327 0.998587i \(-0.483079\pi\)
0.0531327 + 0.998587i \(0.483079\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 9.00000 + 15.5885i 0.317999 + 0.550791i
\(802\) 0 0
\(803\) 3.00000 5.19615i 0.105868 0.183368i
\(804\) 0 0
\(805\) 20.0000 6.92820i 0.704907 0.244187i
\(806\) 0 0
\(807\) 0.500000 0.866025i 0.0176008 0.0304855i
\(808\) 0 0
\(809\) 8.00000 + 13.8564i 0.281265 + 0.487165i 0.971697 0.236232i \(-0.0759127\pi\)
−0.690432 + 0.723398i \(0.742579\pi\)
\(810\) 0 0
\(811\) −2.00000 −0.0702295 −0.0351147 0.999383i \(-0.511180\pi\)
−0.0351147 + 0.999383i \(0.511180\pi\)
\(812\) 0 0
\(813\) −17.0000 −0.596216
\(814\) 0 0
\(815\) −4.00000 6.92820i −0.140114 0.242684i
\(816\) 0 0
\(817\) 12.0000 20.7846i 0.419827 0.727161i
\(818\) 0 0
\(819\) −8.00000 6.92820i −0.279543 0.242091i
\(820\) 0 0
\(821\) 19.5000 33.7750i 0.680555 1.17876i −0.294257 0.955726i \(-0.595072\pi\)
0.974812 0.223029i \(-0.0715945\pi\)
\(822\) 0 0
\(823\) 16.0000 + 27.7128i 0.557725 + 0.966008i 0.997686 + 0.0679910i \(0.0216589\pi\)
−0.439961 + 0.898017i \(0.645008\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) 41.0000 1.42571 0.712855 0.701312i \(-0.247402\pi\)
0.712855 + 0.701312i \(0.247402\pi\)
\(828\) 0 0
\(829\) 2.00000 + 3.46410i 0.0694629 + 0.120313i 0.898665 0.438636i \(-0.144538\pi\)
−0.829202 + 0.558949i \(0.811205\pi\)
\(830\) 0 0
\(831\) 4.00000 6.92820i 0.138758 0.240337i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 1.00000 1.73205i 0.0346064 0.0599401i
\(836\) 0 0
\(837\) 2.50000 + 4.33013i 0.0864126 + 0.149671i
\(838\) 0 0
\(839\) 24.0000 0.828572 0.414286 0.910147i \(-0.364031\pi\)
0.414286 + 0.910147i \(0.364031\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) 3.00000 + 5.19615i 0.103325 + 0.178965i
\(844\) 0 0
\(845\) 1.50000 2.59808i 0.0516016 0.0893765i
\(846\) 0 0
\(847\) 4.00000 + 3.46410i 0.137442 + 0.119028i
\(848\) 0 0
\(849\) 9.00000 15.5885i 0.308879 0.534994i
\(850\) 0 0
\(851\) 32.0000 + 55.4256i 1.09695 + 1.89997i
\(852\) 0 0
\(853\) 14.0000 0.479351 0.239675 0.970853i \(-0.422959\pi\)
0.239675 + 0.970853i \(0.422959\pi\)
\(854\) 0 0
\(855\) −4.00000 −0.136797
\(856\) 0 0
\(857\) −7.00000 12.1244i −0.239115 0.414160i 0.721345 0.692576i \(-0.243524\pi\)
−0.960461 + 0.278416i \(0.910191\pi\)
\(858\) 0 0
\(859\) −7.00000 + 12.1244i −0.238837 + 0.413678i −0.960381 0.278691i \(-0.910099\pi\)
0.721544 + 0.692369i \(0.243433\pi\)
\(860\) 0 0
\(861\) −20.0000 + 6.92820i −0.681598 + 0.236113i
\(862\) 0 0
\(863\) 11.0000 19.0526i 0.374444 0.648557i −0.615799 0.787903i \(-0.711167\pi\)
0.990244 + 0.139346i \(0.0445001\pi\)
\(864\) 0 0
\(865\) 1.00000 + 1.73205i 0.0340010 + 0.0588915i
\(866\) 0 0
\(867\) 17.0000 0.577350
\(868\) 0 0
\(869\) 33.0000 1.11945
\(870\) 0 0
\(871\) 12.0000 + 20.7846i 0.406604 + 0.704260i
\(872\) 0 0
\(873\) 8.50000 14.7224i 0.287681 0.498279i
\(874\) 0 0
\(875\) −4.50000 + 23.3827i −0.152128 + 0.790479i
\(876\) 0 0
\(877\) 16.0000 27.7128i 0.540282 0.935795i −0.458606 0.888640i \(-0.651651\pi\)
0.998888 0.0471555i \(-0.0150156\pi\)
\(878\) 0 0
\(879\) −9.50000 16.4545i −0.320427 0.554996i
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 0 0
\(885\) −2.50000 4.33013i −0.0840366 0.145556i
\(886\) 0 0
\(887\) −24.0000 + 41.5692i −0.805841 + 1.39576i 0.109881 + 0.993945i \(0.464953\pi\)
−0.915722 + 0.401813i \(0.868380\pi\)
\(888\) 0 0
\(889\) −3.50000 + 18.1865i −0.117386 + 0.609957i
\(890\) 0 0
\(891\) 1.50000 2.59808i 0.0502519 0.0870388i
\(892\) 0 0
\(893\) 20.0000 + 34.6410i 0.669274 + 1.15922i
\(894\) 0 0
\(895\) 12.0000 0.401116
\(896\) 0 0
\(897\) 32.0000 1.06845
\(898\) 0 0
\(899\) 7.50000 + 12.9904i 0.250139 + 0.433253i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 15.0000 5.19615i 0.499169 0.172917i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −8.00000 13.8564i −0.265636 0.460094i 0.702094 0.712084i \(-0.252248\pi\)
−0.967730 + 0.251990i \(0.918915\pi\)
\(908\) 0 0
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) −6.00000 −0.198789 −0.0993944 0.995048i \(-0.531691\pi\)
−0.0993944 + 0.995048i \(0.531691\pi\)
\(912\) 0 0
\(913\) −10.5000 18.1865i −0.347499 0.601886i
\(914\) 0 0
\(915\) 5.00000 8.66025i 0.165295 0.286299i
\(916\) 0 0
\(917\) 30.0000 + 25.9808i 0.990687 + 0.857960i
\(918\) 0 0
\(919\) −8.00000 + 13.8564i −0.263896 + 0.457081i −0.967274 0.253735i \(-0.918341\pi\)
0.703378 + 0.710816i \(0.251674\pi\)
\(920\) 0 0
\(921\) 8.00000 + 13.8564i 0.263609 + 0.456584i
\(922\) 0 0
\(923\) −40.0000 −1.31662
\(924\) 0 0
\(925\) −32.0000 −1.05215
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −21.0000 + 36.3731i −0.688988 + 1.19336i 0.283178 + 0.959067i \(0.408611\pi\)
−0.972166 + 0.234294i \(0.924722\pi\)
\(930\) 0 0
\(931\) 22.0000 17.3205i 0.721021 0.567657i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 19.0000 0.620703 0.310351 0.950622i \(-0.399553\pi\)
0.310351 + 0.950622i \(0.399553\pi\)
\(938\) 0 0
\(939\) −9.00000 −0.293704
\(940\) 0 0
\(941\) 6.50000 + 11.2583i 0.211894 + 0.367011i 0.952307 0.305141i \(-0.0987035\pi\)
−0.740413 + 0.672152i \(0.765370\pi\)
\(942\) 0 0
\(943\) 32.0000 55.4256i 1.04206 1.80491i
\(944\) 0 0
\(945\) −2.00000 1.73205i −0.0650600 0.0563436i
\(946\) 0 0
\(947\) −24.0000 + 41.5692i −0.779895 + 1.35082i 0.152106 + 0.988364i \(0.451394\pi\)
−0.932002 + 0.362454i \(0.881939\pi\)
\(948\) 0 0
\(949\) −4.00000 6.92820i −0.129845 0.224899i
\(950\) 0 0
\(951\) −13.0000 −0.421554
\(952\) 0 0
\(953\) −38.0000 −1.23094 −0.615470 0.788160i \(-0.711034\pi\)
−0.615470 + 0.788160i \(0.711034\pi\)
\(954\) 0 0
\(955\) 8.00000 + 13.8564i 0.258874 + 0.448383i
\(956\) 0 0
\(957\) 4.50000 7.79423i 0.145464 0.251952i
\(958\) 0 0
\(959\) −35.0000 + 12.1244i −1.13021 + 0.391516i
\(960\) 0 0
\(961\) 3.00000 5.19615i 0.0967742 0.167618i
\(962\) 0 0
\(963\) −5.50000 9.52628i −0.177235 0.306980i
\(964\) 0 0
\(965\) 27.0000 0.869161
\(966\) 0 0
\(967\) −3.00000 −0.0964735 −0.0482367 0.998836i \(-0.515360\pi\)
−0.0482367 + 0.998836i \(0.515360\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.5000 21.6506i 0.401144 0.694802i −0.592720 0.805408i \(-0.701946\pi\)
0.993864 + 0.110607i \(0.0352793\pi\)
\(972\) 0 0
\(973\) −11.0000 + 57.1577i −0.352644 + 1.83239i
\(974\) 0 0
\(975\) −8.00000 + 13.8564i −0.256205 + 0.443760i
\(976\) 0 0
\(977\) 11.0000 + 19.0526i 0.351921 + 0.609545i 0.986586 0.163242i \(-0.0521952\pi\)
−0.634665 + 0.772787i \(0.718862\pi\)
\(978\) 0 0
\(979\) 54.0000 1.72585
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) 22.0000 + 38.1051i 0.701691 + 1.21536i 0.967872 + 0.251442i \(0.0809046\pi\)
−0.266181 + 0.963923i \(0.585762\pi\)
\(984\) 0 0
\(985\) −13.0000 + 22.5167i −0.414214 + 0.717440i
\(986\) 0 0
\(987\) −5.00000 + 25.9808i −0.159152 + 0.826977i
\(988\) 0 0
\(989\) −24.0000 + 41.5692i −0.763156 + 1.32182i
\(990\) 0 0
\(991\) 7.50000 + 12.9904i 0.238245 + 0.412653i 0.960211 0.279276i \(-0.0900944\pi\)
−0.721966 + 0.691929i \(0.756761\pi\)
\(992\) 0 0
\(993\) 4.00000 0.126936
\(994\) 0 0
\(995\) 12.0000 0.380426
\(996\) 0 0
\(997\) 13.0000 + 22.5167i 0.411714 + 0.713110i 0.995077 0.0991016i \(-0.0315969\pi\)
−0.583363 + 0.812211i \(0.698264\pi\)
\(998\) 0 0
\(999\) 4.00000 6.92820i 0.126554 0.219199i
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 336.2.q.e.289.1 2
3.2 odd 2 1008.2.s.f.289.1 2
4.3 odd 2 168.2.q.a.121.1 yes 2
7.2 even 3 2352.2.a.g.1.1 1
7.3 odd 6 2352.2.q.f.1537.1 2
7.4 even 3 inner 336.2.q.e.193.1 2
7.5 odd 6 2352.2.a.u.1.1 1
7.6 odd 2 2352.2.q.f.961.1 2
8.3 odd 2 1344.2.q.o.961.1 2
8.5 even 2 1344.2.q.d.961.1 2
12.11 even 2 504.2.s.d.289.1 2
21.2 odd 6 7056.2.a.bk.1.1 1
21.5 even 6 7056.2.a.t.1.1 1
21.11 odd 6 1008.2.s.f.865.1 2
28.3 even 6 1176.2.q.g.361.1 2
28.11 odd 6 168.2.q.a.25.1 2
28.19 even 6 1176.2.a.c.1.1 1
28.23 odd 6 1176.2.a.g.1.1 1
28.27 even 2 1176.2.q.g.961.1 2
56.5 odd 6 9408.2.a.p.1.1 1
56.11 odd 6 1344.2.q.o.193.1 2
56.19 even 6 9408.2.a.cf.1.1 1
56.37 even 6 9408.2.a.cq.1.1 1
56.51 odd 6 9408.2.a.ba.1.1 1
56.53 even 6 1344.2.q.d.193.1 2
84.11 even 6 504.2.s.d.361.1 2
84.23 even 6 3528.2.a.q.1.1 1
84.47 odd 6 3528.2.a.i.1.1 1
84.59 odd 6 3528.2.s.p.361.1 2
84.83 odd 2 3528.2.s.p.3313.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.q.a.25.1 2 28.11 odd 6
168.2.q.a.121.1 yes 2 4.3 odd 2
336.2.q.e.193.1 2 7.4 even 3 inner
336.2.q.e.289.1 2 1.1 even 1 trivial
504.2.s.d.289.1 2 12.11 even 2
504.2.s.d.361.1 2 84.11 even 6
1008.2.s.f.289.1 2 3.2 odd 2
1008.2.s.f.865.1 2 21.11 odd 6
1176.2.a.c.1.1 1 28.19 even 6
1176.2.a.g.1.1 1 28.23 odd 6
1176.2.q.g.361.1 2 28.3 even 6
1176.2.q.g.961.1 2 28.27 even 2
1344.2.q.d.193.1 2 56.53 even 6
1344.2.q.d.961.1 2 8.5 even 2
1344.2.q.o.193.1 2 56.11 odd 6
1344.2.q.o.961.1 2 8.3 odd 2
2352.2.a.g.1.1 1 7.2 even 3
2352.2.a.u.1.1 1 7.5 odd 6
2352.2.q.f.961.1 2 7.6 odd 2
2352.2.q.f.1537.1 2 7.3 odd 6
3528.2.a.i.1.1 1 84.47 odd 6
3528.2.a.q.1.1 1 84.23 even 6
3528.2.s.p.361.1 2 84.59 odd 6
3528.2.s.p.3313.1 2 84.83 odd 2
7056.2.a.t.1.1 1 21.5 even 6
7056.2.a.bk.1.1 1 21.2 odd 6
9408.2.a.p.1.1 1 56.5 odd 6
9408.2.a.ba.1.1 1 56.51 odd 6
9408.2.a.cf.1.1 1 56.19 even 6
9408.2.a.cq.1.1 1 56.37 even 6